PAGES MISSING
WITHIN THE
BOOK ONLY
(255,256)
CO >- DO
164146
CQ CO
OSMANIA UNIVERSITY
Call No. I'* ' ^ ' - " ' Accession No. * 7 *
Author U > x.V\ x /
Title " ,' .^ u - / : i>A ^ . ._
This book should be returned on or before the date
las^ marked bclo*v.
ELEMENTS OF ALGEBRA
THE MACM1LLAN COMPANY
NKVV YORK - BOSTON CHICAGO
PAI-I.AS SAN FRANCISCO
MACMILLAN & CO, LIMITKU
LONDON HOMBAY CALCUTTA
MELUCK'KNK
THE MACMILLAN CO. OF CANADA, LTD.
TORONTO
ELEMENTS OF ALGEBRA
BY
ARTHUR SCJBULIi/TZE, PH.D.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS, NKW YORK ITNIVEKSITT
HEAD OF THK MATHEMATICAL DKI'A KTM EN T, HIH
SCHOOL OF COMMERCE, NEW 1 ORK CUT
THE MACMILLAN COMPANY
1917
All rights reserved
COPYRIGHT, 1910,
BY THE MACMILLAN COMPANY.
Set up and electrotyped. Published May, 1910. Reprinted
September, 1910 ; January, 1911; July, IQJS ; February, 1913,'
January, 1915; May, September, 1916; August, 1917.
. J. 8. Cushlng Co. Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
IN this book the attempt is made to shorten the usual course
in algebra, while still giving to the student complete familiarity
with all the essentials of the subject. While in many respects
similar to the author's " Elementary Algebra," this book, owing
to its peculiar aim, has certain distinctive features, chief among
which are the following :
1. All unnecessary methods and "cases" are omitted. These
omissions serve not only practical but distinctly pedagogic
ends. Until recently the tendency was to multiply " cases "
as far as possible, in order to make every example a social
case of a memorized method. Such a large number of methods,
however, not only taxes a student's memory unduly but in va-
riably leads to mechanical modes of study. The entire study
of algebra becomes a mechanical application of memorized
rules, while the cultivation of the student's reasoning power
and ingenuity is neglected. Typical in this respect is the
treatment of factoring in many text-books In this book all
methods which are of real value, and which are applied in
advanced work are given, but "cases" that are taught only
on account of tradition, short-cuts that solve only examples
specially manufactured for this purpose, etc., are omitted.
2. All parts of the theory whicJi are beyond the comprehension
of the student or wliicli are logically unsound are omitted. All
practical teachers know how few students understand and
appreciate the more difficult parts of the theory, and conse-
vi PREFACE
quently hardly ever emphasize the theoretical aspect of alge
bra. Moreover, a great deal of the theory offered in the aver-
age text-book is logically unsound ; e.g. all proofs for the sign
of the product of two negative numbers, all elementary proofs
of the binomial theorem for fractional exponents, etc.
3. TJie exercises are slightly simpler than in the larger look.
The best way to introduce a beginner to a new topic is to offer
Lim a large number of simple exercises. For the more ambi-
tious student, however, there has been placed at the end of
the book a collection of exercises which contains an abundance
of more difficult work. With very few exceptions all the exer
cises in this book differ from those in the "Elementary Alge-
bra"; hence either book may be used to supplement the other.
4. Topics of practical importance, as quadratic equations and
graphs, are placed early in the course. This arrangement will
enable students who can devote only a minimum of time to
algebra to study those subjects which are of such importance
for further work.
In regard to some other features of the book, the following
may be quoted from the author's "Elementary Algebra":
"Particular care has been bestowed upon those chapters
which in the customary courses offer the greatest difficulties to
the beginner, especially problems and factoring. The presen-
tation of problems as given in Chapter V will be found to be
quite a departure from the customary way of treating the sub-
ject, and it is hoped that this treatment will materially dimin-
ish the difficulty of this topic for young students.
" The book is designed to meet the requirements for admis-
sion to our best universities and colleges, in particular the
requirements of the College Entrance Examination Board.
This made it necessary to introduce the theory of proportions
PREFACE vii
and graphical methods into the first year's work, an innovation
which seems to mark a distinct gain from the pedagogical point
of view.
" By studying proportions during the first year's work, the
student will be able to utilize this knowledge where it is most
needed, viz. in geometry ; while in the usual course proportions
are studied a long time after their principal application.
" Graphical methods have not only a great practical value,
but they unquestionably furnish a very good antidote against
'the tendency of school algebra to degenerate into a mechani-
cal application of memorized rules.' This topic has been pre-
sented in a simple, elementary way, and it is hoped that some
of the modes of representation given will be considered im-
provements upon the prevailing methods. The entire work in
graphical methods has been so arranged that teachers who wish
a shorter course may omit these chapters."
Applications taken from geometry, physics, and commercial
life are numerous, but the true study of algebra has not been
sacrificed in order to make an impressive display of sham
applications. It is undoubtedly more interesting for a student
to solve a problem that results in the height of Mt. McKinley
than one that gives him the number of Henry's marbles. But
on the other hand very few of such applied examples are
genuine applications of algebra, nobody would find the length
of the Mississippi or the height of Mt. Etna by such a method,
and they usually involve difficult numerical calculations.
Moreover, such examples, based upon statistical abstracts, are
frequently arranged in sets that are algebraically uniform, and
hence the student is more easily led to do the work by rote
than when the arrangement is based principally upon the alge-
braic aspect of the problem.
viii PREFACE
It is true that problems relating to physics often offer a field
for genuine applications of algebra. The average pupil's knowl-
edge of physics, however, is so small that an extensive use of
such problems involves as a rule the teaching of physics by the
teacher of algebra.
Hence the field of genuine applications of elementary algebra
suitable for secondary school work seems to have certain limi-
tations, but within these limits the author has attempted to
give as many simple applied examples as possible.
The author desires to acknowledge his indebtedness to Mr.
William P. Manguse for the careful reading of the proofs and
for many valuable suggestions.
ARTHUR SCHULTZE.
NEW YORK,
April, 1910.
CONTENTS
CHAPTER I
PAGB
INTRODUCTION 1
Algebraic Solution of Problems 1
Negative Numbers 3
Numbers represented by Letters 6
Factors, Powers, and Hoots ....... 7
Algebraic Expressions and Numerical Substitutions ... 10
CHAPTER II
ADDITION, SUBTRACTION, AND PARENTHESES 15
Addition of Monomials 15
Addition of Polynomials ........ 10
Subtraction .... 22
Signs of Aggregation 27
Exercises in Algebraic Expression 29
CHAPTER III
MULTIPLICATION ... 31
Multiplication of Algebraic Numbers 31
Multiplication of Monomials 34
Multiplication of a Polynomial by a Monomial .... 35
Multiplication of Polynomials 36
Special Cases in Multiplication 39
CHAPTER IV
DIVISION 46
Division of Monomials 46
Division of a Polynomial by a Monomial 47
Division of a Polynomial by a Polynomial 48
Special Cases in Division 61
ix
X CONTENTS
CHAPTER V
PAGE
LINEAR EQUATIONS AND PROBLEMS ,63
Solution of Linear Equations 55
Symbolical Expressions .....,. 67
Problems leading to Simple Equations 63
CHAPTER VI
FACTORING 76
Type I. Polynomials, All of whose Terms contain a Com-
mon Factor 77
Type II. Quadratic Trinomials of the Form x' 2 -f px -f q . 78
Type III. Quadratic Trinomials of the Form px 2 -f qx + r . 80
Type IV. The Square of a Binomial x 2 Ixy -f /^ . . . 83
Type V. The Difference of Two Squares .... 84
Type VI. Grouping Terms 86
Summary of Factoring 87
CHAPTER VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE . . 89
Highest Common Factor 89
Lowest Common Multiple 91
CHAPTER VIII
FRACTIONS 93
Reduction of Fractions 93
Addition and Subtraction of Fractions 97
Multiplication of Fractions 102
Division of Fractions 104
Complex Fractions * , * . 105
CHAPTER IX
FRACTIONAL AND LITERAL EQUATIONS ...... 108
Fractional Equations 108
Literal Equations 112
Problems leading to Fractional and Literal Equations . .114
CONTENTS XI
CHAPTER X
PAGE
RATIO AND PROPORTION ......... 120
Ratio 120
Proportion 121
CHAPTER XI
SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE .... 129
Elimination by Addition or Subtraction 130
Elimination by Substitution 133
Literal Simultaneous Equations 138
Simultaneous Equations involving More than Two Unknown
Quantities 140
Problems leading to Simultaneous Equations .... 143
CHAPTER XII
GRAPHIC REPRESENTATION OF FUNCTIONS AND EQUATIONS . . 148
Representation of Functions of One Variable .... 164
Graphic Solution of Equations involving One Unknown Quantity 168
Graphic Solution of Equations involving Two Unknown Quan-
tities 160
CHAPTER XIII
INVOLUTION 165
Involution of Monomials 165
Involution of Binomials 166
CHAPTER XIV
EVOLUTION ... 169
Evolution of Monomials . 170
Evolution of Polynomials and Arithmetical Numbers . . 171
CHAPTER XV
QUADRATIC EQUATIONS INVOLVING ONB UNKNOWN QUANTITY . 1*78
Pure Quadratic Equations 178
Complete Quadratic Equations 181
Problems involving Quadratics 189
Equations in the Quadratic Form 191
Character of the Roots 193
xii CONTENTS
CHAPTER XVI
PAGK
THE THEORT OP EXPONENTS 195
Fractional and Negative Exponents 195
Use of Negative and Fractional Exponents .... 200
CHAPTER XVII
RADICALS 205
Transformation of Radicals 206
Addition and Subtraction of Radicals 210
Multiplication of Radicals .212
Division of Radicals 214
Involution and Evolution of Radicals ..... 218
Square Roots of Quadratic Surds 219
Radical Equations 221
CHAPTER XVIII
THE FACTOR THEOREM 227
CHAPTER XIX
SIMULTANEOUS QUADRATIC EQUATIONS ...... 232
I. Equations solved by finding x +/ and x / . . . 232
II. One Equation Linear, the Other Quadratic . . . 234
III. Homogeneous Equations 236
IV. Special Devices 237
Interpretation of Negative Results and the Forms i -, . . 241
Problems 243
CHAPTER XX
PROGRESSIONS . 246
Arithmetic Progression 24(j
Geometric Progression 251
Infinite Geometric Progression 263
CHAPTER XXI
BINOMIAL THEOREM . . . . . . .. . . 255
BEVIEW EXERCISE . 268
ELEMENTS OF ALGEBRA
ELEMENTS OF ALGEBRA
CHAPTER I
INTRODUCTION
1. Algebra may be called an extension of arithmetic. Like
arithmetic, it treats of numbers, but these numbers are fre-
quently denoted by letters, as illustrated in the following
problem.
ALGEBRAIC SOLUTION OF PROBLEMS
2. Problem. The sum of two numbers is 42, and the greater
is five times the smaller. Find the numbers.
Let ' x the smaller number.
Then 5 x = the greater number,
and 6 x the sum of the two numbers.
Therefore, 6 x = 42,
x = 7, the smaller number,
and 5 x = 35, the greater number.
3. A problem is a question proposed for solution.
4. An equation is a statement expressing the equality of two
quantities; as, 6 a? = 42.
5. In algebra, problems are frequently solved by denoting
numbers by letters and by expressing the problem in the form
of an equation.
6. Unknown numbers are usually represented by the last
letters of the alphabet ; as, x, y, z, but sometimes other letters
are employed.
B 1
2 ELEMENTS OF ALGEBRA
EXERCISE 1
Solve algebraically the following problems :
1. The sum of two numbers is 40, and the greater is four
times the smaller. Find the numbers.
2. A man sold a horse and a carriage for $ 480, receiving
twice as much for the horse as for the carriage. How much
did he receive for the carriage ?
3. A and B own a house worth $ 14,100, and A has in-
vested twice as much capital as B. How much has each
invested ?
4. The population of South America is 9 times that of
Australia, and both continents together have 50,000,000 in-
habitants. Find the population of each.
5. The rise and fall of the tides in Seattle is twice that in
Philadelphia, and their sum is 18 feet. Find the rise and fall
of the tides in Philadelphia.
6. Divide $ 240 among A, B, and C so that A may receive
6 times as much as C. and B 8 times as much as C.
7. A pole 56 feet high was broken so that the part broken
off was 6 times the length of the part left standing. .Find the
length of the two parts.
8. The sum of the sides of a triangle equals 40 inches.
If two sides of the triangle are equal, and each is twice the
A remaining side, how long is each side ?
A
9. The sum of the three angles of any
triangle is 180. If 2 angles of a triangle
are equal, and the remaining angle is 4
times their sum, how many degrees are
there in each ?
B G 10. The number of negroes in Africa
is 10 times the number of Indians in America, and the sum of
both is 165,000,000. How many are there of each ?
INTRODUCTION 3
11. Divide $280 among A, B, and C, so that B may receive
twice as much as A, and C twice as much as B.
12. Divide $90 among A, B, and C, so that B may receive
twice as much as A, and C as much as A and B together.
13. A line 20 inches long is divided into two parts, one of
which is equal to 5 times the other. How long are the parts ?
14. A travels twice as fast as B, and the sum of the dis-
tances traveled by the two is 57 miles. How many miles did
each travel ? 4
15. A, B, C, and D buy $ 2100 worth of goods. How much
does A take, if B buys twice as much as A, C three times as
much as B, and D six times as much
NEGATIVE NUMBE
EXERCISE 2
1. Subtract 9 from 16.
2. Can 9 be subtracted from 7 ?
3. In arithmetic why cannot 9 be subtracted from 7 ?
"* \
4. The temperature at noon is 16 ami at 4 P.M. it is 9
What is the temperature at 4 P.M.? State this as an
of subtraction.
5. The temperature at 4 P.M. is 7, and at 10 P.M. it is 10
less. What is the temperature at 10 P.M. ?
6. Do you know of any other way of expressing the last
answer (3 below zero) ?
7. What then is 7 -10?
8. Can you think of any other practical examples which
require the subtraction of a greater number from a smaller
one?
7. Many practical examples require the subtraction of a
greater number from a smaller one, and in order to express in
a convenient form the results of these, and similar examples,
4 ELEMENTS OF ALGEBRA
it becomes necessary to enlarge our concept of number, so as to
include numbers less than zero.
8. Negative numbers are numbers smaller than zero; they
are denoted by a prefixed minus sign ; as 5 (read " minus 5 ").
Numbers greater than zero, for the sake of distinction, are fre-
quently called positive numbers, and are written either with a
prefixed plus sign, or without any prefixed sign ; as -f- 5 or 5.
The fact that a thermometer falling 10 from 7 indicates 3
below zero may now be expressed
7 -10 = -3.
Instead of saying a gain of $ 30, and a loss of $ 90 is equal to a
loss of $ 60, we may write
$30 -$90 = -$60.
9. The absolute value of a number is the number taken
without regard to its sign.
The absolute value of 5 is 6, of -f 3 is 3.
10. It is convenient for many discussions to represent the
positive numbers by a succession of equal distances laid off on
a line from a point 0, and the negative numbers by a similar
series in the opposite direction.
, I I lit I I I I I I I I y
-6 -5 -4 -3 -2 -1 +\ +2 + 3 + 4 4-5 +6
Thus, in the annexed diagram, the line from to 4- 6 represents 4- 5,
the line from to 4 represents 4, etc. The addition of 3 is rep-
resented by a motion of "three spaces toward the right, and the subtrac-
tion of 8 by a similar motion toward the left.
Thus, 5 added to 1 equals 4, 5 subtracted from 1 equals 6, etc.
EXERCISE 3
1. If in financial transactions we indicate a man's income by
a positive sign, what does a negative sign indicate ?
2. State in what manner the positive and negative signs may
be used to indicate north and south latitude, east and west
longitude, motion upstream and downstream.
INTRODUCTION 5
3. If north latitude is indicated by a positive sign, by what
is south latitude represented ?
4. If south latitude is indicated by a positive sign, by what
is north latitude represented ?
5. What is the meaning of the year 20 A.D. ? Of an east-
erly motion of 6 yards per second ?
6. A merchant gains $ 200, and loses $ 350. (a) What is
his total gain or loss ? (b) Find 200 - 350.
7. If the temperature at 4 A.M. is 8 and at 9 A.M. it is 7
higher, what is the temperature at 9 A.M. ? What, therefore,
is - 8 + 7 ?
8. A vessel starts from a point in 25 north latitude, and
sails 38 due south, (a) Find the latitude at the end of the
journey. (6) Find 25 -38.
9. A vessel starts from a point in 15 south latitude, and
sails 22 due south, (a) Find the latitude at the end of the
journey, (b) Subtract 22 from 15.
10. From 30 subtract 40. 18. To 6 add 12.
11. From 4 subtract 7. 19. To 2 add 1.
12. From 7 subtract 9. 20. To 1 add 2.
13. From 19 subtract 34. 21. From 1 subtract 2.
14. From subtract 14. 22. To - 8 add 9.
15. From 12 subtract 20. 23. To 7 add 4.
16. From 2 subtract 5. 24. From 1 subtract 2.
17. From 1 subtract 1. 25. Add 1 and 2.
26. Solve examples 16-25 by using a diagram similar to
the one of 10, and considering additions and subtractions as
motions.
27. Which is the greater number :
(a) lor -1? (b) -2 or -4?
28. By how much is 7 greater than 12 ?
ELEMENTS OF ALGEBRA
29. Determine from the following table the range of tempera-
ture in each locality :
HIGHEST
LOWEST
Flagstaff, Ariz
93
20
Chicago ......
103
-23
Washington, D.C. .
104
15
Springfield, 111
107
-24
New York City ....
100
- 6
Key West
100
41
Boston ......
102
-13
NUMBERS REPRESENTED BY LETTERS
11. For many purposes of arithmetic it is advantageous to
express numbers by letters. One advantage was shown in 2 ;
others will appear in later chapters ( 30).
EXERCISE 4
1. If the letter t means 1000, what is the value of 5t?
2. What is the value of 3 6, if b = 3 ? if b = 4 ?
3. What is the value of a + &, if a = 5, and 6 = 7? if a = 6,
and b = 4 ?
4. What is the value of 17 c, if c = 5? ifc = -2?
5. If a boy has 9c? marbles and wins 4c marbles, how
many marbles has. he ?
6. Is the last answer correct for any value of d ?
7. A merchant had 20 m dollars and lost 11 m dollars.
How much has he left ?
8. What is the sum of 8 & and G b ?
9. Find the numerical value of the last answer if b = 15.
10. If c represents a certain number, what represents 9 times
that number ?
INTRODUCTION 1
11. From 26 w subtract 19 m.
12. What is the numerical value of the last answer if m = 2 ?
if m = -2?
13. From 22m subtract 25m, and find the numerical value
of the answer if m = 1 2.
14. Add 13 p, 3p, 6p, and subtract 24 p from the sum.
15. From 10 q subtract 20 q. 17. From subtract 26 x.
16. Add -lOgand +20 q. 18. Add - 6 x and 8 x.
19. From 22# subtract 0. 20. From Wp subtract 10^).
21. If a = 20, then 7 a = 140. What sign, therefore, is
understood between 7 and a in the expression 7 a ?
FACTORS, POWERS, AND ROOTS
12. The signs of addition, subtraction, multiplication, division,
and equality have the same meaning in algebra as they have
in arithmetic.
13. If there is no sign between two letters, or a letter and a
number, a sign of multiplication is understood.
6 x a is generally written 6 a ; m x n is written win.
Between two figures, however, a sign of multiplication
(either x or ) has to be employed ; as, 4x7, or 4 7.
4x7 cannot be written 47, for 47 means 40 -f 7.
14. A product is the result obtained by multiplying together
two or more quantities, each of which is a factor of the product.
Since 24 = 3 x 8, or 12 x 2, each of these numbers is a factor of 24.
Similarly, 7, a, 6, and c are factors of 7 abc.
15. A power is the product of two or more equal factors ;
thus, aaaaa is called the " 5th power of a," and written a 5 ;
aaaaaa, or a 6 , is " the 6th power of a," or a 6th.
The second power is also called the square, and the third
power the cube; thus, 12 2 (read "12 square") equals 144.
8 ELEMENTS OF ALQEBEA
16. The base of a power is the number which is repeated
as a factor.
The base of a 3 is a.
17. An exponent is the number which indicates how many
times a base is to be used as a factor. It is placed a little
above and to the right of the base.
The exponent of m 6 is 6 ; n is the exponent of a n .
EXERCISE 5
1. Write and find the numerical value of the square of 7,
the cube of 6, the fourth power of 3, and the fifth power of 2.
Find the numerical values of the following powers :
2. 7 2 . 6. 4 2 . 10. (i) 8 . 14. 25 1 .
3. 2*. 7. 2*. 11. O 9 . 15. .0001 2 .
4. 5 2 . 8. 10 6 . 12. (4|) 2 . 16. l.l 1 .
5. 8 3 . 9. I 30 . 13. (1.5) 2 . 17. 2 2 + 3 2 .
If a=3, 6=2, c=l, and d=^ find the numerical values of:
18. ci 3 . 20. c 10 . 22. a*. 24. (2 c) 3 . 26. 2 at).
19. b 2 . 21. d\ 23. (6cf) 2 . 25. ab. 27. (4 bdf.
28. If a 3 = 8, what is the value of a?
29. If m 2 = -jJg-, what is the value of m ?
30. If 4 = 64, what is the value of a ?
18. In a product any factor is called the coefficient of the
product of the other factors.
In 12 win 8 /), 12 is the coefficient of mw 8 p, 12 m is the coefficient of n*p.
19. A numerical coefficient is a coefficient expressed entirely
in figures.
In 17 aryx, 17 is the numerical coefficient.
When a product contains no numerical coefficient, 1 is under-
stood ; thus a = 1 a, a B b = 1 a*b.
INTRODUCTION 9
20. When several powers are multiplied, the beginner
should remember that every exponent refers only to the num-
ber near which it is placed.
3 2 means 3 aa, while (3 ) 2 = 3 a x 3 a.
9 afty 3 = 9 abyyy.
2* xyW = 2-2.2.2. xyyyzz.
1 abc* 7 abccc.
EXERCISES
If a = 4, b = 1, c = 2, and x = ^, find the numerical values of :
1.
4 a.
6.
a 8 .
11.
4 ab\
16.
12 ab*<?x.
2.
6x.
7.
6 10 .
12.
4 a 3 6 2 .
17.
3.
abc.
8.
4a6 2 .
13.
2bcx*.
18.
4.
5.
abcx.
6 be.
9.
10.
5bc 2 x.
9 a 8 .
14.
15.
3 abc*.
2 a*bc 2 .
19.
a 2
c 8 *
21. A root is one of the equal factors of a power. Accord-
ing to the number of equal factors, it is called a square root, a
cube root, a fourth root, etc.
3 is the square root of 9, for 3 2 = 9.
6 is the cube root of 125, for 6 8 = 125.
a is the fifth root of a 5 , the nth root of a".
The nth root is indicated by the symbol >/""; thus Va is the
fifth root of a, A/27 is the cube root of 27, \/a, or more simply
Va, is the square root of a.
Using this symbol we may express the definition of root by
(Va) n = a.
22. The index of a root is the number which indicates what
root is to be taken. It is written in the opening of the radical
sign.
In v/a, 7 is the index of the root.
23. The signs of aggregation are : the parenthesis, ( ) ; the
bracket, [ ] ; the brace, j j ; and the vinculum, .
10 ELEMENTS OF ALGEBRA
They are used, as in arithmetic, to indicate that the expres*
sions included are to be treated as a whole.
Each of the forms 10 x (4 -f 1), 10 x [4 + 1], 10 x 4"+T indicates that
10 is to be multiplied by 4 + 1 or by 5.
(a b) is sometimes read "quantity a b."
EXERCISE 7
If a = 2, b = 3, c = 1, d 0, x 9, find the numerical value of:
1. Vff. 7. Val 13. 4(a + &).
2. V36". 8. -\fi?. 14. 6(6 + c).
3. V2a. 9. 4V3~6c. 15. (c-f-d) 2 .
4. v'Ta. 10. 5Vl6c. 16. [6-c] 3 .
6. \/c. 11. aVc^. 17.
6. V^a6. 12.
ALGP:BRAIC EXPRESSIONS AND NUMERICAL
SUBSTITUTIONS
24. An algebraic expression is a collection of algebraic sym-
bols representing some number ; e.g. 6 a 2 6 7 Vac 2 -f 9.
25. A monomial or term is an expression whose parts are not
separated by a sign -f- or ; as 3 cue 2 , 9 Vx, ~* ^-
o c
(6 + c + d} is a monomial, since the parts are a and (6 + c -f d).
26. A polynomial is an expression containing more than one
term.
y, !^-f\/0-3 3 ft, and a 4 + M -f c 4 -f- d 4 are polynomials.
27. A binomial is a polynomial of two terms.
a 2 + 6 2 , and | - \/a are binomials.
28. A trinomial is a polynomial of three terms.
V3 are trinomials.
INTRODUCTION 11
29. In a polynomial each term is treated as if it were con-
tained in a parenthesis, i.e. each term has to be computed
before the different terms are added and subtracted. Otherwise
all operations of addition, subtraction, multiplication, and divi-
sion are to be performed in the order in which they are written
from left to right.
E.g. 3 _|_ 4 . 5 means 3 4- 20 or 23.
Ex. 1. Find the value of 4 2 8 + 5 3 2 - *^.
= 32 + 45-27
= 50.
Ex. 2. If a = 5, b = 3, c = 2, d = 0, find the numerical value
of 6 ab 2 - 9 aWc + f a 3 b - 19 a 2 6cd
6 aft 2 - 9 a& 2 c + f a 3 6 - 19 a 2 bcd
= 6 5 - 3 2 - 9 5 3 2 2 + ^ . 5 8 3 - 19 . 5 2 . 3 2
= 6. 5- 9-9. 6- 9- 2 + I-126- 3-0
= 270 - 810 + 150
= - 390.
EXERCISE 8*
If a=4, 5=3, c=l, d=Q, x=^, find the numerical value of:
1. a + 26+3 c. 9. 5c6 2 +-6ac 3 17c 3 -hl2o;.
2. 3a + 56 2 . 10. a 3 -f & 3 -f c 8 - d s .
3. a 2 -6. ' 11. 3a& 2 + 3a 2 6-a&c 2 .
4. a 2 -5c 2 +-d 2 . 12.
5. 5a 2 -46c-f2^^ 13. (a
6. 2 a 2 + 3 a& +- 4 6^9 ad. 14. (a -f b) 2 c + (a + 6)c 2 .
7. 6a 2 +4a6 2 ~6c' l -+12a(i 4 . *15. a 2 -f (2 6 + a 2 ).
8. 27 c 3 - 5 ax 50 a6cd. 16. 4a6-fVa-V2^.
* For additional examples see page 268,
12 ELEMENTS OF ALGEBRA
17 & 18
* '
8
Find the numerical value of 8 a 3 12 cr6 -f- 6 a6 2 6 s , if :
21. a = 2, 6 = 1. 26. a = 3, 6 = 3.
22. a = 2, 6 = 2. 27. a = 4, 6 = 5.
23. a =3, 6=2. 28. a =4, 6 = 6.
24. a=3, 6 = 4. 29. a = 3, 6 = 6.
25. a = 3, 6 = 5. 30. a = 4, 6 = 7.
Express in algebraic symbols :
31. Six times a plus 4 times 6.
32. Six times the square of a minus three times the cube of 6,
33. Eight x cube minus four x square plus y square.
34. Six w cube plus three times the quantity a minus 6,
35. The quantity a plus 6 multiplied by the quantity a 2
minus 6 2 .
36. Twice a 3 diminished by 5 times the square root of the
quantity a minus 6 square.
37. Read the expressions of Exs. 2-6 of the exercise.
38. What kind of expressions are Exs. 10-14 of this exercise?
30. The representation of numbers by letters makes it pos-
sible to state very briefly and accurately some of the principles
of arithmetic, geometry, physics, and other sciences.
Ex. If the three sides of a triangle contain respectively
a, 6, and c feet (or other units of length), and the area of the
triangle is S square feet (or squares of other units selected),
then
8 = \ V(a + 6 + c) (a 4- 6 - c) (a - 6 -f c) (6 a + c).
INTRODUCTION 13
E.g. the three sides of a triangle are respectively 13, 14, and
15 feet, then a = 13, b = 14, and c = 15 ; therefore
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= 1 V42-12-14.16
= 84, i.e. the area of the triangle equals
84 square feet.
EXERCISE 9
1. The distance s passed over by a body moving with the
uniform velocity v in the time t is represented by the formula
Find the distance passed over by :
a. A snail in 100 seconds, if v .16 centimeters per second.
b. A train in 4 hours, if v = 30 miles per hour.
c. An electric car in 40 seconds, if v = 50 meters per second
d. A carrier pigeon in 10 minutes, if v 5000 feet per minute.
2. A body falling from a state of rest passes in t seconds
over a space S ^gt 2 . (This formula does not take into ac-
count the resistance of the atmosphere.) Assuming g = 32 feet,
(a) How far does a body fall from a state of rest in 2
seconds ? *
(b) A stone dropped from the top of a tree reached the
ground in 2-J- seconds. Find the height of the tree.
(c) How far does a body fall from a state of rest in T ^ 7 of a
second ?
3. By using the formula
find the area of a triangle whose sides are respectively
(a) 3, 4, and 5 feet.
(b) 5, 12, and 13 inches.
(c) 4, 13, and 15 feet.
14 ELEMENTS OF ALGEBRA
4. If the radius of a circle is H units of length (inches,
meters, etc.), the area $ = 3.14 It 2 square units (square inches,
square meters, etc.). Find the area of a circle whose radius is
(a) 10 meters. (b) 2 inches. (c) 5 miles.
5. If i represents the simple interest of p dollars at r fo in
n years, then i = p n r %> or *
Find by means of this formula :
(a) The interest on $800 for 4 years at ty%.
(b) The interest on $ 500 for 2 years at 4 %.
6. If the diameter of a sphere equals d units of length, the
surface $ = 3.14d 2 (square units). (The number 3.14 is fre-
quently denoted by the Greek letter TT. This number cannot
be expressed exactly, and the value given above is only an
approximation.)
Find the surface of a sphere whose diameter equals :
(a) 8000 miles. (b) 1 inch. (c) 10 feet.
7. If the diameter of a sphere equals d feet, then the
volume 7 n
V= ~ cubic feet.
6
Find the volume of a sphere whose diameter equals:
(a) 10 feet. (b) 3 feet. (c) 8000 miles.
8. If F denotes the number of degrees of temperature indi-
cated on the Fahrenheit scale, the equivalent reading C on the
Centigrade scale may be found by the formula
C y = f(F-32).
Change the following readings to Centigrade readings:
(a) 122 F. (b) 32 F. (c) 5 F.
CHAPTER II
ADDITION, SUBTRACTION, AND PARENTHESES
ADDITION OF MONOMIALS
31. While in arithmetic the word sum refers only to the
result obtained by adding positive numbers, in algebra this
word includes also the results obtained by adding negative, or
positive and negative numbers.
In arithmetic we add a gain of $ 6 and a gain of $ 4, but we
cannot add a gain of $0 and a loss of $4. In algebra, how-
ever, we call the aggregate value of a gain of 6 and a loss of 4
the sum of the two. Thus a gain of $ 2 is considered the sum
of a gain of $ 6 and a loss of $ 4. Or in the symbols of algebra
$4) = + $2.
Similarly, the fact that a loss of $6 and a gain of $4 equals a
loss of $2 may be represented thus
In a corresponding manner we have for a loss of $6 and a loss
of $4 (- $6) + (- $4) = (- $10).
Since similar operations with different units always produce
analogous results, we define the sum of two numbers in such a
way that these results become general, or that
and (+6) + (+4) = + 10.
16
16 ELEMENTS OF ALGEBRA
32. These considerations lead to the following principle :
If two numbers have the same sign, add their absolute values ;
if they have opposite signs, subtract their absolute values and
(always) prefix the sign of the greater.
33. The average of two numbers is one half their sum, the
average of three numbers is one third their sum, and the
average of n numbers is the sum of the numbers divided by n.
Thus, the average of 4 and 8 is 0.
The average of 2, 12, 4 is 3 J.
The average of 2, '- 3, 4, 5, - 0, 10, is 2.
EXERCISE 10
Find the sum of:
1.
2.
3.
13.
-4
+ 5
4. +5
+ 4
7. +11
- 3
10. +6
-6
-5
+ 4
5. -9
+ 7
8. -11
- 3
11. -12
- 3
-5
-4
6. -7
-8
9. - 5
-17
12.
-5
* 'H
4
5
-6
u. -i
+ 2
-3
15. -1
-2
-3
16. -12
+ 15
Find the values of:
17. (-17) + (-14). 20. l-f(-2).
18. 15 + (-9). 21. (_
19. + -12. 22.
In Exs. 23-26, find the numerical values of a + b -f c-j-c?, if :
23. a = 2, 6 = 3, c = 4, d = 5.
24. a = 5, 6 = 5, c = 5, d = 0.
ADDITION, SUBTRACTION, AND PARENTHESES 1?
25. a = 22, & = -23, c=14, d = l.
26. = -13, & = 15, c = 0, d = 3.
27. What number must be added to 9 to give 12? -'
28. What number must be added to 12 to give 9 ? - -
29. What number must be added to 3 to give 6 ? C*
30. What number must be added to 3 to give 6? **j
31. Add 2 yards, 7 yards, and 3 yards. }/
32. Add 2 a, 7 a, and 3 a. \\ '
33. Add 2 a, 7 a, and 3 a.
Find the average of the following sets of numbers:
34. 3 and 25. ^ ' 37. 2, 3, 4, - 7, and 13.
35. 5 and - 13. 38. - 3, - 4, - 5, 6, - 7, and 1.
36. 12, 13, and 2.
39. Find the average of the following temperatures : 4 F.,
- 8 F., 27 F., and 3 F.
40. Find the average temperature of New York by taking
the average of the following monthly averages : 30, 32, 37,
48, 60, 09, 74, 72, 66, 55, 43, 34.
41. Find the average gain per year of a merchant, if his
yearly gain or loss during 6 years was : $ 5000 gain, $3000 gain,
$1000 loss, $7000 gain, $500 loss, and $4500 gain.
42. Find the average temperature of Irkutsk by taking the
average of the following monthly temperatures : 12, 10,
-4, 1, 6, 10, 12, 10, 6, 0, - 5, -11 (Centigrade).
34. Similar or like terms are terms which have the same
literal factors, affected by the same exponents.
6 ax^y and 7 ax' 2 y, or 5 a 2 & and , or 16 Va + b and 2Vo"+~&,
are similar terms.
Dissimilar or unlike terms are terms which are not similar.
4 a 2 6c and 4 a 2 6c 2 are dissimilar terms.
o
18 ELEMENTS OF ALGEBRA
35. The sum of two similar terms is another similar term.
The sum of 3 x 2 and x 2 is f x 2 .
Dissimilar terms cannot be united into a single term. The
sum of two such terms can only be indicated by connecting
them with the -f- sign.
The sum of a and a 2 is a -f- a 2 .
The sum of a and b is a -f ( 6), or a 6.
36. Algebraic sum. In algebra the word sum is used in a
wider sense than in arithmetic. While in arithmetic a b
denotes a difference only, in algebra it may be considered
either the difference of a and b or the sum of a and b.
The sum of a, 2 a&, and 4 ac 2 is a 2 a& -|- 4 ac 2 .
EXERCISE 11
Add:
1. -2 a 2.
+ 3 a ab 7 c 2 dn
-4o
6. 12 xY 7. 1
13 xY \
b s x 7 #y -f- 7 a 2 frc
Find the sum of :
9. 2 a 2 , - 3 a 2 , -f 4 a 2 , 5 a 2 , + 6 a f .
10. 12 wp 2 , - 13 rap 2 , 25 rap 2 , 7 rap 2 .
11. 2(a-f &), 3(a-f-6), 9(a-f-6), 12(a-f b)
12. 5l
13. Vm -f- ii, 5Vm + w, 12Vm-f-n, 14
ADDITION, SUBTRACTION, AND PARENTHESES 19
Simplify :
15. -17c 3 + 15c 8 + 18c 3 + 22c 3 +c 3 .
17. xyz + xyz 12 xyz + 13 xyz + 15 xyz.
Add:
18. ra 19. +m 20. m 2 21. a
ZL """ n n 2 a 8
**, x*
22. 6 23. c 2 ^24. ^ 25. 7
-1 1 i 2 -co*
l^S
26. mn 27. xyz
mri
Simplify the following by uniting like terms:
29. 3a-76 + 5a + 2a-36-10a+116.
30. 2a 2 -4a-4 + 6a 2 -7a 2 -9a-2a + 8.
31.
32.
33. "Vx + y Vaj + y 2 2 Vi + 2/ + 2 Va; + / + 3 Va; + y.
Add, without finding the value of each term :
34. 5x173 + 6x173-3x173-7x173.
35. 4x9'
-36. 10x3 8
ADDITION OF POLYNOMIALS
37. Polynomials are added by uniting their like terms. It
is convenient to arrange the expressions so that like terms may
be in the same vertical column, and to add each column.
20 ELEMENTS OF ALGEBRA
Thus, to add 26 ab - 8 abc - 15 6c, - 12 a& 4- 15 abc - 20 c 2 ,
5 ab 4- 10 6c 6 c 2 , and 7 a&c 4- 4 6c + c 2 , we proceed as
f 110WS: 26 aft- 8 & c~15&c
-12a&4l5a&c -20c 2
- 6a& -f-lO&c 6c a
7 a5c + 4 be 4 c a
9a& 6c 26 c' 2 Sum.
38. Numerical substitution offers a convenient method for
checking the sum of an addition. To check the addition of
3 a -f 4 1) 4- o c and 4- 2 a 26 c assign any convenient
numerical values to a, 5, and c, e.g. a = 1, 6=2, c = 1,
then - 3 a -f 4 ft -f 5 c = - 3 + 8 + 5 = 1 0,
2 0-25- c= 2- 4-1 = -3,
the sum a 4- 2 6 + 4 c = 1 + 4 -f- 4 = 7.
But 7 = 10 3, therefore the answer is correct.
NOTE. While the check is almost certain to show any error, It is not
an absolute test ; e.g. the erroneous answer a406 4c would also
equal 7.
39. In various operations with polynomials containing terms
with different powers of the same letter, it is convenient to
arrange the terms according to ascending or descending powers
of that letter.
7 4. x 4 5 x" 2 + 7 x* 4 5 5 is arranged according to ascending powers
of x. 6 a 7 - 7 a fi & + 4 a 4 6c 8 aW 4 7 a&<d? + 9 6 5 4 e 7 is arranged ac-
cording to descending powers of a.
EXERCISE 12
Add the following polynomials :
1. 2a 364-6 c, 3a 46 7 c, and 4a4-6 2c.
2. 9 q 4- 7 t 5 s, 2 ? 4- - 3 s, and 12 q 5 J 4- 2 s.
V3. 2z 2 -4?/ 2 -f2z 2 , -3ar z -22/ 2 4- 2 , 4 a; 2 - 3 / - 2 z 2 , and
5 0^-9 z 2 .
* For additional examples see page 259.
ADDITION, SUBTRACTION, AND PARENTHESES 2i
4. 14d-15e + 2/, 16e + 17/-90, -18/+6y + d, and
</ - 15 d.
5. -12a 2 4 15& 2 -20c 2 , - 12 6 2 ~5 a 2 - 5 c 2 , ll& 2 -7c 2 ,
and -6 2 4- a 2 .
6. 6 # 3 4 5 z 2 4 2 tf -- 7, 6 # 2 - 1 a 4 1 0, -7ar l + 3B 2 -5,and
^_.Ga 2 4-3x45.
7. 9m 3 48m 2 4- 7v/i-f- 6, 5-6 w- 7m 2 - 8 m 3 , 2m 9 -12,
and m 2 ?ft 4 6.
8. xy3xz + yz, 2xy + 4:XZ-}-5yz,4:xy xz 6yz, and
- 2 #?/ 4 ?/z.
, 9. 6a 8 -5a 2 &47a& 2 -4& 3 , 5 a s + 4 aft 2 - 5 cr& + 7 6 3 , and
10a 8 +lOa 2 6-ll& 2 .
10. a 4 + ^* 3 -f- 1> 3 -h ^ 2 1 a 4 ? 3 2 j and a 2 4- a.
11. a byb c^c <l, d e, e /, and / a.
12. 4ar ! + 50 8 + 62 8 , - 5a^-6 ?/ 3 - 7^ 3 ? 2iB 8 + 2y 8 + 2 8 , and
13. 4(a -f 6) - (b + c) 3(c -f a), 2(6 + c) + (c -f a), and
14. a 4 a 1, a 2 -f a -f- 1, a 2 a, and a 2 + a + 1.
v 15. 6(a + 6) 2 - 5 (a + 6) + 3, 7(a + 5) 2 - 9(a + &) - 12, and
- 12(a 4- 6) 2 + 14(a 4 6) 4 10.
16. 7 a; 4 4 5 x*y 2 y?y* 3 xf, a; 4 4 o^?/ 2 4 y\ and 6 a; 4
4 ajy 2 ?/.
17. 5 Va - VS 4 2 Vc, - Va 4 2 V& 4 6 Vc, - 4 Va - 10 Vc,
and 5 Vb 4 Vc.
18. a 3 4 a 2 - a, a 2 4 a - 1, - a 3 4 a 4 1, and 1 4 a 3 - a 2 .
19. 3 a 4 9 y\ 3 afy - 3 ay 8 , 6 afy + 6 ay/ 3 4 10 ?/ 4 , and - 3 a) 4
-y 4 -^/.
v/20. w* 4 3 m 2 n 4 3 m?i 2 4 rz 3 , in 8 3 m 2 n 4- 3 mn 2 n 8 , and
2 w 8 - 2 n 3 .
22 ELEMENTS OF ALGEBRA
21. 4 w 5 + 3 m 3 + 2 m, 4 ?n 4 - 2 m + 2 m 8 , - ra 5 + m, and
3 m 3 7 m.
22. a -f- 6 c, 6 -f c d, c + d e, d + e a, and e -}- a 6.
23. 16m 2 7/-12my 2 + 6y 3 , - 17 1/ 3 + 4 ?nfy - m 3 , 4m 2 ?/-?/ 3 ,
and 2 m 3 5 y 8 .
24. 3 3 -2 8 n + <w 2 ,4 8 -f-3f a n-2<w 2 +n s ,5< 3 4^4.3^* 2n s ,
and 4^ + 3t*n 2tn* + n*.
25. l-2aj Sic 2 , -4-5a-6 2 , -T-8a;-9aj 2 , and
6 + 9 x + 12 a,- 2 .
26. 2-fa 4 -a 2 +7a, a 3 -a 4 3 a-f^, a s -f3o 2, and 5+a\
^ "27. 3 SM/Z + 2 a:?/ -f x y bxyz~lx, $ xy 12 xyz, and 3^2
- 11 xy + 12.
SUBTRACTION
EXERCISE 13
1. If from the five negative units 1, 1, 1, 1, 1,
three negative units are taken, how many negative units re-
main ? What is therefore the remainder when 3 is taken
from - 5 ?
2. Instead of subtracting in the preceding example, what
number may be added to obtain the same result ?
3. The sum total of the units -f 1, -f 1, -f- 1, -f 1, + 1, 1,
1, and 1, is 2. What is the value of the sum if two neg-
ative units are taken away ? If three negative units are taken
away ?
4. What is therefore the remainder when 2 is taken from
2 ? When - 3 is taken from 2 ?
5. What other operations produce the same result as the
subtraction of a negative number?
6. If you diminish a person's debts, does he thereby become
richer or poorer ?
ADDITION, SUBTRACTION, AND PARENTHESES 23
7. State the other practical examples which show that the
subtraction of a negative number is equal to the addition of a
positive number.
40. Subtraction is the inverse of addition. In addition, two
numbers are given, and their algebraic sum is required. In
subtraction, the algebraic sum and one of the two numbers is
given, the other number is required. The algebraic sum is
called the minvend, the given number the subtrahend, and the
required number the difference.
Therefore any example in subtraction may be stated in a
different form ; e.g. from 5 take 3, may be stated : What
number added to 3 will give 5 ? To subtract from a the
number b means to find the number which added to b gives a.
Or in symbols, a-b = x,
if x + b a.
Ex. 1. From 5 subtract 3.
The number which added to 3 gives 5 is evidently 8.
Hence, 6 -(-3) = 8.
Ex. 2. From 5 subtract - 3.
The number which added to 3 gives 5 is 2.
Hence, (- 6) - ( - 3) = - 2.
Ex. 3. From 5 subtract + 3.
This gives by the same method,
41. The results of the preceding examples could be obtained
by the following
Principle. To subtract, change the sign of the subtrahend and
add.
NOTE. The student should perform mentally the operation of chang-
ing the sign of the subtrahend ; thus to subtract 8 2 6 from 6 a 2 fc,
change mentally the sign of 8 a 2 6 and find the sum of 6 a 2 6 and
24 ELEMENTS OF ALGEBRA
42. To subtract polynomials we change the sign of each term
of the subtrahend and add.
Ex. From _6ar 3 -3z 2 + 7 subtract 2 x - 3 x* - 5 x + 8.
Check, If x = l
-6ar 3 -3o 2 +7 = -t- 2
2 or 3 - 3 r*-5o;-f 8
EXERCISE 14
Subtract the lower
number
from the upper one :
1.
5
11.
12
21. 17 31. a
7
12
-17 b
2.
-5
12.
12
22. 14 n 8 32. a
7
-19
15 n 2 -6
3.
o
13.
23. 14 n 2 33. a
-7
7
- 15 n 2 +b
4.
4
14.
-7
24. 16n 2 m 34. a
-2
- 16 n 2 m - b
5.
-5
15.
25. 7x 5 y 3 35. a 2
6
+ 9
-7 ajy <L
6.
-9
16.
-7
26. mpq 2 36 ' a *
5
-6
02 &
7.
-9
17.
-6
07 1
97 Q <n 2 / ' *
^6 I . ~~ *J JJ ov o
8
-7
... fjf
8.
-9
18.
40
, 38. ab
28 CLOC
-8
41
2abc
39 tt 2 6
9.
9
19.
2
29. O Wl< X a
Q
20
a
13 wi'\c "
10.
-5
20.
-2
30. 4a 6Va + &
-5
-3
_7 a _6^/ a _|_5
ADDITION, SUBTRACTION, AND PARENTHESES 25
41. Subtract the sum of 2 m and 7 m from 10m.
42. From 10 a 12 & -f 6 c subtract 14 a -f 12 b -f 5 c, and
check the answer.
43. From $ 7 x 2 ?/ 5 a/ + ?/ subtract ar -f 7 a 2 ?/ - 5 #?/ 7/ 3 ,
and check the answer.
44. From - a 3 -f 2 a a - 7 a -f- 2 subtract a 3 + 2 a 2 - 7 a - 2.
45. From 5a-(>& + 7c 8d take 11 a 6 6 11 c-f 17 d.
46. From 2 x 2 8 a?y + 2 ?/ 2 take 2 :c 2 8 o# 2 ?/ 2 .
47. From mn -f ??/> + qt mt subtract mn -f wp -f- w>t.
48. From a 3 1 subtract a 2 -f a + 1.
49. From 6 a -f 9 6 c subtract 10 b c -f d.
50. From 1 -f & take 1 -f b 2 -f- & b s .
51. From 6 a; 4 + 3 x -f- 12 take 3 ar 3 -f- 4 x + 11.
52. From 2 a -f & take a -f- & -j- c.
53. From a 3 subtract 2 a 3 -f- a 2 -j- a 1.
54. From 3 or 2 2 a:// + ?/ 2 subtract 3 #?/ 2 y 2 .
55. From 5 a 2 2 ab ?/-' subtract 2 a 2 + 2ab 2y 2 .
56. From 6 subtract l-t-2a-f3& + 4<7.
57. From 16 + a 3 subtract 8 2 a + a 2 -f a 3 .
58. From a 4 - 4 a*& + 6 a 2 & 2 - 4 a^ 4- & 4 subtract a 4 + 4 8 6
4- 6 a-& 2 + 4 a& 3 -f 6 4 .
v 59. From 6(a-f- 6)-f- 5(6 + c) 4(c + a) subtract 7(a-f&)
REVIEW EXERCISES
1. From the sum of a 4- b -h c and a & -f c subtract
a _ 6 _ 2 c.
2. From the sum of x 2 4 x -f- 12 and 3 a 2 3 # 3 sub-
tract 4 x 2 + 2 a; 7.
x 3. From a 3 + 2 a 2 4 a subtract the sum of a 3 -}- a 2 2 a
and a 2 a + 4.
26 ELEMENTS OF ALGEBRA
4. From the difference between a? + 5 a: 2 + 58+1 and 4 a? 2
+ 4 x subtract a? 3 -j- # + 1.
5. Subtract the sum of x 2 + 2 and iE 3 + cc 2 from x 3 + a^
6. Subtract the sum of 5 a 2 6 a + 7 and 2a 2 + 3a 4
from 2 a 2 + 2 a + 2.
7. Subtract the sum of 6 m 8 + 5 m 2 + 6 m and 4m*
5 m 2 + 4 m from 2 ra s + 7 m.
8. Subtract the difference of a and a + 6 + c from a + b + c
9. Subtract the sum of # 2 + f and a 2 y 2 from 2 ar* + 2 */ 2
10. To the sum of 2a + 66 + 4c and a 4 6 2 c add the
sum of9ci-66 + c and 10 a + 5 b 2 c.
11. To the sum a 3 4- a 2 + 1 and a 2 + a add the difference
between 5 a 3 + 4 and 4 a 3 + a 2 a.
12. What expression must be added to 7 a 3 + 4 a 2 to pro-
duce 8a 3 -2a-7?
13. What expression must be added to 3a + 56 cto pro-
duce ~2a-6 + 2c?
14. What expression must be subtracted from 2 a to produce
-a+6?
,15. What must be added to 4^ + 4^ + 2 to produce 0?
v
b = x +g z, c = x 2y + z, find :
16. a + 6. 18. a + 6 + c.
17. a 6. 19. a 6 + c.
20. A is n years old. How old will he be 10 years hence ?
2 m n years hence ?
21. A is 2 a years old. How old was he a b years ago?
a + b c years ago ?
ADDITION, SUBTRACTION, AND PARENTHESES 27
SIGNS OF AGGREGATION
43. By using the signs of aggregation, additions and sub-
tractions may be written as follows:
a -f ( 4- & c + d) = a + b c + d.
Hence it is obvious that parentheses preceded by the -f or
the sign may be removed or inserted according to the fol-
lowing principles :
44. I. A sign of aggregation preceded by the sign -f may be
removed or inserted without changing the sign of any term.
II. A sign of aggregation preceded by the sign may be re-
moved w inserted provided the sign of evei'y term inclosed is
changed. E.g. a+(b-c) = a +b - c.
o + ( 6 + c) = a
a ( 4- 6 c) = a & -f- c.
a -- b c = a -f- & -f c.
45. If there is no sign before the first term within a paren*
thesis, the sign -f- is understood.
a (b c) = a 6 4- c.
46. If we wish to remove several signs of aggregation, one
occurring within the other, we may begin either at the inner-
most or outermost. The beginner will find it most convenient
at every step to remove only those parentheses which contain
no others.
Ex. Simplify 4 a - f (7 a + 5&)-[-6& +(-25- a^6)] } .
4a-{(7a + 6&)-[-6&-f(-2&- a~^~6)]}
= 4 a -{7 a -f- 6 b -[- 6 b -f (- 2 b - a -
= 4a 7a 06 66 2&-a + 6
sss 4 a 12 6. Answer.
28 ELEMENTS OF ALGEBRA
EXERCISE 15*
Simplify the following expressions :
1. x + (2y-z). 6. a (-a + 6).
2. a-(3b 2c). f 7. a -f (a - 6) - (a + 6).
3. a 3 + (2a 3 -6 3 + c 3 ). 8. a-(- a + 6)-f (a-2 b).
4. a 2 -(a 2 + 26 2 -c 2 ). 9. 2m
5. 2a 2 -(4a 2 -26 2 +c 2 ). 10. 4a-f- 2a;-y (60;- ?
11. 2a 2 + 5a-(7-f 2a 2 )-f (5-5a).
13.
14. (m -f- 7i -h jp) (m n p) m~n-\-p.
. ___
15. m + n (m ?*,) + M> + w ( m -f ft)-
16. a 2 + [# (r 5 a;)].
17.
18. a (6 c) + [3 a 26 6a].
19. a [36+ {3c (c a)} -2c].
21. 7 (a 6)+ {a [a-
22. By removing parentheses, find the numerical value of :
1422 - [271 - { 271 + (814 - 1422) J ] .
47. Signs of aggregation may be inserted according to 43.
Ex. 1. In the following expression inclose the second and
third, the fourth and fifth terms respectively in parentheses,:
Ex. 2. Inclose in a parenthesis preceded by the sign the
last three terms of
See page 260.
ADDITION, SUBTRACTION, AND PARENTHESES 29
EXERCISE 16
In each of the following expressions inclose the last three
terms in a parenthesis :
1. a-\-l> c + d. 3. 5 a 2 7 x* 4-9 x + 2.
2. > 2m-n + 2q-3t. 4. 4 xy - 2 tf - 4 y* - 1.
In each of the following expressions inclose the last three
terms in a parenthesis preceded by the minus sign :
5. m 2 -27i 2 -3^ 2 + 4r/. 8 . 5^2 _ r )X - 7-fa.
6. x y -f- z + d.
7. p + q + r-s.
EXERCISES IN" ALGEBRAIC EXPRESSION
EXERCISE 17
Write the following expressions :
I. The sum^)f m and n. 2. The difference of a and 6.
NOTE. The minuend is always the first, and the subtrahend the second, '
of the two numbers mentioned.
3. The sum of tKe squares of a and b.
4. The difference of the cubes of m and n.
5. The difference of the cubes of n and m.
6. The sum of the fourth powers of a and 6.
7. The product of m and n.
8. The product of the cubes of m and n.
9. Three times the product of the squares of m and n.
10. The cube of the product of m and n.
II. The square of the difference of a and b.
12. The product of the sum and the difference of m and n.
13. Nine times the square of the sum of a and 6 diminished
by the product of a and b.
30 ELEMENTS OF ALGEBRA
14. The sum of the squares of a and b increased by the
square root of x.
15. x cube minus quantity 2 x 2 minus 6 x plus 6.
16. The sum of the cubes of a, b, and c divided by the dif-
ference of a and d.
Write algebraically the following statements:
V 17. The sum of a and b multiplied by the difference of a and
b is equal to the difference of a 2 and b 2 .
18. The difference of the cubes of a and b divided by the
difference of a and 6 is equal to the square of a plus the prod-
uct of a and 6, plus the square of b,
s -19. The difference of the squares of two numbers divided by
the difference of the numbers is equal to the sum of the two
numbers. (Let a and b represent the numbers.)
CHAPTER III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS
EXERCISE 18
In the annexed diagram of a balance, let us consider the
forces produced at A by 3 Ib. weights, applied at A and JB, and
let us indicate a downward pull at A by a positive sign.
1. By what sign is an upward pull at A represented ?
2. What is the sign of a 3 Ib. weight at A ?
3. What is the sign of a 3 Ib. weight at B ?
4. If the two loads balance, what force is produced by the
addition of five 3 Ib. weights at A ? Express this as a multi-
plication example.
5. If the two loads balance, what force is produced by
taking away 5 weights from A ? What, therefore, is 5 X 3 ?
6. If the two loads balance, what force is produced by the
addition of 5 weights at B ? What, therefore, is 5 x ( 3) ?
7. If the two loads balance, what force is produced by tak-
ing away 5 weights from B ? What therefore is ( 5) x ( 3) ?
31
32 ELEMENTS OF ALGEBRA
8. If the signs obtained by the method of the preceding
examples were generally true, what would be the values of
5x4, 5x(-4), (-5)X4, (
9 x 11, (- 9) x 11, 9 x (- 11), (- 9) x (- 11) ?
9. State a rule by which the sign of the product of two fac-
tors can be obtained.
48. Multiplication by a positive integer is a repeated addition;
thus, 4 multiplied by 3, or 4x3 = 44-44-4 = 12, 4 multi-
plied by 3, or
(_4) X 3=(-4)+(-4)+(-4)=-12.
The preceding definition, however, becomes meaningless if
the multiplier is a negative number. To take a number 7
times is just as meaningless as to fire a gun 7 times.
Consequently we have to define the meaning of a multiplica-
tion if the multiplier is negative, and we may choose any defi-
nition that does not lead to contradictions. Practical examples^
however, such as given in the preceding exercise, make it con-
venient to accept the following definition :
49. Multiplication by a negative integer is a repeated sub-
traction.
Thus, 4 x(-8) = ~(4)-(4)-(4)=:-12,
(- 4) x ( ~ 3> = -(- 4)-(- 4)-(-4) = + 12.
NOTE. This definition has the additional advantage of leading to alge-
braic laws for negative numbers which are identical with those for posi-
tive numbers, a result that would not be obtained by other assumptions.
In multiplying integers we have therefore four cases illus-
trated by the following examples :
4x3 = 4-12. 4x(-3)=-12.
MULTIPLICATION 33
50. We shall assume that the law illustrated for positive
and negative integers is true for all numbers, and obtain thus
the
Law of Signs: TJie product of two numbers with like signs in
positive; the product of two numbers with unlike signs is negative.
Thus, (-a)(+6) = -a&; (- a)(- &) = + a&; etc.
EXERCISE 19
Find the values of the following products :
1. 6 X(-5). 4. 6x-7.
2. (-7) X (-12). 5. (-2)x9.
3. (-2)X 6. 6. (-4)X(-15).
NOTE. If no misunderstanding is possible, the parenthesis about fac-
tors is frequently omitted.
7. -5x-3. 13. (-2) 2 .
8. 4 . - 7. 14. (- 3) 8 .
9. _3.(-4J). 15. (-4)'.
10. _2. -. 16. (-1) 7 .
11. 3. _2^ -3. 17. (-10) 4 .
12. 6.-2--f 18. -1- -2- -3- -4. +5.
19. Formulate a law of signs for a product containing an
even number of negative factors.
20. Formulate a law of signs for a product containing an
odd number of negative factors.
If a = 2, b = 3, c = 1, x = 0, and y = 4, find the numeri-
cal values of:
21. 4a6c. 25. 3 aW. 29. - (a&c) 2 .
22. 3 a 2 ?/ 2 . 26. 2a6 8 c 4 . 30. 4a 2 -f-26 2 .
23. 2a 2 6c. 27. 11 aWcx. 31. 2a + 3& 2 -6c*
24. Ua z b s x. 28. (c#) 8 . 32. 4 a 2 - 2 f + x 2 .
34 ELEMENTS OF ALGEBRA
Find the numerical value of 8 a 8 - 1 2 a 2 6 -f 6 aW - 6 3 , if
33. a=2, & = -3. 35. a = 2, 5 = 2.
34. a = 1, 6 = 4. 36. a = 3, 6 = 1.
MULTIPLICATION OF MONOMIALS
51. By definition, 2 3 = 2 . 2 - 2, and 2 5 = 2- 2 -2 - 2 2.
Hence 2 3 x 2 a = 2 2 2 x 2 2 2 2 2 =2 8 , *.<?., 2 3 + 5 . Or in
general, if m and n are two positive integers,
a m X a n = (a a a - to m factors) (a a a to n factors)
= a a to (m -f n) factors.
fl*" X fl w = fl /w + w .
This is known as
52. The Exponent Law of Multiplication : The exponent of the
product of several powers of the same base is equal to the sum oj
the exponents of the factors.
Ex. 1. 6 aWc x - 7 &*# =(6 - 7) (a 2 a 8 ) . (ft 8 . &*) c d*.
53. In multiplying a product of several factors by a number,
only one of the factors is multiplied by the number.
Ex. 2. 2 x (2* 5 . 7 2 )= 26 . 5 . 7 2 .
Ex. 3. 4 x (2 25) = 8 25, or 2 . 100, i.e., 200.
EXERCISE 20
Express each of the following products as a power :
1. m*.m 4 . 3. 2 3 -2 6 . 5. 5 3 5 2 . 5 . 6"
2. a 3 - a 2 . 4. 3 2 . 3 3 4 . 6. 127 3 127 9 127 U .
7. (a6) 2 -(a5) 9 . 9. a 5 - (-a) 7 .
B. (^ + 2/) 8 -(a? + 2/) 4 -(aj + 2/). 10. 7 8 .(-7).7.
11. 12 14 .(-12) U .12 U .
Perform the operation indicated :
12. 3a-7abc. 14. 2(7.3-5), 16. 5(7-11.2).
13. 4- (2- 25- 7). IB. 50(11-2.3). 17. 2(14.50-3),
M UL TIP LIC A TION 35
18. 7(6- f- 2). 27. -7p*q 4 r*.-7pqt.
19. 11(3- 6- A). 28. _4aft.-4a#.
20. 4 aft -5 aft 2 . 29. - 7 w'W 2 (-8 n^W).
21. 19 mV 5 - 2 ran 4 . 30. 5 aft 3 ftc ( 2ac).
22. - 5 xy 3 tfy 2 z*. 31. 9 afy ( 2 a 3 ?/ 2 ).
23. ' 19 aW (- 2 aft 4 ). 32. (- 4 a 2 ft 3 ) 2 .
24. lla 2 ft 3 c-(-4a 2 ftc 2 ). 33. (2 ax 3 /) 2 .
25. ( 5 aft 3 ) (- C a 2 ftc). 34. (- 3 win 4 ) 1 .
26. 6 e/ 2 a (- 6 e 8 /). 35. (- 2 a 2 ) 3 .
MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL
54. If we had to multiply 2 yards and 3 inches by 3, the
results would obviously be 6 yards and 9 inches. Similarly the
quadruple of a 4 2 b would be 4 a -f 8 ft, for
= (a + 26)+(a + 2 ft) -f (a 4- 2 ft) + (a + 2 ft)
55. This principle, called the distributive law, is evidently
correct for any positive integral multiplier, but we shall as-
sume it for any number.
Thus we have in general
a(b -f c) = ab +ac.
56. To multiply a polynomial by a monomial, multiply each
tet^m by the monomial.
- 3 a 2 6(6 a*bc + 2 be - 1) = - 18 a 4 6 2 c - 6 a 2 6 2 c -f 8 a 2 6.
EXERCISE 21
Find the numerical values of the following expressions, by
first multiplying, and then adding :
1. 2(5-fl5-f25). 3. 3(124342). 5. 12(| + 1 4 i).
2. 6(104-20430). 4. 2(645410). 6. 17(10041042).
7. 23(10004100420).
36 ELEMENTS OF ALGEBRA
Express as a sum of several powers :
8. 5(5 + 5 2 + 5 7 ). 11. 4 13 (4 9 -4 5 -4).
9. 6 2 (6 + 6 2 + 6 3 ). 12.
10. 7 3 (7 3 -f-7 2 + 7 10 ).
Perform the multiplications indicated:
13. 2 m(m-hn -\-p). 16. 4 %Pq\ pqr + 5 pr 7).
14. ~2mn(m 2 +n 2 -p 2 ). 17. -5 x\- 5 x 2 - 5 x- 5).
19. 5 aW( 2 aW + 3 a 5 6 2 c - 6 a6).
20. 7 a 3 6 2 c(- ^ c 2 + 2 ?/ 2 - 3 aftc).
21. - 2 mn(9 mV - 5 w*V -f 7 wn).
22.
23. -:
24. By what expression must be multiplied to give
25. Express 3a^ 4o; 2 -f7a;asa product.
26. Find the factors of 3x + 3 y + 3z.
27. Find the factors of 6 or* -f 3 x* -f 3 a 4 .
28. Find the factors of 2 ofy 4 arty + 8 a; 4 .
29. Find the factors of 5 a 2 6 - 60 a& 2 10 aft.
30. Find the factors of 6 ary - 3 x 2 y 2 + 3 xy.
MULTIPLICATION OF POLYNOMIALS
57. Any polynomial may be written as a monomial by in-
closing it within a parenthesis. Thus to multiply a b by
x + y z,we write (a b) (x + y z) and apply the distributive
law.
(a - 6) (x -f y z ) = x(a b) + y(a b) z(a b)
= (ax bx) -f (ay by) (az bz)
by az + bz.
M UL TIP LIC A TION 37
58. To multiply two polynomials, multiply each term of one by
each term of the other and add the partial products thus formed.
The most convenient way of adding the partial products is
to place similar terms in columns, as illustrated in the follow-
ing example :
Ex.1. Multiply 2 a - 3 b by a 5 b.
2a-3b
a-66
2 a 2 - 3 ab
- 10 ab + 15 6 2
2 a 2 - 13 ab + 15 6 2 Product.
59. If the polynomials to be multiplied contain several
powers of the same letter, the work becomes simpler and more
symmetrical by arranging these expressions according to either
ascending or descending powers.
Ex. 2. Multiply 2 + a 3 -a- 3 a 2 by 2 a a 2 + l.
Arranging according to ascending powers :
Check. If a = I
2 a - 3 a 2 + a 8 = - 1
a =2
2 a - 3 a 2 + a 8
4- 4.a - 2 a 2 6 a 8 -f 2 a*
- 2" a 2 + a 8 + 3 * - a 6
-7 a 2 4 a 8 + 5 a* - a 6 =2
60. Examples in multiplication can be checked by numerical
substitution, 1 being the most convenient value to be substi-
tuted for all letters. Since all powers of 1 are 1, this method
tests only the values of the coefficients and not the values of
the exponents. Since errors, however, are far more likely to
occur in the coefficients than anywhere else, the student should
apply this test to every example.
38 ELEMENTS OF ALGEBRA
EXERCISE 22*
Perform the following multiplications and check the results ;
1. (2s 3y)(3a? + 2y). 17. (2 x* x - 4) (x + 1).
2. (4a-f 76)(2tt 36). 18. (4ra 2 -f-ra l)(ra-f 2).
3. (5c-2d)(2c-3d). 19.
4. (2w 3n)(7m + 8n). 20.
5. (6p -f-6<7)(5^) -f- 3<7). 21.
6. (2 a 5c)(2a-6c). 22.
7. (9m-2n)(4m + 7tt). 23. (a 2 a-l)(2a?-fl).
8. (aj-f6y)(aj 7y). 24.
9. (6i-7n)(llJ-n). 25.
10. (13 A; -2) (3 A: -1). 26.
11. (6xy + 2z)(2xy 4 2). 27.
12. (8r-7*)(6r-39. 28. (2m 2 - 1 - 2m)(l -m).
13. (llr + l)(12r 1). 29. (a 36) 2 .
14. (rcya 12)(a?^2-|-l). 30. (4 a I) 2 .
15. (a&c + 7)(2a&c-3). 31. (6a~7) 2 .
16. (a 2 -|-2a + 2)(a-3). 32. (6a&c-5) 2 .
33. (4a 2 6 2 -3a6-f-2)(2a6~l).
35. (m?n?p 2 -^ 2 mnp -f- 4) (mnp 4- 2).
36. (x + & + 1-f a^faj -1).
37. (a 2 -
OQ //)4
OO. lA/ '
QQ //j.2
O7. I ^/
40. (m-fn)(m-4- n)(m n)(m n).
41. (a-^-26) 8 .
* For additional examples see page 261.
M UL TIP LIC A TION
39
SPECIAL CASES IN MULTIPLICATION
61. The product of two binomials which have a common term.
62. The product of two binomials which have a common term
in equal to the square of the common term, plus the sum of the two
unequal terms multiplied by the common term, plus the product
of the two unequal terms.
(5 a 6 ft) (5 a 9 ft) is equal to the square of the common term, 25 a 2 ,
plus the sum of the unequal terms multiplied by the common terms, i.e.
( 16 ft) (5 a) = 75 ab, plus the product of the two unequal terms, i.e.
-f 54 ft 2 . Hence the product equals 25 a' 2 75 ab + 54 ft 2 .
EXERCISE 23
Multiply by inspection :
1. (a + 2) (a -f 3).
2. (a-3)(a + 2).
3. ( a _3)( a _4).
4. (a; + 3) (a -7).
6. (-!)(* -5).
6. (p-12)(p + ll).
7. (wi + 9)(m+9).
8. (6 -12) (6 -f- 13).
9. (2,-25)(y+4).
10. (J + 60)(f-2).
11. (*- !!)( + 21).
12. (a - 2 6) (a -f 6).
13. (a -2 6) (a -3 6).
14. (a + 2 6) (a 6).
29. Find two binomials
15. (a -9) (a + 9).
16. (ra- n)(w-f w).
17. (a 2 - 5 b z ) (a 2 -f 4 ft 2 ).
18. (ofy* -f 3) (tfy* - 4).
19. (a5 2 2^*-12)(ajy
20. (a 3 -7) (a 3 -8).
21. (100 +2) (100 + 3).
22. (10+ 1) (10 + 2).
23. (1000 + 5) (1000 + 4).
24. (100-1) (100 + 2).
25. (1000 -2) (1000 + 3).
26. 102 x 103.
27. 1005x1004.
28. 99 X 102.
whose product equals a? 2 3 x + 2.
40 ELEMENTS OF ALGEBRA
Find two binomial factors of each, of the following expres-
sions :
30. ar'-Sz + G. 34. w 2 + 2 w - 15.
31. a 2 + 6 a + 8. 35. m 2_ 3m _ 4
32. a 2 7 a + 10. 36. n 2 10ii+16.
33. w 2 2 ro - 15. 37. p 2 -p- 30.
63. Some special cases of the preceding type of examples
deserve special mention :
II.
III. (a-
Expressed in general language :
I. 77ie square of the sum of two numbers is equal to tlie square
of the first, plus twice the product of the first and the second, plus
the square of the second.
II. 71ie square of the difference of two numbers is equal to the
square of the Jirst, minus twice the product of the first and the
second, plus the square of the second.
III. The product of the sum and the difference of two numbers
is equal to the difference of their squares.
The student should note that the second type (II) is only a
special case of the first (I).
Ex. (4 x 3 + 7 i/ 2 )' 2 is equal to the square of the first, i.e. 16 y* t plus twice
the product of the first and the second, i.e. oft x 3 y' 2 , plus the square of the
second, i.e. 49 y*. Hence the required square equals 16 xP -f- 66 s; 8 j/ 2 + 49 y 4 .
EXERCISE 24
Multiply by inspection :
1. (a + 6) 2 . 4. (a-2) a . 7. <J>-7) J .
2. (a + 2) 2 . 5. (p + 3) 2 . 8. (a-26) 2 .
3. (a -a) 2 6. (*-5) 2 . 9. (x+3i/) 2 .
MULTIPLICATION 41
10. (2x-3yy. 13. G> 2 +5g)*. 16. (2a6-c) 2 .
11. (4a-36) 2 . 14. (3p 2 -9) 2 . 17. (2a# + 3z) 2 .
12. (a 2 -3) 2 . 15. (6a 2 -7& 2 ) 2 . 18. (4 a 2 6 2 -5c 2 ) 2
19. (6afy 2 2 2 -5) 2 . 25. (a; 2 -11 # 2 ) 2 .
20. (m -f n)(ra w )- 26 - (^ 2 + 11 2/ 2 ) 2 -
21. (2m + 3)(2m-3). 27. (5 r 2 - 2 Z 2 ) (5 r 2 -f 2 J 2 ).
22. ( tt 2 -7)(a 2 -f 7). 28. (5 r*-2t 2 ) 2 .
23. (c-d 2 -5)(c 2 d 2 + 5). 29. ( a + 5)(5+a).
24. (^-.ll^X^+lly 2 ). 30. (m 2 -27i 5 )(m 2 + 2n 5 ).
31. (100 + 1) 2 . 34. 104 2 . 37. 991
32. (100 + 2) 2 . 35. (1000 -I) 2 . 38. (20 -f- 1) 2 .
33. 103 2 . 36. ,998 2 . 39. 22 2 .
40. (100 + 2) (100 -2). 41. 99x101. 42. 998x1002.
Extract the square roots of the following expressions :
43. x*+2xy+y\ 45. m 2 -2m-hl. 47. n*-6n+9.
44. a 2 -2a6 + & 2 . 46. n 2 -f4n+4. 48. a 2 -8a6+166 2 .
49. a 2 + 10 ab -f 25 b\
Pind two binomial factors of each of the following expres-
sions :
50. y?-f. 51. a 2 -9. 52. m 2 16. 53. 6 2 -25n 2
54. 9a 2 -496 2 . 56. 25 a 4 -9.
55. 16aW-25. 57. 9 a 2 - 30 ab + 25 6 2 .
64. The product of two binomials whose corresponding terms
are similar.
By actual multiplication, we have
3x + 2y
5x 4 y
2xy-Sy*
42 ELEMENTS OF ALGEBRA
The middle term of the result is obtained by adding the
product of 5 x 2 y and 4 y 3 x. These products are fre-
quently called the cross products, and are represented as
follows:
or
Wxy-12xy
Hence in general, the product of two binomials whose corre-
sponding terms are similar is equal to the product of the first two
terms, plus the sum of the cross products, plus the product of the
last terms.
EXERCISE 25
Multiply by inspection :
1. (2a-3)(a + 2). 8. (2a 2 6 2 -7)(a 2 & 2 + 5).
2. (3m + 2)(m-l). 9. (2x 2 y 2 + z 2 )(ary + 2z 2 ).
3. (2m-3)(3m + 2). 10. (6
4. (5a-4)(4a-l). 11. (-
5. (4s + y)(3-2y). 12. (5a?
6. (5a6-4)(5a&-3). 13. (10 + 2) (10 4-3).
7. (x 2 i- 5 ft 2 ) (2 x 2 -3 6 s ). 14. (100 + 3)(100 + 4).
65. The square of a polynomial.
(a 4- & + c) 2 = a 2 + tf + c 2 ,-f 2 a& -f 2 ac + 2 &c.
7%e square of a polynomial is equal to the sum of the squares
of each term increased by twice the product of each term with
each that follows it.
The student should note that the square of each term is
always positive, while the product of the terms may have plus
or minus signs.
M UL TIP LIC A TION 43
EXERCISE 26
Find by inspection :
1. (m-f n+p) 2 . 6.
2. (x-y+z)*. 7. (.r 2 _
3. (a + 6-5) 2 . 8. (2a-36 + 5c) 2 .
4. (,i-2&-c) 2 . 9. (3 4y s-f n) 2 .
5. (u-4& + 3c'. 10.
Find the square root of :
11. s? + y 2 + z z + 2xy + 2yz + 2 xz.
12. m 2 -+- n 2 "-f- jp 2 -f 2 mn 2 ?wp 2 np.
13.
66. In simplifying a polynomial the student should remem.
ber that a parenthesis is understood about each term. Hence,
after multiplying the factors of a term, the beginner should
inclose the product in a parenthesis.
Ex. Simplify (x + 6) (a - 4) - (x - 3) (x - 5).
Check. If x = 1,
( + 6)( - 4) - (>-.3)(z- 5) = (7 - - 3) - (- 2 -4) = - 20
= [ X a + 2 a; - 24] - [a? - 8 x + 15]
- X 2 + 2 x - 24 - y? + 8 a; - 1 5
= 10 x - 39. = 10 - 39. = - 29.
EXERCISE 27
Simplify the following expressions, and check the answers :
!. 6( a -2)-6. 3. 4(* + 2)-5(-3).
2. 6~2(a + 7). 4. 4(aj-2)-h3(-7).
5. 2(m n) 3(m + n)H- (m n).
6. 3(6 3 + 6 2 )-2(6 2 + &)~(&4-& 8 ).
7. (m-f n)(m+2)-3m(n + m).
8. (a-2)(a-3)~(a-l)(a-4).
44 ELEMENTS OF ALGEBRA
9.
10.
11. 4(m + 2) + 5(w 3)
12. (a? 5)(oj-2) (a;-
13. (n -f 5) (w - 2) + (n - 7) (n + 4) - 2 (n* - 2)
14. 6(p+2)-7(p-9)-2(i> + l)(p-l).
15. (5 x- 2 y)(3 x -f 2 y) - (4 - y) (a-
16. 3 (a -f 6) 2 - 4 (a + &) (a -f 2 6) + (a -
17.
18.
19.
20.
21.
22. (a 2 -fa-f 1) ( a - 1) - (a 8 + 1) (a - 1).
CHAPTER IV
DIVISION
67. Division is the process of finding one of two factors if
their product and the other factor are given.
The dividend is the product of the two factors, the divisor is
the given factor, and the quotient is the required factor.
Thus to divide 12 by + 3, we must find the number which multiplied
_ 12
by -f 3 gives 12. But this number is 4 ; hence r - =4.
+ 3
68. Since -f a - -f b = -f ab
-fa b = ab
_ a -f- b = ab
and a b = -f- ab,
it follows that = + b
4-a
ab
a
ab
a
69. Hence the law of signs is the same in division as in
multiplication : Like signs produce plus, unlike signs minus.
70. Law of Exponents. It follows from the definition that
a 8 -5- a 5 = a 3 , for a 3 X a 5 = a 8 .
Or in general, if m and n are positive integers, and m is
greater than n, a m -f- a" = a'"-", for a m ~ n a n = a m <
45
46 ELEMENTS OF ALGEBRA
71. TJie exponent of a quotient of two powers with equal bases
equals the exponent of the dividend diminished by the exponent
of the divisor.
DIVISION OF MONOMIALS
72. To divide 10x 7 y 3 z by 2x 6 y 2 , we have to find the
number which multiplied by 2x*y z gives 10 x^ifz. This
number is evidently
Therefore, , = - 5 a*yz.
*
Hence, the quotient of two monomials is a monomial whose
coefficient is the quotient of their coefficients, preceded by the proper
sign, and whose literal part is the quotient of their literal parts
found in accordance with the law of exponents.
73. In dividing a product of several factors by a number,
only one of these factors is divided by that number. Thus
(8 12 - 20)-?-4 equals 2 - 12 . 20, or 8 - 3 . 20 or 8 - 12 . 5.
EXERCISE 28
Perform the divisions indicated :
'
2 . 76-H-15. ' 3" ' 7'
3. -39-*- 3. 7 -4* 9 5 11
4. 2 15 -j-2 12 . ' 4 2 6 8
5. 3 19 -j-3
10. (3 5 -2 4 )^(3 4 .2 2 ). 11. (2 3 .3*.5 7 )-f-(
12 38 - 56 ' 2 V 14 36 a 2
' ' '
3 5 -5.2 5 ' -12 a ' 2abc
13 y-ffl-g 15 -42^ -56aW
'' ' ' UafiV
DIVISION 47
lg -^1^. 20> _Z^L4L. 22.
16 w 7 7i 9
10 132 a 1 V* 14 01 -240m 40 6c fl
iy. - * /5i.
120m- 3J) c
23. (15- 25. a 2 ) -=- 5. 25. (18 . 5 . 2a 2 )-f-9a.
24. (7- 26 a 2 ) -f- 13. 26. (
DIVISION OF POLYNOMIALS BY MONOMIALS
74. To divide ax-}- fr.e-f ex by x we must find an expression
which multiplied by x gives the product ax + bx -J- ex.
But x(a + b -\- e) ax + bx + ex.
TT aa? + bx -f ex . , .
Hence = a 4- b + c.
a?
To divide a polynomial by a monomial, cfc'wde each term of the
dividend by the monomial and add the partial quotients thus
formed.
3 xyz
EXERCISE 29
Perform the operations indicated :
1. (5* _5* + 52) -5. 5 2 . 3. (2 4 -2 5 -2 <? )^-2 2 .
2. (G^-G^-G^-i-G 97 . 4. (8- 3 +8- 5 + 8- 7) -*-8.
5. (11- 2 + 11 -3 + 11 -5)-*- 11.
fl 18 aft- 27 oc Q 5a 5 +4a s -2a 2
o. - y. -
9a -a
-14gV+21gy 15 a*b - 12 aW + 9 a 2
Itf 3 a 2
48 ELEMENTS OF ALGEBRA
, 22 m 4 n - 33 m s n 2 -f 55 mV
4,
- 49 aW + 28 a -W - 14 g 4 6 4 c
- 39 afyV + 26 arVz 3 -
15. (115 afy -f 161 afy 2 - 69 a; 4 ?/ 3 - 23 ofy 4 ) -5- 23 x 2 y.
16. (52 afyV - 39 oryz 4 - 65 zyz 3 - 26 tf#z) -5- 13 xyz.
, 17. (85 tf - 68 x 4 ?/ + 51 afy 2 - 34 xy* -f 1 7 a;/) -f- - 17 as.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL
75. Let it be required to divide 25 a - 12 -f 6 a 3 - 20 a 2 by
2 a 2 -f 3 4 a, or, arranging according to descending powers of
a, divide
6a 3 -20a 2 -f 25a-12 by 2a 2 -
The term containing the highest power of a in the dividend (i.e. a 8 ) is
evidently the product of the terms containing respectively the highest
power of a in the divisor and in the quotient.
Hence the term containing the highest power of a in the quotient is
If the product of 3 a and 2 2 4 a + 3, i.e. 6 a 3 12 a 2 -f 9 a, be sub-
tracted from the dividend, the remainder is 8 a 2 -f 16 a 12.
This remainder obviously must be the product of the divisor and the
rest of the quotient. To obtain the other terms of the quotient we have
therefore to divide the remainder, 8 a 2 -f- 16 a 12, by 2 a 2 4 a + 3.
We consequently repeat the process. By dividing the highest term in
the new dividend 8 a 2 by the highest term in the divisor 2 a 2 , we obtain
4, the next highest term in the quotient.
Multiplying 4 by the divisor 2 a 2 4 a + 3, we obtain the product
8 a 2 -I- 16 a 12, which subtracted from the preceding dividend leaves
no remainder.
Hence 3 a 4 is the required quotient.
DIVISION 49
The work is usually arranged as follows :
a 3 - 20 * 2 + 25 a - 12 I 2 a 2 - 4 a + 3
3 - 20 * 2 + 25 a - 12 I 2 a 2 - 4
3 - 12 a 2 + {) a _ I 8 a - 4
0a-
- 8 a? 4- 16 a- 12
76. The method which was applied in the preceding ex-
ample may be stated as follows :
1. Arrange dividend and divisor according to ascending or
descending powers of a common letter.
2. Divide the first term of the dividend by the first term of the
divisor, and write the result for the first term of the quotient.
3. Multiply this term of the quotient by the whole divisor, and
subtract the result from the dividend.
4. Arrange the remainder in the same order as the given
expression, consider it as a new dividend, and proceed as before.
5. Continue the process until a remainder zero is obtained, or
until the highest poiver of the letter according to which the dividend
was arranged is less than the highest poiver of the same letter in
the divisor.
77. Checks. Numerical substitution constitutes a very con-
venient, but not absolutely reliable check.
An absolute check consists in multiplying quotient and
divisor. The result must equal the dividend if the division
was exact, or the dividend diminished by the remainder if the
division was not exact.
Ex. 1. Divide 8 a 3 -f 8 a - 4 + 6 a 4 - 11 a 2 by 3 a - 2.
Arranging according to descending powers, , , ^ _ , _ ,
6 a 4 + 8 a 8 - 11 a 2 -f 8 a 4 I 3 a - 2 = 7-r-l,
6 a 4 4 a 3 2 a 8 -f 4 a 2 a . _+ 2 = 7
+ 12 a 8 -11 a 2
- 3 a 2 + 8 a
- 3 a' 2 + 2 a
-4
+ 6a - 4
50 ELEMENTS OF ALGEBRA
Ex. 2. Divide a 4 - 4 6 4 - 6 a 3 6 -f- 9 2 6 2 by 26 2 -3a& + a l .
Arranging according to descending powers of a, we have
a<- 6 a 3 6 -f 9 a 2 6 2 -46* I a 2 - 8 ab + 2 6^
a 4 - 3 a 8 fr -f 2 a 2 6 2 | a* - 3 ab - 2 6 2
2 -46*
- 3 a^ + 9a 2 6 2 - 6 ab 8
+ 6 a& a - 4 6 4
- 2 a^a + 6 aft - 4 ft*
Check. The numerical substitution a = 1, & = 1, cannot be used in this
example since it renders the divisor zero. Hence we have either to use a
larger number for a, or multiply.
(a 2 - 8 ab + 2 & 2 ) ( a _ 3 ab - 2 6 2 )
= [(a 2 - 3 aft) + 2 6 2 ] [(a 2 - 3 a&) - 2 6 2 ]
= (a 2 -3 aft) 2 -4 6*
= a 2 - 6 8 6 + 9 a 2 6 2 - 4 5*.
EXERCISE 30 *
Perform the operations indicated and check the answers :
2. (jf_2y-15)-i-<y-6).
3. (15 a 2 - 46 a# -f 16 i/ 2 ) -5- (5 a-
4. (5 m 2 _ 26 mn 4- 5 n 2 ) -*- (m 5 w).
5.
6.
7. (6^-53^ + 40)^(6^-5).
8. (56 a; 2 -f- 19 x -15) --(8 -3).
9.
10.
11.
12.
13. (25 a 2 - 36 ft 2 ) -j- (5 a -f- 6 6)
* See page 263.
DIVISION 51
14. (6a 2 & 2 + 23a& + 20)-*-(2a& + 6).
15. (8xy + lo-22x' 2 y)-+(2x 2 y-3).
16. (3 a 3 - 11 a 2 + 9 a - 2) -f- (3 a - 2).
v /17. (1 -f- 13 m + 47 m 2 + 35 w 3 ) -5- (5 m -f 1) .
18. (a? -8) -*-( 2).
19. (aj s -3aj-2)-^(oj-2).
20. (81 m 4 + 1 - 18 m 2 ) -f- (1 G m -f 9 m 2 ).
SPECIAL CASES IN DIVISION
78. Division of the difference of two squares.
Since (a -f b) (a V) = a 2 b 2 ,
, .
a b a -f b
I.e. the difference of the squares of two numbers is divisible
by the difference or by the sum of the two numbers.
Ex.l.
EXERCISE 31
Write by inspection the quotient of :
2 -- 6
x 1 '
3 c 2 -^. v 7 169 a<6 2 - 81 c 8 f
c + 3* ' ISVft-Qc 8
a 2 -166 2 64 a? 10 - 1
' ' ' '
52 ELEMENTS OF ALGEBRA
Find exact binomial divisors of each of the following
expressions :
9. w 4 -!. 13. 36 a 4 ?/ 4 - 49.
10. a 4 -b. f 14. 121a 100 -9& 2 .
11. aW 1. 15. a 16 -100ry.
, 12. a; 12 -r/ 16 . 16. 1,000,000-1.
CHAPTER V
LINEAR EQUATIONS AND PROBLEMS
79. The first member or left side of an equation is that part
of the equation which precedes the sign of equality. The sec-
ond member or right side is that part which follows the sign of
equality.
Thus, in the equation 2 x + 4 x 9, the first member is 2 x + 4, the
second member is x 0.
80. An identity is an equation which is true for all values
of the letters involved.
Thus, (a + ft) (a b) a 2 6 2 , no matter what values we assign to a
and b. The sign of identity sometimes used is = ; thus we may write
(rt+6)(a-ft) = 2 - b' 2 .
81. An equation of condition is an equation which is true
only for certain values of the letters involved. An equation
of condition is usually called an equation.
.r -f9 = 20 is true only when a; =11; hence it is an equation of
condition.
82. A set of numbers which when substituted for the letters
in an equation produce equal values of the two members, is
said to satisfy an equation.
Thus x 12 satisfies the equation x + 1 13. x 20, y = 7 satisfy
the equation x y = 13.
83. An equation is employed to discover an unknown num-
ber (frequently denoted by x, y y or z) from its relation to
known numbers.
63
54 ELEMENTS OF ALGEBRA
84. If an equation contains only one unknown quantity, an^
value of the unknown quantity which satisfies the equation is
a root of the equation.
9 is a root of the equation 2 y + 2 = 20.
85. To solve an equation is to find its roots.
86. A numerical equation is one in which all the known quan
tities are expressed in arithmetical numbers ; as (7 x) (x -f 4)
= a; 2 - 2.
87. A literal equation is one in which at least one of the
known quantities is expressed by a letter or a combination of
letters ; as x -f a = bx c.
88. A linear equation or an equation of the first degree is one
which when reduced to its simplest form contains only the
first power of the unknown quantity; as 9ie 2 = 6#-f7.
A linear equation is also called a simple equation.
89. The process of solving equations depends upon the fol-
lowing principles, called axioms :
1. If equals be added to equals, the sums are equal.
2. If equals be subtracted from equals, the remainders are
equal.
3. If equals be multiplied by equals, the products are equal.
4. If equals be divided by equals, the quotients are equal.
5. Like powers or like roots of equals are equal.
NOTE. Axiom 4 is not true if the divisor equals zero. E.g. 0x4
= 0x5, but 4 does not equal 5.
90. Transposition of terms. A term may be transposed from
one member to another by changing its sign.
Consider the equation x + a=.b.
Subtracting a from both members, x b a. (Axiom 2)
I.e. the term a has been transposed from the left to thQ
right member by changing its sign.
LINEAR EQUATIONS AND PROBLEMS 55
Similarly, if x a = b.
Adding a to both members, x b + a. (Axiom 1)
The result is the same as if we had transposed a from the
first member to the right member and changed its sign.
91. The sign of every term of an equation may be changed
without destroying the equality.
Consider the equation x-\- a= 6-fc.
Multiplying each member by 1, x a = b c. (Axiom 3)
SOLUTION OF LINEAR EQUATIONS
92. Ex. 1. Solve the equation Qx 5 = 4 a? -f 1.
Adding 5 to each term, 6# = 4x + l + 6.
Subtracting 4 x from each term, Qx 4x 1 + 6.
Uniting similar terms, 2 x = 6.
Dividing both members by 2, x = 3. (Axiom 4)
Check. When x = 3.
The first member, 6a-5 = 18-5 = 13.
The second member, 4-fl = 12-fl = 13
Hence the answer, x = 3, is correct.
93. To solve a simple equation, transpose the unknown terms
to the first member, and the known terms to the second. Unite
similar terms, and divide both members by the coefficient of the
unknown quantity.
Ex.2. Solve the equation (4 y) (5 y) = 2 (11 3 y) + #*.
Simplifying, 20 - 9 y + y 2 = 22 - 6 y -f y\
Transposing, - 9 y + 6 y = 20 -f 22.
Uniting, 3 y - 2
Dividing by - 8, y = f .
Check. If y - - f
The first member, (4-y)(6- y) = C4 + })(5-f i)^ V= JI 4 1 = 26 i
The second member, 2(11 - 3 y) + y 2 = 2(11 + 2) + | = 26 -f f = 26$
56 ELEMENTS OF ALGEBRA
Ex. 3. Solve the equation | (x 4) = \ (x + 3).
Simplifying, \x 2-^x-fl.
Transposing, x x = 2 -f- 1 .
Uniting, x = 3.
Dividing by J, x = 18.
Cfcecfc. If x = 18.
The left member {(x - 4) = \ x 14 = 7.
The right member (x + 3) = x 21 = 7.
NOTE. Instead of dividing by \ botli members of the equation \ x = 3,
it would be simpler to multiply both members by 0.
BXEECISB 32*
Solve the following equations by using the axioms only :
1. 5# = 15+2a;. 5. Xx 7 = 14.
2. 7a? = 5a?+18. 6. 4a + 5 = 29.
3. 3 a; = 60 -7 a?. 7. 17 a? + 16 = 16 a? + 17.
4. 7 a; = 16 + 5 a?. 8. 7 a; 3 = 17 3 a?.
Solve the following equations by transposing, etc., and
check the answers :
9. 4 y 11 = 2 ?/- 7. 12. 9 ?/ - 17 + 4y = 36.
10. 13a? 9a? = 7 3a?. 13. 13 y -99 = 7 y- 69.
11. 24-7y = 68-lly. 14. 3-2 = 26-4.
15. 17 + 5a;-7a: = 39-4a; + 22.
16. 17 -9 x + 41 = 12 -8 a? -50.
17. 14y = 59-(24y + 21).
18.
19.
20. 87-
21. 9(5 x -3) = 63. v23. 7 (6 x + 24) = 6 (10 x + 13).
22. 6(3 a? -3)= 9(3 a; -16). 24. 7 a; + 7(3 aj + 1) =63.
* See page 264.
LINEAR EQUATIONS AND PROBLEMS 57
25. 73-4* = 13*~2(5*-12).
26. 6(6a;-5)-5(7a>-8)=4(12-3a5) + l.
27. 7(7 x + 1) -8(7-5 a?) +24 = 12 (4 a? - 5) + 199.
28. y -
29.
30. (a; 5) (as + 3) = (a; -7) (a; + 4).
31. (a - 1) (a? - 5) = (a; + 7) (.7; - 3) + 14.
.32.
33.
34. (aj-
35.
36. (4 t - 12) (2 t + 5) - (2 t + 6) (4 1 - 1 0) = 0.
37. (5 x - 7) (7 x + 4) - (14 x + 1) (a? + 7) = 285 + 21 a*
38. (z + 2) 2 -(a-5) 2 :=2.
. 39. (x + I) 2 + (x + 2) 2 = (x - 3) 2 + (a? - 4) 2 .
40. 2(* + l) 2 -(2J-3)( + 2) = 12.
41. (6 u - 2) (M - 3) - 5(2 u - 1) (u - 4) + 4 w 2 - 14 = 0.
42. |aj = 5 -Jaj. * 44. | a? + 6 =
43.
SYMBOLICAL EXPRESSIONS
94. Suppose one part of 70 to be a?, and let it be required to
find the other part. If the student finds it difficult to answer
this question, he should first attack a similar problem stated
in arithmetical numbers only, e.g. : One part of 70 is 25 ; find
the other part. Evidently 45, or 70 25, is the other part.
Hence if one part is a?, the other part is 70 x.
WJienever the student is unable to express a statement in alge-
braic symbols, he should formulate a similar question stated in
arithmetical numbers only, and apply the method thus found to the
algebraic problem.
58 ELEMENTS OF ALGEBRA
Ex. 1. What must be added to a to produce a sum b ?
Consider the arithmetical question : What must be added to 7 to pro-
duce the sum of 12 ?
The answer is 5, or 12 7.
Hence 6 a must be added to a to give 5.
Ex. 2. x -f- y yards cost $ 100 ; find the cost of one yard.
$> 100
If 7 yards cost one hundred dollars, one yard will cost - --
100
Hence if x -f y yards cost $ 100, one yard will cost - dollars.
EXERCISE 33
1. By how much does a exceed 10 ?
2. By how much does 9 exceed x ?
3. What number exceeds a by 4 ?
4. What number exceeds m by n ?
5. What is the 5th part of n ?
6. What is the nth part of x ?
7. By how much does 10 exceed the third part of a?
6. By how much does the fourth part of x exceed b ?
9. By how much does the double of b exceed one half of c ?
10. Two numbers differ by 7, and the smaller one is p.
Find the greater one.
11. Divide 100 into two parts, so that one part equals a.
12. Divide a into two parts, so that one part is 10.
13. Divide a into two parts, so that one part is b.
14. The difference between two numbers is d, and the
smaller one is s. Find the greater one.
15. The difference between two numbers is c?, and the
greater one is g. Find the smaller one.
16. What number divided by 3 will give the quotient a? 2 ?
17. What is the dividend if the divisor is 7 and the quotient
is a? 2 ?
LINEAR EQUATIONS AND PROBLEMS 59
18. What must be subtracted from 2 b to give a?
19. The smallest of three consecutive numbers is a. Find
the other two.
20. The greatest of three consecutive numbers is x. Find
the other two.
21. A is # years old, and B is y years old. How many years
is A older than B ?
22. A is y years old. How old was he 5 years ago ? How
old will he be 10 years hence ?
23. If A's age is x years, and B's age is y years, find the
sum of their ages 6 years hence. Find the sum of their ages
5 years ago.
24. A has ra dollars, and B has n dollars. If B gave A 6
dollars, find the amount each will then have.
25. How many cents are in d dollars ? in x dimes ?
26. A has a dollars, b dimes, and c cents. How many cents
has he ?
27. A man had a dollars, and spent 5 cents. How many
cents had he left ?
28. A room is x feet long and y feet wide. How many
square feet are there in the area of the floor ?
29. Find the area of the floor of a room that is 2 feet longer
and 3 feet wider than the one mentioned in Ex. 28.
30. Find the area of the floor of a room that is 3 feet shorter
and 4 feet wider than the one mentioned in Ex. 28.
31. A rectangular field is x feet long and ?/ feet wide. Find
the length of a fence surrounding the field.
32. What is the cost of 10 apples at x cents each ?
33. What is the cost of 1 apple if x apples cost 20 cents ?
34. What is the price of 12 apples if x apples cost 20 cents ?
35. What is the price of 3 apples if x apples cost n cents ?
60 ELEMENTS OF ALGEBRA
36. If a man walks 3 miles per hour, how many miles wil\
he walk in n hours ?
37. If a man walks r miles per hour, how many miles will
he walk in n hours ?
38. If a man walks n miles in 4 hours, how many miles does
he walk each hour ?
39. If a man walks r miles per hour, in how many hours
will he walk n miles ?
40. How many miles does a train move in t hours at the
rate of x miles per hour ?
41. x years ago A was 20 years old. How old is he now ?
42. A cistern is filled by a pipe in x minutes. What frac-
tion of the cistern will be filled by one pipe in one minute ?
43. A cistern can be filled by two pipes. The first pipe
alone fills it in x minutes, and the second pipe alone fills it in
y minutes. What fraction of the cistern will be filled per
second by the two pipes together ?
44. Find 5 % of 100 a. -46. Find a % of 1000.
45. Find 6 % of x. 47. Find x % of 4.
48. Find a; % of m.
49. The numerator of a fraction exceeds the denominator
by 3. If m is the denominator, find the fraction.
-.50. The two digits of a number are x and y. Find the
number.
95. To express in algebraic symbols the sentence: " a exceeds
b by as much as b exceeds 9," we have to consider that in this
statement "exceeds" means minus ( ), and "by as much as"
means equals (=) Hence we have
a exceeds b by as much as c exceeds 9.
a b = c - 9.
LINEAR EQUATIONS AND PROBLEMS 61
Similarly, the difference of the squares of a and b increased
a 2 - b' 2 ' -}-
by 80 equals the excess of a i<5 over 80.
80 = a 3 - 80.
Or, (a 2 -b 2 ) + 80 = a 8 -80.
In many cases it is possible to translate a sentence word by
word in algebraic symbols ; in other cases the sentence has to
be changed to obtain the symbols.
There are usually several different ways of expressing a
symbolical statement in words, thus:
a b = c may be expressed as follows :
The difference between a and b is c.
a exceeds b by c.
a is greater than b by c.
b is smaller than a by c.
The excess of a over b is c, etc.
EXERCISE 34
Express the following sentences as equations :
1. The double of a is 10.
2. The double of x increased by 10 equals c.
3. The sum of a and 10 equals 2 x.
4. One third of x equals c.
5. The difference of x and y increased by 7 equals a.
6. The double of a increased by one third of b equals 100.
7. Four times the difference of a and b exceeds c by as
much as d exceeds 9.
8. The product of the sum and the difference of a and b
diminished by 90 is equal to the sum of the squares of a and
b divided by 7.
9. Twenty subtracted from 2 a gives the same result as 7
subtracted from a.
62 ELEMENTS OF ALGEBRA
10. Nine is as much below a as 17 is above a.
11. #is5%of450. 13. 100 is x% of 700.
12. x is 6 % of m. 14. 50 is x % of a. 15. m is x % of n.
16. If A's age is 2 x, B's age is 3 x 10, and C's age is
4 a; 20, express in algebraic symbols :
(a) A is twice as old as B.
(b) A is 4 years older than B.
(c) Five years ago A was x years old.
(d) In 10 years A will be n years old.
(e) In 3 years A will be as old as B is now.
->.,*(/) Three years ago the sum of A's and B's ages was 50.
(g) In 3 years A will be twice as old as B.
(Ji) In 10 years the sum of A's, B's, and C's ages will be 100.
17. If A, B, and C have respectively 2 a, 3 a; -700, and
x 4- 1200 dollars, express in algebraic symbols :
(a) A has $ 5 more than B.
(6) If A gains $20 and B loses $40, they have equal
amounts.
(c) If each man gains $500, the sum of A's, B's, and C's
money will be $ 12,000.
(d) A and B together have $ 200 less than C.
(e) If B pays to C $100, they have equal amounts.
18. A sum of money consists of x dollars, a second sum. of
5 x 30 dollars, a third sum of 2 x + 1 dollars. Express as
equations :
(a) 5 % of the first sum equals $ 90.
(b) a% of the second sum equals $20.
(c) x c / of the first sum equals 6 % of the third sura.
(d) a % of the first sum exceeds b % of the second sum by
(e) 4 % of the first plus 5 % of the second plus 6 % of the
third sum equals $8000.
00 x % of the first equals one tenth of the third sum.
LINEAR EQUATIONS AND PROBLEMS 63
PROBLEMS LEADING TO SIMPLE EQUATIONS
96. The simplest kind of problems contain only one un-
known number. In order to solve them, denote the unknown
number by x (or another letter) and express the yiven sentence as
an equation. The solution of the equation (jives the value of the
unknown number.
The equation can frequently be written by translating the
sentence word by word into algebraic symbols ; in fact, the
equation is the sentence written in alyebraic shorthand.
Ex. 1. Three times a certain number exceeds 40 by as
much as 40 exceeds the number. Find the number.
Let x = the number.
Write the sentence in algebraic symbols.
Three times a certain no. exceeds 40 by as much as 40 exceeds the no.
3x x -40 = 40- x
Or, 3z-40:r:40-z.
Transposing, 3 x + x = 40 + 40.
Uniting, 4 x = 80.
x = 20, the required number.
Check. 3 x or 60 exceeds 40 by 20 ; 40 exceeds 20 by 20.
Ex. 2. In 15 years A will be three times as old as he was
5 years ago. Find A's present age.
Let x = A's present age.
The verbal statement (1) may be expressed in symbols (2).
(1) In 15 years A will be three times as old as he was 5 years ago.
(2) x + 16 = 3 x (x - p)
Or, x+16 = 3(3-5).
Simplifying, x + 15 = 3 x 15.
Transposing, x 3 x 16 15.
Uniting, -23 =-30.
Dividing, x= 15.
Check. In 15 years A will be 30 ; 6 years ago he was 10 ; but
30 = 3 x 10.
NOTE. The student should note that x stands for the number of
years, and similarly in other examples for number of dollars, number of
yards, etc.
64 ELEMENTS OF ALGEBRA
Ex. 3. 56 is what per cent of 120 ?
Let x = number of per cent, then the problem expressed in symbols
W Uldbe 66 = -*-. 120,
300
or, | x 56.
Dividing, x = 46f.
Hence 5(5 is 40 % of 120.
EXERCISE 35
1. What number added to twice itself gives a sum of 39?
- 2. Find the number whose double increased by 14 equals 44.
3. Find the number whose double exceeds 40 by 10.
4. Find the number whose double exceeds 30 by as much
as 24 exceeds the number.
5. A number added to 42 gives a sum equal to 7 times the
original number. Find the number.
6. 47 diminished by three times a certain number equals
twice the number plus 2. Find the number.
7. Forty years hence A will be three times as old as to-da3 r .
Find his present age.
8. Six years hence a man will be twice as old as he was
12 years ago. How old is he now ?
9. Four times the length of the Suez Canal exceeds 180
miles by twice the length of the canal. How long is the Suez
Canal?
10. 14 is what per cent of 500 ?
11. 50 is 4 % of what number?
12. What number is 7 % of 350?
13. Ten times the width of the Brooklyn Bridge exceeds
800 ft. by as much as 135 ft. exceeds the width of the bridge.
Find the width of the Brooklyn Bridge.
14. A train moving at uniform rate runs in 5 hours 90 miles
more than in 2 hours. How many miles per hour does it run ?
LINEAR EQUATIONS AND PROBLEMS 65
15. A and B have equal amounts of money. If A gains
$200, and B loses $100, then A will have three times as much
as 15. How many dollars has each ?
16. A and B have equal amounts of money. If A gives B
$200, B will have five times as much as A. How many dol-
lars has A now?
17. A has $40, and B has $00. How many dollars must B
give to A to make A's money equal to 4 times B's money ?
18. A man wishes to purchase a farm containing a certain
number of acres. He found one farm which contained 30 acres
too many, and another which lacked 25 acres of the required
number. If the first farm contained twice as many acres as
the second one, how many acres did he wish to buy ?
19. In 1800 the population of Maine equaled that of Vermont.
During the following 90 years, Maine's population increased
by 510,000, Vermont's population increased by 180,000, and
Maine had then twice as many inhabitants as Vermont. Find
the population of Maine in 1800.
97. If a problem contains two unknown quantities, two verbal
statements must be given. Ill the simpler examples these two
statements are given directly, while in the more complex prob-
lems they are only implied. We denote one of the unknown
numbers (usually the smaller one) by x, and use one of the
given verbal statements to express the other unknown number
in terms of x. The other verbal statement, written in algebraic
symbols, is the equation, which gives the value of x.
Ex. 1. One number exceeds another by 8, and their sum is
14. Find the numbers.
The problem consists of two statements :
I. One number exceeds the other one by 8.
II. The sum of the two numbers is 14.
F
66 ELEMENTS OF ALGEBRA
Either statement may be used to express one unknown num-
ber in terms of the other, although in general the simpler one
should be selected.
If we select the first one, and
Let x = the smaller number,
Then x -+- 8 = the greater number.
The second statement written in algebraic symbols produces
the equation , / , o\ -i <
^ #4- (o?-f 8) = 14.
Simplifying, a; + a- -f 8 = 14.
Transposing, x -f x =14 8.
Uniting, 2 x = 6.
Dividing, a? = 3, the smaller number.
x -j- 8 = 11, the greater number.
Another method for solving this problem is to express one unknown
quantity in terms of the other by means of statement II ; viz. the sum of
the two numbers is 14.
Let x the smaller number.
Then, 14 x = the larger number.
Statement I expressed in symbols is (14 x) x = 8, which leads ot
course to the same answer as the first method.
Ex. 2. A has three times as many marbles as B. If A gives
25 marbles to B, B will have twice as many as A.
The two statements are :
I. A has three times as many marbles as B.
II. If A gives B 25 marbles, B will have twice as many as A.
Use the simpler statement, viz. I, to express one unknown quantity in
terms of the other.
Let x = B's number of marbles.
Then, 3 x = A's number of marbles.
To express statement II in algebraic symbols, consider that by the
exchange A will lose, and B will gain.
Hence, x 4- 26 = B's number of marbles after the exchange.
3 x 26 = A's number of marbles after the exchange.
LINEAR EQUATIONS AND PROBLEMS 67
Therefore, x -f 25 = 2(3 x 25). (Statement II)
Simplifying, x + 25 = 6 x 50.
Transposing, x Qx 25 60.
Uniting, - 5 x - - 75.
Dividing, x = 15, B's number of marbles.
3 x = 45, A's number of marbles.
* . ' *
Check. 45 - 25 = 20, 15 + 25 = 40, but 40 = 2 x 20.
98. The numbers which appear in the equation should always
be expressed in the same denomination. Never add the number
of dollars to the number of cents, the number of yards to their
price, etc.
Ex. 3. Eleven coins, consisting of half dollars and dimes,
have a value of $3.10. How many are there of each ?
The two statements are :
I. The number of coins is 11.
II. The value of the half dollars and dimes is $3.10.
Let x = the number of dimes, then, from I,
11 x = the number of half dollars.
Selecting the cent as the denomination (in order to avoid fractions), we
express the statement II in algebraic symbols.
50(11 -)+ 10 x = 310.
Simplifying, 660 50 x + 10 x = 310.
Transposing, 50 x + 10 x - - 550 -f 310.
Uniting, 40 x - - 240.
Dividing, x = 6, the number of dimes.
11 x = 5, the number of half dollars.
Check. 6 dimes = 60 cents, 6 half dollars = 260 cents, their sum is
.$3.10.
EXERCISE 36
v 1. Two numbers differ by 44, and the greater is five times
the smaller. Find the numbers.
v, 2. Two numbers differ by 60, and their sum is 70. Find
the numbers.
w'3. The sum of two numbers is 42, and the greater is
6 times the smaller. Find the numbers.
68 ELEMENTS OF ALGEBRA
4. One number is six times another number, and the
greater increased by five times the smaller equals 22. Find
the number.
5. Find two consecutive numbers whose sum equals 157.
6. Two numbers differ by 39, and twice the greater exceeds
tnree times the smaller by 65. Find the numbers.
7. The number of volcanoes in Mexico exceeds the number
of volcanoes in the United States by 2, and four times the
former equals five times the latter. How many volcanoes are
in the United States, and in Mexico ?
8. A cubic foot of iron weighs three times as much as a
cubic foot of aluminum. If 4 cubic feet of aluminum and
2 cubic feet of iron weigh 1600 Ibs., find the weight of a cubic
foot of each substance.
9. Divide 20 into two parts, one of which increased by
3 shall be equal to the other increased by 9.
10. A's age is four times B's, and in 5 years A's age will be
three times B's. Find their ages.
11. Mount Everest is 9000 feet higher than Mt. McKinley,
and twice the altitude of Mt. McKinley exceeds the altitude of
Mt. Everest by 11,000 feet. What is the altitude of each
mountain ?
12. Two vessels contain together 9 pints. If the smaller
one contained 11 pints more, it would contain three times as
much as the larger one. How many pints does each contain ?
13. A is 14 years older than B, and B's age is as much
below 30 as A's age is above 40. What are their ages ?
14. A line 60 inches long is divided into two parts. Twice
the larger part exceeds five times the smaller part by 15 inches.
How many inches are in each part ?
15. On December 21, the night in Copenhagen lasts 10 hours
longer than the day. How many hours does the day last ?
LINEAR EQUATIONS AND PROBLEMS 69
99. If a problem contains three unknown quantities, three
verbal statements must be given. One of the unknown num-
bers is denoted by x, and the other two are expressed in terms
of x by means of two of the verbal statements. The third
verbal statement produces the equation.
Tf it should be difficult to express the selected verbal state-
ment directly in algebraical symbols, try to obtain it by a
series of successive steps.
Ex. 1. A, B, and C together have $80, and B has three
times as much as A. If A and B each gave $5 to C, then
three times the sum of A's and B's money would exceed C's
money by as much as A had originally.
The three statements are :
I. A, B, and C together have $80.
II. B has three times as much as A.
III. If A and B each gave $5 to C, then three times the sum of A's
and B's money would exceed C's money by as much as A had originally.
Let x the number of dollars A has.
According to II, 3 x the number of dollars B has,
and according to I, 80 4 x = the number of dollars C has.
To express statement III by algebraical symbols, let us consider first
the words ** if A and B each gave $ 5 to C."
x 5 = number of dollars A had after giving $5.
8 x 5 = number of dollars B had after giving $5.
90 4 x = number of dollars C had after receiving $10.
Expressing in symbols :
Three times the sum of A's and B's money exceeds C's money by A's
3 x ( x _5 + 3z-5) - (90-4z) = x.
original amount.
The solution gives x = 8, number of dollars A had.
3 x = 24, number of dollars B had.
80 4 x = 48, number of dollars C had.
Check. If A and B each gave $ 5 to C, they would have 3, 19, and 68,
respectively. 8(8 + 19) or 66 exceeds 58 by 8.
70 ELEMENTS OF ALGEBRA
Ex. 2. A man spent $1185 in buying horses, cows, and
sheep, each horse costing $ 90, each cow $ 35, and each sheep
$ 15. The number of cows exceeded the number of horses by
4, and the number of sheep was twice as large as the number
of horses and cows together. How many animals of each kind
did he buy ?
The three statements are :
I. The total cost equals $1185.
IT. The number of cows exceeds the number of horses by 4.
III. The number of sheep is equal to twice tho number of horses and
cows together.
Let x the number of horses,
then, according to II, x -j- 4 = the number of cows,
and, according to III,
2 (2 x -f 4) or 4 x + 8 = the number of sheep.
Therefore, 90 x = the number of dollars spent for horses,
85 (x + 4) = the number of dollars spent for cows,
and, 15 (4 x + 8) = the number of dollars spent for sheep
Hence statement I may be written,
90 x + 35 (x +-4) -f 15(4z-f 8) = 1185.
Simplifying, 90 x + 35 x 4- 140 + (50 x x 120 = 1 185.
Transposing, 90 x -f 35 x + GO x = 140 1 20 + 1185.
Uniting, 185 a = 925.
Dividing, x = 5, number of horses.
x -f 4 = 9, number of cows.
4 x -f 8 = 28, number of sheep.
Check. 5 horses, 9 cows, and 28 sheep would cost 6 x 90 -f 9 x 35 -f
28 x 15 or 450 + 316 + 420 = 1185; 9 -5 = 4 ; 28 = 2 (9 + 5).
EXERCISE 37
1. Find three numbers such that the second is twice the
first, the third five times the first, and the difference between
the third and the second is 15
2. Find three numbers such that the second is twice the
first, the third exceeds the second by 2, and the sum of the
first and third is 20.
LINEAR EQUATIONS AND PROBLEMS 71
3. Find three numbers such that the second is 4 less than
the first, the third is three times the second, and the sum of
the first and third is 36.
- 4. "Find three numbers such that the sum of the first two
is 4, the third is five times the first, and the third exceeds the
second by 2.
5. Divide 25 into three parts such that the second part is
twice the first, and the third part exceeds the second by 10.
6. Find three consecutive numbers whose sum equals 63.
v - 7. The sum of the three sides of a triangle is 28 inches,
and the second one is one inch longer than the first. If twice
the third side, increased by three times the second side, equals
49 inches, what is the length of each?
8. New York has 3,000,000 more inhabitants than Phila-
delphia, and Berlin has 1,000,000 more than Philadelphia
(Census 1905). If the population of New York is twice that
of Berlin, what is the population of each city ?
9. The three angles of any triangle are together equal to
180. If the second angle of a triangle is 20 larger than the
first, and the third is 20 more than the sum of the second and
first, what are the three angles ?
10. In a room there were three times as many children as
women, and 2 more men than women. If the number of
men, women, and children together was 37, how many children
were present ?
x 11. A is twice as old as B, and A is 5 years younger than
C. Five years ago the sum of B's and C's ages was 25 years.
What are their ages ?
v . 12. Find three consecutive numbers such that the sum of
the first and twice the last equals 22.
13. The gold, the copper, and the pig iron produced in one
year (1906) in the United States represented together a value
72
ELEMENTS OF ALGEBRA
of $ 750,000,000. The copper had twice the value of the gold,
arid the value of the iron was $300,000,000 more than that of
the copper. Find the value of each.
14. California has twice as many electoral votes as Colorado,
and Massachusetts has one more than California and Colorado
together. If the three states together have 31 electoral votes,
how many has each state ?
100. Arrangement of Problems. If the example contains
quantities of 3 or 4 different kinds, such as length, width, and
area, or time, speed, and distance, it is frequently advantageous
to arrange the quantities in a systematic manner.
E.g. A and B start at the same hour from two towns 27 miles
apart, B walks at the rate of 4 miles per hour, but stops 2 hours
on the way, and A walks at the rate of 3 miles per hour with-
out stopping. After how many hours will they meet and how
many miles does A walk ?
TIME
(in hours)
HATE
(miles per hour)
DisTANCK
(miles)
A ....
X
3
3z
B . . . .
x-2
4
4 (x-2)
Explanation. First fill in all the numbers given directly, i.e. 3 and 4.
Let x = number of hours A walks, then x 2 = number of hours B
walks. Since in uniform motion the distance is always the product of
rate and time, we obtain 3 a; and 4 (x 2) for the last column. But the
statement "A and B walk from two towns 27 miles apart until they
meet " means the sum of the distances walked by A and B equals 27 miles.
Hence 3 x + 4 (x 2) = 27.
Simplifying, 3z + 4a:-8 = 27.
Uniting, 7 x = 35.
Dividing, x = 5, number of hours.
8 x = 15, number of miles A walks.
LINEAR EQUATIONS AND PROBLEMS
73
Ex. l. The length of a rectangular field is twiee its width.
If the length were increased by 30 yards, and the width de-
creased by 10 yards, the area would be 100 square yards less.
Find the dimensions of the field.
LKNOTH
(yards)
WIDTH
(yards)
AKEA
(square yards)
First field ....
2*
X
2*2
2 30
y 10
(2x + 30) (r 10)
" The area would be decreased by 100 square yards," gives
(2.x + 00) (a -10) = 2s 2 -100.
Simplify, 2 x 2 + 10 x - 300 = 2 z 2 - 100.
Cancel 2 # 2 and transpose, 10 x = 200.
z = 20.
2 a = 40.
The field is 40 yards long and 20 yards wide.
Check. The original field has an area 40 x 20 =800, the second fid 1
70x10 or 700. But 700 = 800 100.
Ex. 2. A certain sum invested at 5 % brings the same in-
terest as a sum $200 larger at 4%. What is the capital?
PRINCIPAL
(No. of dollars)
HATE %
INTEREST
(No. of dollars)
X
.05
.0535
a; + 200
.04
.04 (x + 200)
Therefore .05 x = .M(x + 200).
Simplify, .053; = .04 x + 8.
Transposing and uniting, .01 x 8.
Multiplying, x = 800; $ 800 = required sum.
Check. $ 800 x .06 = $ 40; $ 1000 x .04 = $ 40.
74 ELEMENTS OF ALGEBRA
EXERCISE 38
1. A rectangular field is 10 yards and another 12 yards
wide. The second is 5 yards longer than the first, and the sum
of their areas is equal to 390 square yards. Find the length
of each.
2. A rectangular field is 2 yards longer than it is wide.
If its length were increased by 3 yards, and its width decreased
by 2 yards, the area would remain the same. Find the dimen-
sions of the field.
3. A certain sum invested at 5 % brings the same interest
as a sum $ 50 larger invested at 4 %. Find the first sum.
4. A sum invested at 5 %, and a second sum, twice as large,
invested at 4 %, together bring $ 78 interest. What are the
two sums ?
5. Six persons bought an automobile, but as two of them
were unable to pay their share, each of the others had to pay
$ 100 more. Find the share of each, and the cost of the auto-
mobile.
6. Ten yards of silk and 30 yards of cloth cost together
$ 42. If the silk cost three times as much per yard as the cloth,
how much did each cost per yard ?
7. A man bought 6 Ibs. of coffee for $ 1.55. For a part he
paid 24 ^ per pound and for the rest he paid 35 ^ per pound.
How many pounds of each kind did he buy ?
8. Twenty men subscribed equal amounts to raise a certain
sum of money, but four men failed to pay their shares, and in
order to raise the required sum each of the remaining men had
to pay one dollar more. How much did each man subscribe ?
9. A sets out walking at the rate of 3 miles per hour, and
two hours later B follows on horseback traveling at the rate of
5 miles per hour. After how many hours will B overtake A,
and how far will each then have traveled ?
LINEAR EQUATIONS AND PROBLEMS 75
v 10. A and B set out walking at the same time in the same
direction, but A has a start of 2 miles. If A walks at the rate
of 2 miles per hour, and B at the rate of 3 miles per hour, how
far must B walk before he overtakes A ?
11. A sets out walking at the rate of 3 miles per hour, and
two hours later B starts from the same point, traveling by coach
in the opposite direction at the rate of 6 miles per hour. After
how many hours- will they be 36 miles apart ?
X 12. The distance from New York to Albany is 142 miles.
If a train starts at Albany and travels toward New York at the
rate of 30 miles per hour without stopping, and another train
starts at the same time from New York traveling at the rate of
41 miles an hour, how many miles from New York will they
meet?
CHAPTER VI
FACTORING
101. An expression is rational with respect to a letter, if,
after simplifying, it contains no indicated root of this letter ;
irrational, if it does contain some indicated root of this letter.
a 2 \- V& is rational with respect to , and irrational with respect
to 6. a
102. An expression is integral with respect to a letter, if
this letter does not occur in any denominator.
-f- db + 6 2 is integral with respect to a, but fractional with respect
6
to b.
103. An expression is integral and rational, if it is integral
and rational with respect to all letters contained in it; as,
a- + 2 ab + 4 c 2 .
104. The factors of an algebraic expression are the quantities
which multiplied together will give the expression.
In the present chapter only integral and rational expressions
are considered factors.
Although Va' J b~ X V <2 Ir a 2 ?> 2 , we shall not, at this
stage of the work, consider vV b' 2 a factor of a 2 6 2 .
105. A factor is said to be prime, if it contains no other
factors (except itself and unity) ; otherwise it is composite.
The prime factors of 10 a*b are 2, 5, , a, a, 6.
76
FACTORING 77
106. Factoring is the process of separating an expression
into its factors. An expression is factored if written in the
form of a product.
(x' 2 4 x + 3) is factored if written in the form (a; 8) (s-1). It
would not be factored if written x(x 4) +3, for this result is a sum,
and not a product.
107. The factors of a monomial can be obtained by inspection
The prime factors of 12 &V 2 are 3, 2, 2, 01, x, x, ?/, y.
108. Since factoring is the inverse of multiplication, it fol-
lows that every method of multiplication will produce a method
of factoring.
E.g. since (a + 6) (a - 6) = a 2 - 6 2 , it follows that a 2 - 6 2 can be
factored, or that a 2 - IP = (a + &)(a - &).
109. Factoring examples may be checked by multiplication
or by numerical substitution.
TYPE I. POLYNOMIALS ALL OF WHOSE TERMS
CONTAIN A COMMON FACTOR
mx + my+ mz~m(x+y + z). ( 55.)
110. Ex. 1. Factor G ofy 2 - 9 x 2 if + 12 xy\
The greatest factor common to all terms is 8 xy' 2 . Divide
6 a% 2 - 9 x 2 y 8 + 12 flcy* by 3 xy\
and the quotient is 2 x 2 3 xy -f 4 1/ 2 .
But, dividend = divisor x quotient.
Hence 6 aty 2 - 9 x 2 ^ + 12 sy* = 3 Z2/ 2 (2 # 2 - 3 sy + 4 y 8 ).
Ex. 2. Factor
14 a* W - 21 a 2 6 4 c 2 + 7 a 2 6 2 c2 7 a 2 6 2 c 2 (2 a 2 - 3 6 a + 1).
78 ELEMENTS OF ALGEBRA
EXERCISE 39
Resolve into prime factors :
1. 6 abx - 12 cdx. 6. 4 tfy -f- 5 x*y 2 6 xy .
2. 3x*-6x*. 7. 17 a? - 51 x 4 + 34 X s .
3. 15 2 &-{-20a 2 & 3 . 8. 8 a*b 2 -f 8 6V - 8 c 2 a 2 .
4. 14a 4 6 4 -7a 2 & 2 . 9. 15 ofyV - 45 afy 3 - 30 aty.
5. Ilro 8 + llm s -llm. 10. a 4 -a : '-J-a 2 .
11. 32 a 4 *?/ - 16 a'V -f 48 ctfa^ 8 .
12.
13.
14. 34 a^c 8 - 51 aW + 68 a6c.
15.
16.
17. q*-q*-q 2 + q. 21. 13- 5 + 13 -8.
18. a(m-f-7i) + & ( m + 7i )- 22 - 2.3.4.5 + 2.3.4.6.
19. 3 a; 2 (a + 6) -3 /(a + 6). 23. 2 3 5-f 2 . 3 5 6.
20.
TYPE IT. QUADRATIC TRINOMIALS OF THE FORM
111. In multiplying two binomials containing a common
term, e.g. (as 3) and (cc-f-5), we had to add 3 and 5 to ob-
tain the coefficient of x, and to multiply 3 and 5 to obtain the
term which does not contain x or (x 3)(x -f 5) = x 2 -f-2 x 15.
In factoring x 2 -f 2 x 15 we have, obviously, to find two
numbers whose product is 15 and whose sum is -f- 2.
Or, in general, in factoring a trinomial of the form x 2 -f-/>#-f q,
we have to find two numbers m and n whose sum is p y and
whose product is g; and if such numbers can be found, the
factored expression is (x -}-m)(x + n).
FACTORING 79
Ex. l. Factor a 2 -4 x - 11.
We may consider 77 as the product of 1 . 77, or 7 11, or 11 7,
or 77 1, but of these only 11 and 7 have a sum equal to 4.
Hence a: 2 - 4 x - 77 = (a;- 11) (a + 7).
Since a number can be represented in an infinite number of
ways as the sum of two numbers, but only in a limited number
of ways as a product of two numbers, it is advisable to consider
the factors of q first. If q is positive, the two numbers have
both the same sign as p. If q is negative, the two numbers
have opposite signs, and the greater one has the same sign as p.
Not every trinomial of this type, however, can be factored.
Ex. 2. Factor a 2 - 11 a + 30.
The two numbers whose product is 30 and whose sum is 11 are 5
and -6.
Therefore a 2 11 a 4- 30 = (a - 5) (a 6).
Check. If a = 1, a 2 - 1 1 a + 30 = 20, and (a - 5) (a - G) = - 4 . - 6 = 20.
Ex. 3. Factor tf + 10 ax - 11 a 2 .
The numbers whose product is 11 a 2 and whose sum is 10 a are 11 a
and a.
Hence fc 2 -f 10 ax 11 a?=(x + 11 a) (a- a).
Ex. 4. Factor x? - 1 afy 8 + 12 /.
The two numbers whose product is equal to 12 yp and whose sum equals
- 7 y are -4 y* and -3 y*. Hence z 6 -? oty+12 if= (x 3 -3 y)(x*-4 y 8 ).
112. In solving any factoring example, the student should first
determine whether all terms contain a common monomial factor.
EXERCISE 40
Besolve into prime factors :
4. tf-
5.
3. m 2 -5m + 6. 6. a 2 -
80 ELEMENTS OF ALGEBEA
7. x*-2x-8. 22. ay -11 ay +24.
8. x 2 + 2x-S. 23. ra 2 + 25ra + 100.
9. y_ 6y 16. 24. 3?/-4 + ?/ 2 .
10. ?/ 2 + 6 y 16. 25. a' 2 -2a&-24& 2 .
11. ?/ 2 -15?/ + 44. 26. n 4 + 60+177> 2 .
12. ?/ 2 -5?/-14. 27. a 6 + 7 a 8 -30.
13. ?/ 2 + 4?/-21. 28. a 8 -7 a 4 -30.
14. a 2 +11 a + 30. 29. a? + 5 a; 2 + 6 a.
15. or - 17 a? + 30. 30. 100 xr - 500 x + 600.
16. ^ 2 -7p-8. 31. 6 a 4 -18 a 3 + 12 a 2 .
17. </ 2 + 5<y 24. 32. x*y 4xy 21y.
18. a 2 ^ 2 + 7ax 18. 33. ra 2 a' 2 4 wia 2 21 a 2 .
19. a 2 -17a& + 7(U 2 . 34. 10 x 2 y 2 - 70 x 2 y - 180 a; 2 .
20. a 2 -9a&-226 2 . 35. 200 x 2 + 400 x + 200.
21. a 4 + 8 a 2 -20. 36. 4 a 2 - 48 aft + 446 2 .
TYPE ITT. QUADRATIC TRINOMIALS OF THE FORM
113. According to 66, .
(4 x + 3) (5 x - 2) = 20 x 2 + 7 x - 6.
20 x 2 is the product of 4 a; and 5 x.
6 is the product of + 3 and 2.
+ 7 a? is the sum of the cross products.
Hence in factoring 6 x 2 13 x + 5, we have to find two bino-
mials whose corresponding terms are similar, such that
The first two terms are factors of 6 x 2 .
The last two terms are factors of 5,
and the sum of the cross products equals 13 x.
By actual trial we find which of the factors of 6 x 2 and 5
give the correct sum of cross products.
FACTORING 81
If we consider that the factors of -f 5 must have like signs,
and that they must be negative, as 13 x is negative, all pos-
sible combinations are contained in the following :
6x-l 6.e-5 3xl 3 a; 5
V V \/ V
A A /\ A
x-5 x-1 2x- 5 2o?-l
- 31 x 11 x - 17 x - 13 a
Evidently the last combination is the correct one, or
G a; 2 - 13 x + 5 = (3 x - 5) (2 x - 1).
114. In actual work it is not always necessary to write down
all possible combinations, and after a little practice the student
should be able to find the proper factors of simple trinomials
at the first trial. The work may be shortened by the follow-
ing considerations :
1. If p and r are positive, the second terms of the factors have
the same sign as q.
2. If p is poxiliw, and r is negative, then the second terms of
the factors have opposite signs.
If a combination should give a sum of cross products, which has the
same absolute value as the term qx, but the opposite sign, exchange the
signs of the second terms of the factors.
3. If py? -\-qx-\-r does not contain any monomial factor, none
of the binomial factors can contain a monomial factor.
Ex. Factor 3 x 2 - 83 x -f- 54.
The factors of the first term consist of one pair only, viz. 3 x and x,
and the signs of the second terms are minus. 64 may be considered the
product of the following combinations of numbers : 1 x 54, 2 x 27, X x 18,
6 x 9, 9 x 6, 18 x 3, 27 x 2, 54 x 1. Since the first term of the first fac-
tor (3 x) contains a 3, we have to reject every combination of factors of
54 whose first factor contains a 3. Hence only 1 x 54 and 2 x 27 need
be considered.
82 ELEMENTS OF ALGEBRA
3x-l 3s-2
X X
x-54 x -27
- 163 x - 83 x
Therefore 3 z 2 - 83 x + 64 = (3 a; - 2) (x - 27).
115. The type pa; 2 -f go; -h r is the most important of the
trinomial types, since all others (II, IV) are special cases of
it. In all examples of this type, the expressions should be
arranged according to the ascending or the descending powers
of some letter, and the monomial factors should be removed.
EXERCISE 41
Kesolve into prime factors :
1. 2x* + 9x-5. 19. SoJ + llay 40*.
2. 4a 2 -9tt + 2. 20. 15 aj* - 77 xy + 10 y 2 .
3. 3x*-Sx + 4. 21. 10a 2 -23afc + 126 2 .
4. 5m 2 -26m -f 5. 22. G a 2 - 13 xy + 6 y 2 .
5. 6n 2 + 5?i-4. 23. 12 x 2 -7 ay- 10 i/ 2 .
6. 3a 2 + 13a; + 4. 24. 4a? + 14oj + 12.
7. Sar' + Sa-G. 25. 2 ar* + 1 1 or 2 + 12 a.
8. 12y 2 -2/-6. 26.
9. 2i/ 2 + 2/-3. x 27.
10. 2 * 2 - 17-9. 28. 100^-200^ + 100^.
11. 10 a 2 - 19 a -f 6. 29. 5 a 6 -9 a 6 - 2 a 4 .
12. 9 y 2 + 32^-16. 30. 90 x*y 2 - 260 xy 2 - 30 y 2 .
13. 2m 2 -t-7w + 3. 31. 90 a 2 - 300 ab -f- 250 fc 2 .
14. 10a? 2 -9a;-7. 32. 8 f - 3 y 2 - 4 y 4 .
15. 12^-17^-1-6. 33. 40a 2 -90aV + 20aV.
16. 6n 2 -f 13w + 2. 34. 144 x 2 - 290 xy -f 144 y*
17. 2.y 2 + 172/-9. 35. 4 x 4 -f- 8 ofy 2 + 3 y 4 .
18. 14 a 2 -fa -4.
FACTORING 83
TYPE IV. THE SQUARE OF A BINOMIAL
Jr 2 2 xy +/.
116. Expressions of this form are special cases of the pre-
ceding type, and may be factored according to the method used
for that type. In most cases, however, it is more convenient
to factor them according to 65.
a 2 - 2 xy + if = (x ?/) 2 .
A trinomial belongs to this type, i.e. it is a perfect square,
when two of its terms are perfect squares, and the remaining
term is equal to twice the product of the square roots of these
terms.
The student should note that a term, in order to be a perfect
square, must have a positive sign.
16 y? 24 xy + 9 y' 2 is a perfect square, for 2VWx 2 x V0y2" = 24 xy.
Evidently 10 & 24 xy + 9 y 2 = (4 x - 3 y) 2 .
To factor a trinomial which is a perfect square, connect the
square roots of the terms which are squares by the sign of the re-
maining term, and indicate the square of the resulting binomial.
EXERCISE 42
Determine whether or not the following expressions are per-
feet squares, and factor whenever possible :
1. m 2 + 2mn + n 2 . 8. 9 + 6 a 2 6 2 -f a 4 .
2. c 2 -2cd-d 2 . 9. a 2 -flOa&4-6 4 .
3. 9 2 -10g-f25. 10. wi 4 -f 6 m*ti -f 9 n*.
4. x* - 10 x -f 16. 11. 2/ 2 -
5. a 2_4 a & + 462. 12. 9
6. a 2 -10a6-25. 13. 25 x> - 20 xy -f 4 y\
7. m 2 -14ww + 49n 2 . 14. 16 a 2 - 26 ab + 9 6 2 .
84 ELEMENTS OF ALGEBRA
15. 16a 2 -24a&4- 9& 2 . 18. 10 a 2 - 20 ab + 10 b 2 .
16. 3<> a; 4 + GO a; 2 + 25. 19. a 4 - 2 ofy + ofy 2 .
17. 225 ofy 2 - 60 a# + 4. 20. m 3 - 6 m* + 9 m.
Make the following expressions perfect squares by supply-
ing the missing terms :
21. u 2 -6& + ( ). 26. 100w 4 +( )+49.
22. x*-Sx + ( ). 27. 64 a 4 -48 a 2 +( ).
23. a 2 + 6a + ( ). 28. 4m 2 20 m -f- ( ).
24. 9a 2 -( ) + 16&*. 29. 4a 2 + 12a + ( ).
- 25. 144 a 2 + ( )-f816 2 . 30. !Gar 9 -( )+25.
TYPE V. THE DIFFERENCE OF TWO SQUARES
JT 2 -/.
117. According to 65,
i.e. ^//c difference of the squares of two numbers is equal to the
product of the sum and the difference of the two numbers.
Ex. 1. 4 aV - 9 z* = (2 ary + 3 z 3 ) (2 ^ 4 - 3 * 3 ).
Ex. 2. 1G a 8 - 64 6 10 = 16(a 8 - 4 6 10 )
= lG(tt 4 +2Z> 5 )(a 4 -26 5 ).
Ex. 3. a 4 - 6 4 = (a 2 -f b 2 ) (a 2 - b 2 )
= (a* + b*)(a + b)(a-b).
NOTE, a 2 -f 6 2 is prime.
EXERCISE 43
Resolve into prime factors :
1. tf-y\ 4. 4a 2 -l. 7. 100a 2 -6 8 .
2. a 2 -9. 5. 1-49 a 2 . 8. a 2 & 2 -121.
3. 36 -6 2 . 6. 81 -* 2 . 9. 9a 2
FACTORING 85
10.
25 ory-
- 81 z 2 .
16.
a 4 - 6 4 .
22.
2 -2/ 8 -
11.
49 a 2 ?/ 2 -
-64.
17.
aV> 4 - 81.
23.
75 ary
-48Z?/ 8 .
12,
2f>a 4 -
1.
18.
1 - .x- 8 .
24.
242 x*y
r -2?/.
13.
100 a^ 2
c 4 .
19.
10 a 2 - 10 6 2 .
25.
144 o,* 2 -
-^y.
14.
169 a 4 -
-100.
20.
13 rrcc - 13 l> 2 x.
26.
100 2 -
3 2 .
15.
225 a 2 - 16 6 4 .
21.
*-*.
27.
9991.
118. One or both terms are squares of polynomials.
Ex. 1. Factor a 2 - (c 4- d) 2 .
a 2 - (c + d) 2 = (a + c + cZ) (a - c - (I) .
Ex. 2. Resolve into prime factors and simplify
EXERCISE 44
Resolve into prime factors :
1. (m-f-n) 2 _p 2 . 8. (m 3n) 2 (
2. (m-7?) 2 -y. 9. (2a-5&) 2 -(5c-9ef) 2 .
3. (m -f n) 2 16p 2 . 10. (a -f- 6) 2 6 2 .
4. # 2 (?/ 4- 2:) 2 . 11. x? (x y)*.
5. 16 cc 2 (y -f a:) 2 . 12. (x -f 3 ?/) 2 9 2/ 2 .
6. 25a 2 -(&-c) 2 . - 13. (2a + 5) 2 -(3a-4) 2 .
T. (m-h2n) 2 36|> 2 . 14. (2s
86 ELEMENTS OF ALGEBRA
TYPE VI. GROUPING TERMS
119. By the introduction of parentheses, polynomials can
frequently be transformed into bi- and trinomials, which may
be factored according to types I- VI.
A. After grouping the terms, ive find that the new terms con-
tain a common factor.
Ex. 1. Factor ax -f- bx -f ay -f by.
ax + bx + ay + by = x(a + &) + y(a + 6)
Ex. 2. Factor or 5 5 x 2 x -f 5.
x 8 - 6z 2 x + 5 = z 2 (.r, - 5) - (x - 5)
EXERCISE 45
Resolve into prime factors :
1. ax + bx + ay+by. 7.
2. ma ?*a + m& nb. 8. raV + nV m 2 ?/ n
3. 2an-3&n + 2ag-3&?. 9. 3 a 2 ic - 4 6 2 x -f 3 a 2 y -
4. 4:cx + 4cy--5dx 5dy. 10. a 3 a 2 6 + ab 2 6 3 .
5. 10ax-5ay-6bx + 3by. 11. c 3 - 7 c 2 + 2c - 14.
6. a ? + x 2 + 2x + 2. 12. a 5 - a 4 a - ab + bx.
B. By grouping, the expression becomes the difference of two
squares.
Ex.1. Factor 9 x*-y*-4:Z 2 -f 4 yz.
= (3 x + y - 2 ) (3 x - y + 2 2).
FACTORING 87
Ex. 2. Factor 4 a 2 - 6 2 + 9 tf - 4 f - 12 aaj -f- 4 6y.
Arranging the terms,
4 a 2 - 6 2 + 9 a;* _ 4 */2 _ 12 ax + 4 6y
= 4 a 2 - 12 ax + 9 a 2 ft 2 + 4 &t/ 4 y 2
= (4 a 2 - 12 z + 9 x2)_ (&2 _ 4 ty + 4 ^2)
EXERCISE 46
Kesolve into prime factors :
1. x* + 2xy + y*-q*. 4. 36
2. l~a 2 -2a5-6 2 . 5. 9 m 2 - 6 ww + n 2 - <
3. a 2 -4a6 + 46 2 -25. 6.
7. a 2 -10a6 + 256 2 -
8. x 4 -ar 2 -2a;-l.
SUMMARY OF FACTORING
I. First find monomial factors common to all terms.
II. Binomials are factored by means of the formula
a 2 -6 2 = (a + 6)(a-6).
III. Trinomials are factored by the method of cross products,
although frequently the particular cases II and IV are more con-
venient.
IV. Polynomials are reduced to the preceding cases by grouping
terms.
EXERCISE 47
MISCELLANEOUS EXAMPLES*
Resolve into prime factors :
!. m 2 16. 4. 6a 4 + 37a 2 + 6. 7. a 8 - 9 a 2 .
2. 8ra 2 + 16. 6 6a 4 -12a 2 + 6. 8. 4 v* - 10 xy + 4 y\
3. w 2 -m 2. 6. 2a3/ + c 2 + 2/ 2 . $- m 2 -Gw + 9-n 2 .
* See page 266.
88 ELEMENTS OF ALGEBRA
10. x*-xif. 26. tt 4 -2a 8 + a*-l.
11. 10 a 2 -32 aft + 6 ft 2 . 27. 42 x 2 - 85 xy + 42 y z .
12. 4 a 4 4ft 4 . 28. 10 w 6 43 w :J 9.
13. 49 a 4 -42 a 3 + 9 a 2 . 29. 25 a 2 ft 2 + 25 aft - 24.
14. 20a 4 ft 4 -90a 2 ft 2 -50. 30. 13 c 2 - 13 c - 156.
16. 2 or 3 7# 2 4 a; + 14. 32. 20 >r + 2 ?<s 42 s 2 .
17. 1 _ w . 33. (^ _|_ ft)2 __ G4.
18. ?v 8 n Q y 2 . 34. 256 (a; 2 ?/) 2 .
19. 5a' 2 -50^ + 45. 35. 2 a 4 -128.
20. a 6 a 3 156. 36. any V 51 xyz + 50.
22. 3 a 2 + 6 aft + 3 ft 2 48. 38. a 5 2 (
23. 80 a 4 - 310 x 2 - 40. 39. a 1
24. a; 5 a. 40. 3
25. a 3 + a 2 + a + l. 41. 3# 4 -3a 2 -36.
CHAPTER VII
HIGHEST COMMON FACTOR AND LOWEST COMMON
MULTIPLE
HIGHEST COMMON FACTOR
120. The highest common factor (IT. C. F.) of two or more
expressions is the algebraic factor of highest degree common
to these expressions ; thus a 6 is the II. C. F. of a 7 and a e b 7 .
Two expressions which have no common factor except unity
are prime to one another.
121. The H. C. F. of two or more monomials whose factors
are prime can be found by inspection.
The H. C. F. of a 4 and a 2 b is a 2 .
The H. C. F. of aW, aW, and cfiW is a 2 /) 2 .
The H. C. F. of (a + ft) 8 and (a + fc) 2 (a - ft) 4 is (a + 6) 2 .
122. If the expressions have numerical coefficients, find by
arithmetic the greatest common factor of the coefficients, and
prefix it as a coefficient to H. C. F. of the algebraic expressions.
Thus the H. C. F. of 6 sfyz, 12 tfifz, and GO aty 8 is 6 aty.
The student should note that the power of each factor in the
H. C. F. is the lowest power in which that factor occurs in any
of the given expressions.
EXERCISE 48
Find the H. C. F. of :
4. 3 3 2 2 , 3 2 - 2 3 , 3 2 4 .
2. 15 aW, 25 W. 5. 2 5 2 3 3 , 2 2 . 5 s 3 4 , 2 s 5 4 - 3 2 .
3. 13 aty 8 , 39 afyV. 6. 7 3 5 s , 7 2 5, 7 . II 2 .
89
90 ELEMENTS OF ALGEBEA
7. 6 rarcV, 12 w*nw 8 , 30 mu\
8. 39 afyV, 52 oryz 4 , 65 zfyV.
9. 38 #y, 95 2/V, 57 a>V.
10. 225 aWd, 75 a&X -15 bed 2 .
11. 4 a 9 , 8 a 10 , 16 a 11 , 24 a 8 6.
12. 4(m -f ?i) 3 , 5(w + w) 2 , 7(m + n}\m ri).
13. 6(m+l) 3 (m+2), 8(?/i-f-l) 2 O + 3), 4(m+l) 2 .
14. 6 a; 3 (a7 - ?/) 5 , 9 aj*(a? - y)\ 12 0^(0; - y) 3 .
123. To find the H. C. F. of polynomials, resolve each poly-
nomial into prime factors, and apply the method of the
preceding article.
Ex. 1. Find the H. C. F. of ^-4^ + 4 if, x 2
and tf -7 xy + 10 f.
x* - 3 xy + 2 y* = (x - 2 ?/) (x - y) .
x 2 - 7 xy + 10 7/ 2 = (x - 2 y) (a; - 5 y).
Hence the H. C. F. = x 2 y.
EXERCISE 49
Find theH. C. F. of:
1. 4 a 3 6 4 , 8 a 6 6 3 - 12 a s 6 6 . 5. 6 a 2 - 6 a&, 5 a6 - 5 ^ 3 .
2. 15 x-y^ 2 , 10 arV - 5 x 3 ?/ 2 . 6. y? - # 2 , 4 afy -f 4 a;?/ 2 .
3. 25 m 2 7i, 15 7/i 3 n 2 10 mV. 7. a 2 - 6 2 , a' 2 + 2 a& + 6 2 .
4. 3ao; 4 -3^ 4 , 6 mx 4 - 6 ?io; 4 . 8. ar* - 5 a; + 6, ^-
9.
10. ^-707 + 12, 0^-80:4-16, ^-
11. a 2 + 5^ + 6,^-9, ^-f a;-6.
12. a 2 - 8 a + 16, a 3 -16 a, a 2 -3a-4.
13. a 2 + 2a-3, a 2 + 7a-f!2, a 3 -9a.
14. y + 3y-64,y + y-42,
15. 2a 2 -f5a-f 2, 4a 2 -f 4a-
16. a 2 -
LOWEST COMMON MULTIPLE 91
LOWEST COMMON MULTIPLE
124. A common multiple of two or more expressions is an
expression which can be divided by each of them without a
remainder.
Common multiples of 3 x 2 and 6 y are 30 x z y, 60 x^y' 2 , 300 z 2 y, etc.
125. The lowest common multiple (L. C. M.) of two or more
expressions is the common multiple of lowest degree; thus,
ory is the L. C. M. of tfy and xy*.
126. If the expressions have a numerical coefficient, find by
arithmetic their least common multiple and prefix it as a coef-
ficient to the L. C. M of the algebraic expressions.
The L. C. M. of 3 aW, 2 a^c 8 , 6 c 6 is C a*b*c*.
The L. C. M. of 12(a + ft) 3 and (a + &)*( - &) 2 is 12(a + &)( - 6)2.
127. Obviously the power of each factor in the L. C. M. is
equal to the highest power in which it occurs in any of the
given expressions.
128. To find the L. C. M. of several expressions which are
not completely factored, resolve each expression into prime
factors and apply the method for monomials.
Ex. 1. Find the L. C. M. of 4 a 2 6 2 and 4 a 4 -4 a 2 6 8 .
4 a 2 &2 _
Hence, L. C. M. =4 a 2 6 2 (a 2 - 6 3 ).
Ex. 2. Find the L.C.M. of a s -& 2 , a 2 + 2a&-f b\ and 6-a.
Hence the L.C.M. = (a -f &)' 2 (a - 6) .
NOTE. The L. C. M. of the last example is also - (a + &) 2 (a ft). In
general, each set of expressions has two lowest common multiples, which
have the same absolute value, but opposite signs.
92 ELEMENTS OF ALGEBRA
EXERCISE 50
Find the L. C. M. of:
1. a, a 2 , a 3 . 4. 3 xif, 8 afy, 24 x.
2. afy, xy\ y*. 5. 6 a?b, 5 a 2 ^ 2 , 15 Z> 3 , 30 a.
3. 2 ic 3 , 4 a 2 , 8 a. 6. T a 3 , 3 a 2 , G a, a.
7. 4 a 5 6cd, 20 afc'cd 2 , 40 abJ, 8 d 5 .
8. 9 a, 3 ab, 3(a + b). 9. 2(m + 7i) 2 , 3(m + n) 4 m 2
10. (a-2)(a-3) 2 , ( a -3)(a-4) 2 , ( a -4)(a-2) 2 .
11. 3 a, 2 a 2 , 2a?b-'2ab 2 .
12. 6 a; -f 6 y, 5 a? 5 y, ic 2 ?/.
13. a -{- b, a~b, a? b 2 . 14. a 2 4, a 2 -f 4 a +4.
15. a?-b\ a 2 + 2ab + b' 2 , 2a-2b.
16. a,- 2 3 a; -f2, x 2 5 a; + 6.
17. a + 2, a -f 3, a -f- 1.
18. 2 a - 1, 4 a 2 - 1, 4 a -f 2.
19. x 2 -f 3 # + 2, x 2 + 5 a + 6, x 2 + 4 a -f 4,
20. 5 or 2 -f- 5 #, 3 a 2 3, 15 #.
21. a 2 -fa6, a& + & 2 , a 2 ~ab 6b 2 .
22. a 4 -!, a^-1, 1.
23. ic 2 7ic+10, a? 8 lOaj-f-lfi, x* 2 ~5a;-f 6.
24. ax -{-ay ~ bx by, 3 a 3 b, 2 x -\-2 y.
(For additional examples see page 268. )
CHAPTER VIII
FRACTIONS
REDUCTION OF FRACTIONS
129. A fraction is an indicated quotient; thus - is identical
with a -f- b. The dividend a is called the numerator and the
divisor b the denominator. The numerator and the denominator
are the terms of the fraction.
130. All operations with fractions in algebra are identical
with the corresponding operations in arithmetic. Thus, the
value of a fraction is not altered by multiplying or dividing
both its numerator and its denominator by the same number;
the product of two fractions is the product of their numerators
divided by the product of their denominators, etc.
In arithmetic, however, only positive integral numerators
and denominators are considered, but we shall assume that the
arithmetic principles are generally true for all algebraic numbers.
131. If both terms of a fraction are multiplied or divided by
the same number) the value of the fraction is not altered.
rni a ma i mx x
Thus - = , and = -
b mb my y
132. A fraction is in its lowest terms when its numerator
and its denominator have no common factors.
Ex. 1. Reduce ~- to its lowest terms.
Remove successively all common divisors of numerator and denomina-
tor, as 8, a?, j/' 2 , and z 8 (or divide the terms by their H. C. F.
TT 6 .ry 2 2 4 2 z
Hence ^ = --
3 x
94 ELEMENTS OF ALGEBRA
133. To reduce a fraction to its lowest terms, resolve numerator
and denominator into their factors, and cancel all factors that are
common to both. Never cancel terms of the numerator or the
denominator; cancel factors only.
Ex. 2. Keduce a * ~ 6 a ' * 8 a to its lowest terms.
6 a 4 24 a 2
q s _. 6 n 2 + 8 a _ ap 2 - 6 a + 8)
tf a* - 24 a* 6 d\a* - 4)
Ex. 3. Keduce -- ~ 2 62 to its lowest terms.
6 2 a 2
_ Q
2 6
EXERCISE 51*
Reduce to lowest terms :
i 9-5 3 o 12a4 K
*' 3T5"** 3 * T^ 3 " 6 '
2 3 2 7 8 36 arV 39 a 2 6 8 c 4
2.3 3 -7 a ' ' 18 x 2 ^'
* See page 268.
FRACTIONS 95
7- ^-.
22 a 2 bc ' m- 2 m 3
1 4- n h 11
8. g J- 21. ~__ 9n _ 22
9. ^+3*.
10. LJZJ^JL. 23.
9x + .
11 " a " ^ Mtr . 04 !l
9 or 2 6 it*?/ + y 2
2fi _
3 ??i 3 7i
12. '-M f . 25.
26. rt< - 9 2
14. ^ ^ "-^ ' ^ .7 , 27.
3 a 2 - 10 a + 3
12 m 2 7 w,n + ?
x 1 4 xy + 4 ?/ 2 15 m 2 8 wn + n 2
3 a 3 _ 6 a 2 ?i T> a/i 2
15 ' ft< _.*.. //(/ ' -*-7 , OQ
m 2 5 ?tt + 6 *
16. " ^" ^. 29.
nx ny
17. 5^-- 30.
4 a; 10 y
18. rt ^' ^ - 31.
19. "-""-;' 32.
96 ELEMENTS OF 'ALGEBRA
134. Reduction of fractions to equal fractions of lowest common
denominator. Since the terms of a fraction may be multiplied
by any quantity without altering the value of the fraction, we
may use the same process as in arithmetic for reducing frac-
tions to the lowest common denominator.
Ex. 1. Reduce -^-, , and ^ to their lowest com-
T , 6rar 3 a? kalr
mon denominator.
The L. C. M. of //-* 2 , 3 a\ and 4 aW is 12 afo 2 x 2 .
To reduce - to a fraction with the denominator 12 a 3 6 2 x 2 , numerator
and denominator must be multiplied by ^lA^L O r 2 a 3 .
' 22
Similarly, multiplying the terms of - by 4 6' 2 .r 2 , and the terms of
***- by 3 A 2 , we have -M^- , -1^- , and
^ ' 22 ' 2>
135. Tb reduce fractions to their lowest common denominator,
take the L. C.M. of the denominators for the common denominator.
Divide the L.C.M. by the denominator of each fraction, and
multiply each quotient by the corresponding numerator.
Ex - Reduce
to their lowest common denominator.
TheL.C.D. =(z + 3)(z- 3)O - 1).
Dividing this by each denominator, we have the quotients (x 1),
(x + 3), and (a- 8).
Multiplying these quotients by the corresponding numerators and
writing the results over the common denominator, we have
-6 6a;~16
(a + 3) (a -8) (-!)' (z + 3)(s-3)O-l)' (a + 3) (x- 3) (-!)'
NOTE. Since a = - , we may extend this method to integral expressions,
FRACTIONS 97
EXERCISE 52
.Reduce the following to their lowest common denominator .
1. 5?, JL. 3. 2,^1. 6. i i, i.
22 a 2 5 a a? 8 1 5 5c 26 5
^* .T n"> . ., ., * ^' S? > 7^ S* **. 7i ",oj o> 77" o*
' ?y2" m^ m 2 m 3 2 ab* o atf or
o 2aj 5a _ 2a-l 3
/ . o a. ~ . - n.
8 a 3 zl a
' ' 8 ' * '
9a 2 ~l' 3a-l
2 a 8 6 5 4a
a + 6 a-
9 ^ , jj
y j 3.T 3y
Ga-1 2a 2
ax ay 9 bxby ' a 5 ' a -f-5 ; a 2 25
IB. ?--, . g !
18. - a + 2
' a 2 -3a-f 2
ADDITION AND SUBTRACTION OF FRACTIONS
136. Since --{-- = 5L^ (Art. 74), fractions having a common
c c c
denominator are added or subtracted by dividing the sum or the
difference of the numerators by the common denominator.
137. If the given fractions have different denominators,
they must be reduced to equal fractions which have the
lowest common denominator before they can be added (01
subtracted).
98 ELEMENTS OF ALGEBRA
Ga-6
Ex - ' Sim ^ 2^JT) + :
The L. C. D. is 4(2 a - 3 ft)(2 a -f 3 ft).
Multiplying the terms of the first fraction by 2(2 a + 3 ft), the terms of
the second by (2 a - 3 ft), and adding, we obtain
2 a + 3 ft 6 a - ft _ 2(2 a + 3 ft) (2 a + 3 ft) -f (2 a - 3 ft) (6 a -ft)
2(2 a - 3 ft) 4(2 a -f 3 ft) 4(2 a - 3 ft)(2 a + 3 ft)
_ 8 a 2 -f 24 aft -f 18 ft 2 + 12 a 2 - 20 aft -f 3 ft 2
~~ 4(2a-3ft)(2a-f 3ft)
20 a 2 -f 4 aft -f 21 ft 2
138. The results of addition and subtraction should be re-
duced to their lowest terms.
a-3b
T? cr -\-t -r- ,
Ex. 2. Simplify _T__ + _
* ^ ^
a 2 ab a(a ~ ft).
a 2 _ 3 ab + 2 ft2 = ( a _ ft)( _ 2 ft).
a 2 -2 aft :=(- 2 ft).
Hence the L. C. D. = a(a - ft) (a 2 6).
a -f 2 6 2a + ft a - 3 ft
a 2 - aft a2 - 3 aft + 2 ft2 "~ a 2 - 2 aft
_(a + 2ft)(a-2ft) +a (2q + ft)~. (a-8ft)(a~-ft)
a(a - ft) (a 2ft)
= a 2 - 4 ft 2 + (2 a 2 4- aft) Ca 2 - 4 aft + 8 ft 2 )
a(a - ft)(a 2ft)
= a 2 ^. 4 6 + 2qg+6~ag-f4a&-8 ft 2
a(a-ft)(a -2ft)
... 2 a 2 + 5 aft - 7 ft 2 _. (2 a + 7 ft)fa - ft)
a(a - ft)(a - 2 ft) ~ a(a - ft)(a - 2 ft)
a(a - 2 ft) '
NOTE. In simplifying a term preceded by the minus sign, e.g.
(a 3 ft) (a ft), the student should remember that parentheses are
understood about terms ( 66) ; hence he should, in the beginning, write
the product in a parenthesis, as (a 2 4 aft -f- 3 ft 2 ).
Simplify :
2a-4
FRACTIONS
EXERCISE 53*
2.
99
9m + 7n 6m 5n
3.
5 3
2x + 3y 3x y x + 2y
15 45
8.
5a-76
9.
4a 106
6a-116 15a-26
13 a
116
e
'
7.
6 2
3 a 4 6 2 a 3 6
4a 36
3 u 2 v
10. ++.
a 6 c
12.
v 5 wv 8 v
12 uv
18 v
30 u
13. -+-
14. ? + i-
15. A+2_3.
j>0 i> q
16. -1* + *
17.
19. _H_ + _*_.
a+6 a 6
20. 2 + a "" 2 , 6 '
21.
1 -f m 1
1 m 1 -f w
m-f 3
1 1
t-3 m-2*
M. 2L + .
18- -^4-f-
23.
24.
2a 1 a-2
a _2 a + 3*
6 a 4- 5 2 a -7
a-f-1 '
25.
+ 2)
* See page 270.
LOO ELEMENTS OF ALGEBRA
26 - 27 ~. -
x*3x + 2 x-2 ' x + 3y x-3y
5x Gx x 2 2x
,9. +. 30.
a 4 3a 9 a 2 a-f-1
31.
32.
34.
35.
1 _ m 2 i + m i _ w
& 6 36
a 2+ a ^_2&2
x-2
Q /Yl "I "I O
Qfi 3 L \_ L I :_
ou * 7 IT-i ~T~ 7 TTo i '
37 i. _
_ 9 <1 - 1 i 1 '> -.9 79 '
a 2 - 6 2 a + b
38. a-f 1-f n. / IIlNT: Let a + a =
/j. ! 1
39. - + 1. 42. a ?^ -
ic 1 (
40. -_ + a; + y. 43.
a? ?/
41. ^-2-^+6- 44.
m 3 a ^> 2
45 ' x 2 -7x+12~x 2 -l7x + 4:~ } '
FRACTIONS 101
139. To reduce a fraction to an integral or mixed expression.
* = 2 + *- (S74)
ceo v '
5a 2 -15a-7 5 a 2 15a 7 .7
Hence = = a 3
o a 5 a o a 5 a 5 a
v , T, , 4 or 3 2a; 2 + 4tf 17 , . ,
Ex. 1. Reduce - to a mixed expression.
2x 3
4 x 3 - ;
2 x 2 + 2 g 4- 6
4 x 2 + 4 x
4 x 2 - 6 x
+ 10x- 17
Therefore x y 3g - 17 (2^ + 2x -f
5-3 (2x-,'3)
2
EXERCISE 54
Keduce each of the following fractions to a mixed or integral
expression :
a a + 1
9a 2 -6a + 2
3a
m 2 *- 5 m -f 6 7
m 4
n 2 + 7n + 14 fi
102 ELEMENTS OF ALGEBRA
MULTIPLICATION OF FRACTIONS
140. Fractions are multiplied by taking the product of tht
numerators for the numerator, and the product of the denomina-
tors for the denominator; or, expressed in symbols:
a c _ac
b'd~bd'
141. Since - = a, we may extend any principle proved for
fractions to integral numbers, e.g. - x c =
b b
To multiply a fraction by an integer, multiply the numerator by
that integer.
142. Common factors in the numerators and the denominators
should be canceled before performing the multiplication. (In
order to cancel common factors, each numerator and denomi-
nator has to be factored.)
Ex. !. Simplify
1 J
The expreeaion =-
8 6 . 2 a
Ex. 2. Simplify
F J
FRACTIONS 103
EXERCISE 55
Find the following products :
2!v! 8 a 2 ^ 36^ 21m* 17 ab
' 5 2 2 4 " ' 48 ' a s b*' ' 34 ab 2 ' 14m 4 '
.. 4
5 3 *3 8 ' '
14 b* 10 a 8 4a-f-86 q~.
' " ' "
76 5c 36C 2 " 7a-216 4
8.
10 (a 6) 12 a 2 2 ab + fc 2 a
5 # 56 c& 4- 6 ot 2 12 d6 4- 20 b* o,
3a 2 6 / GoA 56 2 c ar + 1 o?-f 2
' 4 ac 2 V ai> V " " " " ' " " ~ '
_ 9m 2 -25n 2 1 ,. a 2 4-3a-4 a 2 -5a-h4
JO. 14. -
3m +&n 3m 5 n a 2 3 a 4 <
15.
x 2 + x (x I) 2
17.
18.
_G a 2 -5a-6
x 7 a; 5
aj 1 a? 2 -f 5 a; 50
104 ELEMENTS OF ALGEBRA
DIVISION OF FRACTIONS
143. To divide an expression by a fraction, invert the divisor
and multiply it by the dividend. Integral or mixed divisors
should be expressed in fractional form before dividing.
144. The reciprocal of a number is the quotient obtained by
dividing 1 by that number.
The reciprocal of a is -
a
The reciprocal of J is 1 -f- | |.
The reciprocal of ? + * is 1 + + * = _*_.
x x a + b
Hence the reciprocal of a fraction is obtained by inverting
the fraction, and the principle of division may be expressed as
follows :
145. To divide an expression by a fraction, multiply the
expression by the reciprocal of the fraction.
Ex. 1. Divide X-n?/ 8 . by
* 2
x* -f xy 2 x*y + y 3
s^jf\ = x* - y3 x*y -f y 8
x' 2 y -f 2/ 3 x 3 + xy* x' 2 ~
EXERCISE 56*
Simplify the following expressions :
2 x * '""*'-*- om - 3 i_L#_-i-17 r ar J
13 a& 2 ' 2 a 2 6 2 u 2
5 ft2 + a . a 4-1
' :
a-b
* See page 272.
FRACTIONS 105
-.- - ^-5^+4 t a^-3^-4
' * ' ' ? 4*
.
a?-~ab > ' a a- 2 4 a 4- 4 ' a: + 3
80 . m-
.
m 2 -f- 1 5 w + 56 w 2 4- 5 ??i 50 m 5
12 a 2 4-g-2 a 2 4-g-20 . g-
a 2 2 a 8 a 2 25 a 2 5 a
6 .T 2 ?/ 4 a*?/ 2 15 #4- 10 ?/ _._ # 4- y
45 in^o ?/ 2 ^y ~ "xy '
14
15 a2 + (Jf fr a b . a 4- 6
a 2 + 064- 6 s
COMPLEX FRACTIONS
146. A complex fraction is a fraction whose numerator or
denominator, or both, are fractional.
Ex. l. Simplify <!
c a
a 2 c 4- ab 2 4- &c*
-L 4.
& c a
a _ 6 , c ac
6 c a
4- a6 2 4- &c 2
~
a 2 c 4- afr 2 4- ^c 2
16 ELEMENTS OF ALGEBRA
147. In many examples the easiest mode of simplification ia
multiply both the numerator and the denominator of the
mplex fraction by the L. C. M. of their denominators.
If the numerator and denominator of the preceding examples
B multiplied by a&c, the answer is directly obtained.
x -}- ?/ . x y
_x^_l y
Ex. 2. Simplify X ~V x + y .
xy x + y
Multiplying the terms of the complex fraction by
(x y), the expression becomes
(x
EXERCISE 57
Simplify :
x
2. 4* 3. JL. 4. i.
y y 2 x* c
X
+ - 7i+~
6. . 7. . 9. -.
& ,a c _^ ,y
^c a 32
6. -n 8. . 10.
a a m ""
FRACTIONS 107
m , o 1 i 1 1 i 2 , -i
11. : 15. ~T" ~ y 19
14. L 18. 1 +
!^-5n * 1 4 ' 5
a "~ 2 - - 20 -
12. & a
a4-6
13.
^n* 17. a + 2 i " f "
m n 1 1 (
a + 1
/*-_i_i
4-
s-y 1+ 1 flg-f-l a?l
ti a;-~l ic+1
(For additional examples see page 273.)
CHAPTER IX
FRACTIONAL AND LITERAL EQUATIONS
FRACTIONAL EQUATIONS
148. Clearing of fractions. If an equation contains frac-
tions, these may be removed by multiplying each term by the
L. C. M. of the denominator.
Ex.1. Solve ^-2^ = 12 -* + *-*.
63 2
Multiplying each term by 6 (Axiom 3, 89),
2 x - 2(x - 3) = 72 - 3 (a; 4- 4) - x.
Removing parentheses, 2x 2 a; + 6 = 72 Bx 12 Qx.
Transposing, 2z-2a;-f3# + C:E=-6-f72-12.
Uniting, 9 x = 64.
x = 6.
Check. If x = 6, each member is reduced to 1.
Ex. 2. Solve 5 14
-I x + 1 x + 3
Multiplying by (x -!)(&+ 1) (x + 3),
5(3 + 1) (a + 3) - 14 (a; - l)(z + 3) = - 9(se + !)( - I).
Simplifying, 5 x 2 + 20 x + 15 14 x 2 - 28 x + 42 = - 9 x 2 + 9.
Transposing, 5 x 2 14 z 2 + z 2 + 20 x - 28 a = 15 - 42 + 9.
Uniting, - 8 x = - 48,
85 = 6.
Check. If x 6, each member is reduced to 1.
108
FRACTIONAL AND LITERAL EQUATIONS 109
EXERCISE 58
Solve the following equations :
^ 3 _ a? + 7
4
--. -
32 '2
3 a? 4 2 a? 4- 1 _ 7 - 7 a;
"T"" 4 " o ""~TiT"
3 ' 3 + 4
10. ^-1 = 9. 12. 5 = 12. 14. 1 + 5 = 19
a/ - - 4 a/ &
1 1 *> ' ^0
11. - = 2. 13. = 2. 15. -^ = 5.
a: 7 a; a? + 1
16. = hi- 18. x +^ + ^' = 11.
xx 2, 3
a? a? a; 4 2 16
20 y + 2 y-2 y-3 == ^
334
on
110 ELEMENTS Of ALGEBRA
24. ?_=_. 29.
y+3~2
25.
26 4a4-l- . 31
26. 4* + l-~. 31.
27 . 2^12 = 2 . 32 . 6 2 20
28 . = . 33.
x+3 x-3 o^-
3 - 2 - 13
34. _J_ = _J3 J_.
_ - . ._ 3x
35. 3
36.
3x-2 3x*-2x 3x-2
51 23 x + 26 22
2^4-3 4^-9 2a?-3
1 1 A *
37.
38 - = - ^^ 39 7 x- 11_4 x-
' '
40.
149. If two or more denominators are monomials, and" the
remaining one a polynomial, it is advisable first to remove the
monomial denominators only, and after simplifying the result-
ing equation to clear of all denominators.
FRACTIONAL AND LITERAL EQUATIONS 111
Ex.1. Solve
10 5# 1 5
Multiplying each term by 10, the L. C. M. of the monomial denomina-
tors, 2( - n
16 x + 3 - ~ x ~ &Q =: 16 x - 2.
Transposing and uniting. 5 = 20 g ~ Jff .
5 a: 1
Multiply ing by 6 a; 1, 26 a; - 5 = 20 x - 60.
Transposing and uniting, 5 x 45.
Dividing, x : = 9.
Check. If a; = 9, each member is reduced to ^.
Solve the following equations :
41
5a;-2
42
,,
43.
44.
~ == - .2 7a;-29
9 18 3 507-12'
9
24 a; -f 13 8#-f 2__ 2x 7
15 5 ~~7-16*
10 x -f 6 __ 4a;-r-7 6a? + l
5 6a-fll~~ 3
6x-flO
5 ' 2a?~25 15
17a?~9
14 28 64-14
112 ELEMENTS OF ALGEBRA
LITERAL EQUATIONS
150. Literal equations ( 88) are solved by the same method
as numerical equations.
When the terms containing the unknown quantity cannot be
actually added, they are united by factoring.
Thus, ax -f- bx = (a -f 6) jr.
x -f- m 2 * mnx = (1 4- m 2 mn) x.
Ex.1.
b a a
Clearing of fractions, ax- a z + bx - IP = 3 & 2 ab.
Transposing, ax -f bx = 2 -f b 2 - 3 6 2 ab.
Uniting, (a -f 6)z = a' 2 - & - 2 6 2 .
Dividing, l = !=?_=^6?
a -f fr
Reducing to lowest terms, a; = a 2 6.
151. It frequently occurs that the unknown letter is not
expressed by x, y, or z.
Ex. 2. If L= = a , find a in terms of b and c.
3a-c 3(a-c) ?
Multiplying by 3 (a - c) (3 a - c)
6(rt-fc)(a-c) = (2a + &)(3a-c).
6 a 2 6 a& - ac + 6 6c = 6 a 2 - 2 ac + 3 aft - be.
Transposing all terms containing a to one member,
6 ab 6 ac + 2 ac 3 ab = 6 6c ~ 5c.
Simplifying, 9 a& 4 ac = 7 6c.
Uniting the a, and multiplying by 1, a(9 b -f 4 c) = 7 &c.
Dividing, = -l^
5> 9 b 4- 4 c
FRACTIONAL AND LITERAL EQUATIONS 113
EXERCISE 59
Solve the following equations :
*, _
2. iw + 3a; = 8 4 #.
3. a + 26+3aj=2o + 6 + 2a?. 21. -4-- = c .
a Z>
4. mx = n.
6. 3(*-
8. 4 (a x) = 3 (6 a). 1 a; 1 -f- a;
9. 3(2a + aj) = 2(3a a). 25 ? + l = 2L l .
10. aaj-ffta? = c. a/ '~~ 1 ;i
11. ^ + 7^ = 0*+^ 26.
m x IIL n
12. ax -f- ^o; = 5.
a? a , x b _
13. a^ + &o; = 6 co?. x!7 - ITo T ~
14. (m -f n) a? = 2 a + (m-?i)a?. 2 8. If s = vt, solve for v.
15. (wi n) x =px + q. 29. If s = rt, solve for t.
16. - = H. 17. - = n. 30. If s = V-t 2 , solve for a.
y
18. - + - = 1. 31. If ^^ = -, solve for a.
xx 1 a c
32. If -=-+!, solve for/.
* * f P q
33. Solve the same equation for^).
34. The formula for simple interest ( 30, Ex. 5) is t =^,
i denoting the interest, p the principal, r the number of $>, and
n the number of years. Find the formula for:
() The principal.
(6) The rate.
(c) The time, in terms of other quantities.
i
114 ELEMENTS OF ALGEBRA
35. (a) Find a formula expressing degrees of Fahrenheit
(F) in terms of degrees of centigrade (<7) by solving the equation
(ft) Express in degrees Fahrenheit 40 C., 100 C., - 20 C.
36. If C is the circumference of a circle whose radius is R,
then = 2 TT#. Find R in terms of C and TT.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL
EQUATIONS
152. Ex. 1. When between 3 and 4 o'clock are the hands of
a clock together ?
At 3 o'clock the hour hand is 15 minute spaces ahead of the minute
hand, hence the question would be formulated : After how many minutes
has the minute hand moved 15 spaces more than the hour hand ?
Let x = the required number of minutes after 3 o'clock,
then x = the number of minute spaces the minute hand moves
over,
and = the number of minute spaces the hour hand moves
12 over.
Therefore x ~ = the number of minute spaces the minute hand moves
more than the hour hand.
Or x ~^ = 15 '
Uniting, !i^=15.
Multiplying by 12, 11 x - 180.
Dividing, x = ^ = 16^- minutes after 3 o'clock.
Ex. 2. A can do a piece of work in 3 days and B in 2 days,
In how many days can both do it working together ?
If we denote the required number of days by x and the piece of work
by 1, then A would do each day ^ and B j, while in x days they would do
respectively ~ and ^, and hence the sentence written in algebraic symbols
ff /- 3 2
FRACTIONAL AND LITERAL EQUATIONS 115
A more symmetrical but very similar equation is obtained by writing
in symbols the following sentence : ** The work done by A in one day plus
the work done by B in one day equals the work done by both in one day."
Let x = the required number of days.
Then
Therefore,
Solving,
- = the part of the work both do in one day.
32 x
x = |, or 1J, the required number of days.
Ex. 3. The speed of an express train is $ of the speed of an
accommodation train. If the accommodation train needs 4
hours more than the express train to travel 180 miles, what is
the rate of the express train ?
TlMR
(hours)
RATB
(miles per hour)
DlSTAN<'E
(miles)
Express train
!> + = wo
6 x
Ox
6
180
Accommodation train
180 + * = !
X
x
180
Therefore,
Clearing,
Transposing,
Hence
180 = 152 + 4
xx*
180 = 100 + 4 x.
4x = 80.
f x = 36 = rate of express train.
(1)
Ox
Explanation : If x is the rate of the accommodation train, then j a
5
the rate of the express train. But in uniform motion Time =
Distance
Rate
Hence the rates can be expressed, and the statement, u The accommo-
dation train needs 4 hours more than the express train," gives the equa-
tion /I).
116 ELEMENTS OF ALGEBRA
EXERCISE 60
1. Find a number whose third and fourth parts added
together make 21.
2. Find the number whose fourth part exceeds its fifth
part by 3.
3. Two numbers differ by 6, and one half the greater ex-
ceeds the smaller by 2. Find the numbers.
4. The sum of two numbers is oO, and one is ^ of the other.
What are the numbers ?
5. Find two consecutive numbers such that J- of the greater
increased by ^ of the smaller equals 9.
6. Two numbers differ by 3, and J of the greater is equal
to l s of the smaller. Find the numbers.
7. Twenty years ago A's age was | of his present age.
Find A's age.
8. The sum of the ages of a father and his son is 50, and
10 years hence the son's age will be -| of the father's age.
Find their present ages.
9 A post is a fifth of its length in the ground, one half of
its length in water, and 9 feet above water. What is the length
of the post ?
10 A man left ^ of his property to his wife, to his daugh-
ter, and the remainder, which was $4000, to his son. How
much money did the man leave ?
11. A man lost f of his fortune and $500, and found that
he had \ of his original fortune left. How much money had
he at first?
12 After spending ^ of his money and $10, a man had
left ^ of his money and $15. How much money had he
at first?
FRACTIONAL AND LITERAL EQUATIONS 117
13. The speed of an accommodation train is f of the speed
of an express train. If the accommodation train needs 1 hour
more than the express train to travel 120 miles, what is the
rate of the express train? ( 152, Ex. 3.)
14. An express train starts from a certain station two hours
after an accommodation train, and after traveling 150 miles
overtakes the accommodation train. If the rate of the express
train is -f of the rate of the accommodation train, what is the
rate of the latter ?
15. At what time between 4 and 5 o'clock are the hands of
a clock together? ( 152, Ex. 1.)
16. At what time between 7 and 8 o'clock are the hands of
a clock together ?
17. At what time between 7 and 8 o'clock are the hands of
a clock in a straight line and opposite ?
18. A man has invested J- of his money at 4%, ^ at 5%, and
the remainder at 6%. How much money has he invested if
his animal interest therefrom is $500?
19. A has invested capital at 4J % and P> has invested $ 5000
more at 4%. They both derive the same income from their
investments. How much money has each invested ?
20. An ounce of gold when weighed in water loses -fa of an
ounce, and an ounce of silver -fa of an ounce. How many
ounces of gold and silver are there in a mixed mass weighing
20 ounces in air, and losing 1-*- ounces when weighed in water?
21. A can do a piece of work in 3 days, and B in 4 days.
In how many days can both do it working together ? ( 152,
Ex. 2.)
22. A can do a piece of work in 2 days, and B in 6 days. In
how many days can both do it working together ?
23. A can do a piece of work in 4 clays, and B in 12 days.
In how many days can both do it working together ?
118 ELEMENTS OF ALGEBRA
153. The last three questions and their solutions differ only
in the numerical values of the two given numbers. Hence, by
taking for these numerical values two general algebraic num-
bers, e.g. m and n, it is possible to solve all examples of this
type by one example. Answers to numerical questions of this
kind may then be found by numerical substitution. The prob-
lem to be solved, therefore, is :
A can do a piece of work in m days and B in n days. In how
many days can both do it working together ?
If we let x = the required number of days, and apply the
method of 170, Ex. 2, we obtain the equation -- = -.
m n x
Solving, 3;=
m -f- n
Therefore both working together can do it in mn days.
m -f- n
To find the numerical answer, if A can do this work in 6 days
ft Q
and B in 3 days, make m 6 and n = 3. Then = 2. i.e.
6 I 3
they can both do it in 2 days.
Solve the following problems :
24. In how many days can A and B working together do a
piece of work if each alone can do it in the following number
ofdavs: (a) A in 5, B in 5.
(6) A in 6, B in 30.
(c) A in 4, B in 16.
(d) A in 6, B in 12.
25. Find three consecutive numbers whose sum is 42.
26. Find three consecutive numbers whose sum is 57.
The last two examples are special cases of the following
problem :
27. Find three consecutive numbers whose sum equals m.
Find the numbers if m = 24 ; 30,009 ; 918,414.
FRACTIONAL AND LITERAL EQUATIONS 119
28. Find two consecutive numbers the difference of whose
squares is 11.
29. Find two consecutive numbers -the difference of whose
squares is 21.
30. If each side of a square were increased by 1 foot, the
area would be increased by 19 square feet. Find the side of
the square.
The last three examples are special cases of the following
one:
31. The difference of the squares of two consecutive numbers
is ?n ; find the smaller number. By using the result of this
problem, solve the following ones :
Find two consecutive numbers the difference of whose squares
is (a) 51, (b) 149, (c) 16,721, (d) 1,000,001.
32. Two men start at the same hour from two towns, 88
miles apart, the first one traveling 3 miles per hour, and
the second 5 miles per hour. After how many hours do they
meet, and how many miles does each travel ?
33. Two men start at the same time from two towns, d miles
apart, the first traveling at the rate of m, the second at the
rate of n miles per hour. After how many hours do they
meet, and how many miles does each travel ?
Solve the problem if the distance, the rate of the first, and
the rate of the second are, respectively :
(a) 60 miles, 3 miles per hour, 2 miles per hour.
(b) 35 miles, 2 miles per hour, 5 miles per hour.
(c) 64 miles, 3J miles per hour, 4J- miles per hour.
34. A cistern can be filled by two pipes in m and n minutes
respectively. In how many minutes can it be filled by the
two pipes together ? Find the numerical answer, if m and n
are, respectively, (a) 20 and 5 minutes, (b) 8 and 56 minutes,
(c) 6 and 3 hours.
CHAPTER X
RATIO AND PROPORTION
11ATTO
154. The ratio of two numbers is the quotient obtained by
dividing the first number by the second.
Thus the ratio of a and b is - or a * b. The ratio is also frequently
b
written a : b, the symbol : being a sign of division. (In most European
countries this symbol is employed as the usual sign of division.) The
ratio of 12 : 3 equals 4, 6 : 12 = .5, etc.
155. A ratio is used to compare the magnitude of two
numbers.
Thus, instead of writing " a is 6 times as large as ?>," we may write
a : b = 6.
156. The first term of a ratio is the antecedent, the second
term the consequent.
In the ratio a : ft, a is the antecedent, b is the consequent. The
numerator of any fraction is the antecedent, the denominator the
consequent.
157. The ratio - is the inverse of the ratio -.
a b
158. Since a ratio is a fraction, all principles relating to
fractions may be af)plied to ratios. E.g. a ratio is not changed
if its terms are multiplied or divided by the same number, etc.
Ex. 1. Simplify the ratio 21 : 3|.
A somewhat shorter way would be to multiply each term by 6.
120
RATIO AND PROPORTION 121
Ex. 2. Transform the ratio 5 : 3J so that the first term will
equal 1. 5 33 3
*~5 : 5 ~ '4*
EXERCISE 61
Find the value of the following ratios :
1. 72:18. 3. 62:16. 5. $24: $8.
2. J:l. 4. 4|-:5f 6. 5 f hours : 8^- hours.
Simplify the following ratios :
7. 3:4. 9. 7|:4 T 4 T . 11. 16 x*y : 24 xif.
8. 3:1}. 10. 27 06: 18 a6. 12. 64 x*y : 48 a-y 3 .
Transform the following ratios so that the antecedents equal
unity :
15. 16:64. 16. 7f:6J, 17. : 1. 18. 16a 2 :24a&.
159. A proportion is a statement expressing the equality of
two ratios.
| = |or:6=c:(Z are proportions.
160. The first and fourth terms of a proportion are the
extremes, the second and third terms are the means. The last
term is the fourth proportional to the first three.
In the proportion a : b = c : c?, a and d are the extremes, b and c the
means. The last term d is the fourth proportional to a, b, and c.
161. If the means of a proportion are equal, either mean
is the mean proportional between the first and the last terms,
and the last term the third proportional to the first and second
terms.
In the proportion a : b = b : c, b is the mean proportional between a
and c, and c is the third proportional to a and b.
122 ELEMENTS OF ALGEBRA
162. Quantities of one kind are said to be directly proper
tional to quantities of another kind, if the ratio of any two of
the first kind, is equal to the ratio of the corresponding two
of the other kind.
If 4 ccm. of iron weigh ,30 grams, then G ccm. of iron weigh 45 grams,
or 4 ccm. : 6 ccm. = 30 grams : 45 grams. Hence the weight of a mass of
iron is proportional to its volume.
NOTE. Instead of u directly proportional " we may say, briefly, " pro-
portional.'*
Quantities of one kind are said to be inversely proportional to
quantities of another kind, if the ratio of any two of the first
kind is equal \o the inverse ratio of the corresponding two of
the other kind.
If 6 men can do a piece of work in 4 days, then 8 men can do it in
3 days, or : 8 equals the inverse ratio of 4 : 3, i.e. 3 : 4. Hence the num-
ber of men required to do some work, and the time necessary to do it, are
inversely proportional.
163. In any proportion t/ie product of the means is equal to the
product of the extremes.
Let a : b = c : d,
!-;
Clearing of fractions, ad = be.
164. The mean proportional bettveen two numbers is equal to
the square root of their product.
Let the proportion be a : b = b : c.
Then 6 2 = ac.__(163.)
Hence b = Vac.
165. If the product of two numbers is equal to the product of
two other numbers^ either pair may be made the means, and the
other pair the extremes, of a proportion. (Converse of 163.)
If mn = pq, and we divide both members by nq, we have
?^~ E.
q~~ n
PATIO AND PROPORTION 123
Ex. 1. Find x, if 6 : x = 12 : 7.
12x = 42. (163.)
Hence a? = f f = 3 J.
Ex. 2. Determine whether the following proportion is true
rn t: 8:6 = 7 : 4|.
8 x 4$ = 35, and 5 x 7 = 35 ; hence the proportion is true.
166. If a : 6 = c : d, then
I. 6 : a = d : c. (Frequently called Inversion.)
II. a:c=b:d. (Called Alternation.)
III. a + b:b = c + d:d. (Composition.)
Or a + b:a = c + d:c.
IV. a b : b = c d : d. (Division.)
V. a + b : a b = c-)-d:c d. (Composition and Division.)
Any of these propositions may be proved by a method which
is illustrated by the following example :
To prove
b d
This is true if ad - bd = be - bd.
Or if ad = be.
But ad = be. ( 163.)
Hence ^ =^'
o d
167. These transformations are used to simplify proportions.
I. Change the proportion 4 : 5 = x : 6 so that x becomes the
last term.
By inversion 5 : 4 = 6 : x.
124 ELEMENTS OF ALGEBRA
IT. Alternation shows that a proportion is not altered when
its antecedents or its consequents are multiplied or divided by
the same number.
E.g. to simplify 48:21=32:7x, divide the antecedents by 16, the
consequents by 7, 3:3 = 2:3.
Or 1:1 = 2:x, i.e. x = 2.
III. To simplify the proportion 5:6 = 4 x : x.
Apply composition, 11 : 6 = 4 : x.
IV. To simplify the proportion 8 : 3 = 5 -f x : x.
Apply division, 5 : 3 = 5 : jr.
Divide the antecedents by 5, 1 : 3 = 1 : x.
V. To simplify ? = + *.
m 3n mx
Apply composition and division, = ^-
tin 2x
Or .!=!*.
3n x
Dividing the antecedents by m, = -
JJ n x
NOTE. A parenthesis is understood about each term of a proportion.
EXERCISE 62
Determine whether the following proportions are true :
1. 5^:8 = 2:3. 4. 11 : 5 = 12 : 5ft.
2. 3J.:J = 7:2f 5. 8ajy:17 = i^:l-^.
3. 15:22=101:15.
Simplify the following proportions, and determine whether
they are true or not :
6. 120:42 = 20:7. 8. 18:19 = 24:25.
7. 72:50 = 180:125. 9. 6 : 13 = 5f : llf
10. m 2 n 2 : (m n) 2 = (m + rif : m 2 n 2 .
RATIO AND PROPORTION 125
Determine the value of x :
11. 40:28 = 15:0;. 15. 21 : 4z = 72 : 96.
12. 112:42 = 10:a. 16. 2.8:1.6 = 35:*.
13. 03:a?=135:20. 17. 4 a*:15ab = 2a:x.
14. a?:15 = l^:18. 18. 16 n* : x = 28 w : 70 ra.
Find the fourth proportional to:
19. 1, 3, 5. 21. 3, 3t, f. 23. ra 2 , rap, rag.
20. 2, 4, 6. 22. ra, w,j>.
Find the third proportional to :
24. 9 and 12. 26. 14 and 21. 28. 1 and a.
25. 16 and 28. 27. a 2 and ab. 29. a and 1.
Find the mean proportional to :
30. 4 and 16. 32. 2 a and 18 a. 34. ra + landra 1.
31. |- and 2 /. 33. 8 a 2 and 2 b 2 .
35. Form two proportions commencing with 5 from the
equation 6 x 10 = 5 x 12.
36. If ab = xy, form two proportions commencing with b.
Find the ratio of x : y, if :
37. 6x = 7y. 41. (a + fyx = cy. 45. 7iy = 2:x.
38. 9 x = 2 y. 42. x:5 = y:2. 46. y : b = x : a.
39. 6 x = y. 43. x : m = y : n. 47. y : 1 = x : a 2 .
40. mx = ny. 44. 2 : 3 = y : #.
Transform the following proportions so that only one terra
contains x:
48. 2:3 = 4- x: x. 51. 22: 3 = 2 + x: x.
49. 6:5 = 15-o;:ff. 52. 19 : 18 = 3 4- a? : a?.
50. a : 2 = 5 x : x. 53. 2 : 5 = a; : 3 + x.
126 ELEMENTS OF ALGEBEA
54. State the following propositions as proportions :
(a) Triangles (7 T and T) of equal altitudes are to each, othei
as their basis (b and b').
(6) The circumferences (C and C 1 ) of two circles are to each
other as their radii (R and A").
(c) The volume of a body of gas (V) is inversely propor-
tional to the pressure (P).
(d) The areas (A and A') of two circles are to each other as
the squares of their radii (R and R').
(e) The number of men (m) is inversely proportional to the
number of days (d) required to do a certain piece of work.
55. State whether the quantities mentioned below are
directly or inversely proportional :
(a) The number of yards of a certain kind of silk, and the
total cost.
(b) The time a train needs to travel 10 miles, and the speed
of the train.
(c) The length of a rectangle of constant width, and the area
of the rectangle.
(d) The sum of money producing $60 interest at 5%, and
the time necessary for it.
(e) The distance traveled by a train moving at a uniform
rate, and the time.
56. A line 11 inches long on a certain map corresponds to
22 miles. A line 7^- inches long represents how many miles ?
57. The areas of circles are proportional to the squares of
their radii. If the radii of two circles are to each other as
4 : 7, and the area of the smaller circle is 8 square inches,
what is the area of the larger?
58. The temperature remaining the same, the volume of a
gas is inversely proportional to the pressure. A body of gas
under a pressure of 15 pounds per square inch has a volume of
16 cubic feet. What will be the volume if the pressure is
12 pounds per square inch ?
RATIO AND PROPORTION 127
69. The number of miles one can see from an elevation of
h miles is very nearly the mean proportional between h and the
diameter of the earth (8000 miles). What is the greatest dis-
tance a person can see from an elevation of 5 miles ? From
the Metropolitan Tower (700 feet high) ? From Mount
McKinley (20,000 feet high) ?
168. When a problem requires the finding of two numbers
which are to each other as m : n, it is advisable to represent these
unknown numbers by mx and nx.
Ex. 1. Divide 108 into two parts which are to each other
as 11 : 7.
Let 11 x = the first number,
then 7 x = the second number.
Hence 11 x -f 7 x = 108,
or 18 x = 108.
Therefore x = 6.
Hence 11 x = 66 is the first number,
and 7 x = 42 is the second number.
Ex. 2. A line AB, 4 inches long, 4
is produced to a point C, so that r ' i 1
(AC): (BO) =7: 5. Find^K7and BO. A B
Let AC=1x.
Then BG = 5 x.
Hence AB = 2 x.
Or 2 x = 4.
x=2.
Therefore 7 = 14 = AC.
128 ELEMENTS OF ALGEBRA
EXERCISE 63
1. Divide 44 in the ratio 2 : 9.
2. Divide 45 in the ratio 3 : 7.
3. Divide 39 in the ratio 1 : 5.
4. A line 24 inches long is divided in the ratio 3 : 5. What
are the parts ?
5. Brass is an alloy consisting of two parts of copper and
one part of zinc. How many ounces of copper and zinc are in
10 ounces of brass ?
6. Gunmetal consists of 9 parts of copper and one part of
tin. How many ounces of each are there in 22 ounces of gun-
metal ?
7. Air is a mixture composed mainly of oxygen and nitro-
gen, whose volumes are to each other as 21 : 79. How many
cubic feet of oxygen are there in a room whose volume is 4500
cubic feet?
8. The total area of land is to the total area of water as
7 : 18. If the total surface of the earth is 197,000,000 square
miles, find the number of square miles of land and of water.
9. Water consists of one part of hydrogen and 8 parts of
oxygen. How many grams of hydrogen are contained in 100
grams of water?
10. Divide 10 in the ratio a : b.
11. Divide 20 in the ratio 1 : m.
12. Divide a in the ratio 3 : 7.
13. Divide m in the ratio x: y %
14. The three sides of a triangle are 11, 12, and 15 inches,
and the longest is divided in the ratio of the other two. How
long are the parts ?
15. The three sides of a triangle are respectively a, 6, and c
inches. If c is divided in the ratio of the other two, what are
its parts ?
(For additional examples see page 279.)
CHAPTER XI
SIMULTANEOUS LINEAR EQUATIONS
169. An equation of the first degree containing two or more
unknown numbers can be satisfied by any number of values of
the unknown quantities.
If 2oj-3y = 6, (1)
,-L 2 a? 5 /0 \
then y = - (2)
I.e. if x = 0, y = - -.
If =,y=--|.
If x = 1, y = 1, etc.
Hence, the equation is satisfied by an infinite number of sets
of values. Such an equation is called indeterminate.
However, if there is given another equation, expressing a
different relation between x and y, such as
* + = 10, (3)
these unknown numbers can be found.
From (3) it follows y 10 x y and since the equations have
to be satisfied by the same values of x and y, the two values of
y must be equal.
Hence 2s -5 = 10 _ ^ ( 4)
o
The root of (4) is x = 7, which substituted in (2) gives y = 3.
Therefore, if both equations are to be satisfied by the same
values of x and y, there is only one solution.
K 129
130 ELEMENTS OF ALGEBRA
170. A system of simultaneous equations is a group of equa
tions that can be satisfied by the same values of the unknown
numbers.
x -H 2 y 6 and 7 x 3 y = I are simultaneous equations, for they are
satisfied by the values x = I, y 2. But 2 x y 6 and 4 x 2 y = 6 are
not simultaneous, for they cannot be satisfied by any value of x and y.
The first set of equations is also called consistent, the last set inconsistent.
171. Independent equations are equations representing differ-
ent relations between the unknown quantities ; such equations
cannot be reduced to the same form.
6 x -f 5 y ~ 50, and 3 x + 3 y =. 30 can be reduced to the same form ;
viz. x -f y 10. Hence they are not independent, for they express the
same relation. Any set of values satisfying 5 x + 6 y = 60 will also satisfy
the equation 3 x -f- 3 y = 80.
172. A system of two simultaneous equations containing two
unknown quantities is solved by combining them so as to obtain
one equation containing only one unknown quantity.
173. The process of combining several equations so as to
make one unknown quantity disappear is called elimination.
174. The two methods of elimination most frequently used
I. By Addition or Subtraction.
II. By Substitution.
ELIMINATION BY ADDITION OR SUBTRACTION
175. E,X. Solve
-y=-
Multiply (1) by 2, 6 x -f 4 y - 26. (3)
Multiply (2) by 3, 6 x - 21 y = - 24, (4)
Subtract (4) from (3), 26 y = 60.
Therefore, y = 2.
SIMULTANEOUS LINEAR EQUATIONS
131
Substitute this value of y in either of the given equations, preferably
the simpler one (1), 3 x + 4 = 13
Therefore x = 3.
y = 2.
In general, eliminate the letter whose coefficients have the lowest
common multiple.
Check. 3. 8 + 2.2 = 9 + 4 = 13,
2. 3-7- 2 = 6- 14 =-8.
Multiply (1) by 5,
Multiply (2) by 3,
Add (3) and (4),
Therefore
Substitute (6) in (1),
Transposing,
Therefore
Check.
25 x - 15 y = 235.
39 x + 15 y = 406.
64 x = 040.
x = 10.
60 - 3 y = 47.
3 y = 3.
y = 1.
x = 10.
5 . 10 - 3 1 = 47,
13 10 + 5 1 = 135.
(3)
(4)
(6)
176. Hence to eliminate by addition or subtraction :
Multiply y if necessary y the equations by such numbers as will
make the coefficients of one unknown quantity equal.
If the signs of these coefficients are like, subtract the equations;
if unlike, add the equations.
EXERCISE 64
Solve the following systems of equations and check the
answers:
'
ELEMENTS OF ALGEBRA
5. -I ^ 13-
i 3 a; v = ll.
I 7 x + 3 y = 50.
6-1
l7a; + 2/ = 24. 17.
7 ' 1fi fl ,4.1ft = 6. is fl<>* + 22/ = 40,
1 r A O t K
8.
I oj 5 y = 17.
19< I a;-f2/ = 50.
9- 1 r '
20. I , _.
[2o; + 3?/ = 41,
{ 3 x -f 2 y = 39.
f 3 # ?/ = 0,
11. I x ~ y~~>
22.
12. ] ^ , v '
23.
13.
f 3 X y = 1U, 94
14. J * ' ^ 4 '
15 ' ^ - -60. 25 * i 3.9 *- 3.5 y = -2.3.
SIMULTANEOUS LINEAR EQUATIONS 133
ELIMINATION BY SUBSTITUTION
177. Solve (2-7, 8, (1)
I3ar + 2y = 13. (2)
Transposing 7 y in (1) and dividing by 2, x ^""
Substituting this value in (2) , 3 ( 7 ?/ t " 8 ) + 2 y = 13.
Clearing of fractions, 21 y 24 + 4 y = 26.
25 y = 60.
Therefore y = 2.
This value substituted in either (1) or (2) gives x 3.
178. Hence to eliminate by substitution :
Find in one equation the value of an unknown quantity in
terms of the other. Substitute this value for one unknown quan-
tity in the other equation, and solve the resulting equation.
EXERCISE 65
Solve by substitution :
f5aj = 2y + 10,
l3a; = 4#-8.
134 ELEMENTS OF ALGEBRA
179. Whenever one unknown quantity can be removed with-
out clearing of fractions, it is advantageous to do so ; in most
cases, however, the equation must be cleared of fractions and
simplified before elimination is possible.
(1)
(2)
Ex. Solve 3
2 7
Multiplying (1) by 12 and (2) by 14,
43 + 8-f-3y + 9 = 36. (3)
7z + 21-2y-4 = 14. (4)
From (3), 4* + 3y = 19. (6)
From (4), 7x_2y=-3. (6)
Multiplying (6) by 2 and (6) by 3,
Sx + 6y = 3S. (7)
21z-6y=-9. (8)
Adding (7) and (8) , 29 x = 29.
x = l.
Substituting in (6) , 7 2 y = - 3.
y = 6.
1 + 8 _
2 7
EXERCISE 66
Solve by any method, and check the answers:
(4t(x-\-) + 5(y + 5) = 64. ""^IT
3. \
f8(z-8)-9(y-9) = 26,
' \6(a;-6)-7(y-7)==18.
SIMULTANEOUS LINEAR EQUATIONS 135
3x
4.
"25 6 ' tsjj
' 4(5 x - 1) + 5(6 y - 1) = 121,
r4(5-
' l2(3-
8.
9. .
10.
11.
12.
13.
a; + y , a; ff _13
2 "*" 4 ~ 2'
4
2a?-5
4^ 3 ~
3a?-2^4
4~2v 3 1
14. J
15 8
15.
16.
; 10
17.
18.
y-M
a;-f-2
2
= 2,
= 3.
136
19.
ELEMENTS OF ALGEBRA
-4_1 <X + l_3
20.
2'
2 a; -f y - Q ^ 4
2/4-1 4'
21.
23
22.
((*
* ((
, {;
?~y , 3x-\- 1
"
24.
180. In many equations it is advantageous at first not to
consider x and y as unknown quantities, but some expressions
involving x, and y, e.#. - and -
x y
SIMULTANEOUS LINEAR EQUATIONS
Ex. 1. Solve .
2x(2),
x y
x y
Clearing of fractions,
Substituting x = 3 in (1),
Therefore
137
(1)
(2)
(3)
a;
33 = 11 x.
y=4.
Examples of this type, however, can also be solved by the regular
method.
Clearing (1) and (2) of fractions,
+ 8 x - 3 xy.
15 y - 4 x = 4 xy.
2x(5),
(4) + (G),
Dividing by 11 y,
(4)
(6)
(6)
(7)
3 = #, etc.
Solve :
1.
2.
x y
2'
1.
2*
EXERCISE 67
3.
4.
x y
*
138
ELEMENTS OF ALGEBRA
5.
4 6 K
--- = 5,
x y
U y
6.
7.
253
10 "
12 25
8. 4
9.
M-Oi
a; y
331
13.
21 9
x y
o
--- = 6,
x y
LITERAL SIMULTANEOUS EQUATIONS
181. Ex. 1. Solve
(1) x n,
(2) x 6,
(8) (4),
anx + bny = en.
6w3 + bny = 6p.
anx bmx = en ftp.
(1)
(2)
(3)
(4)
SIMULTANEOUS LINEAR EQUATIONS
139
Uniting,
Dividing,
(1) x w,
(2) x a,
(7) - W,
Uniting,
(an bm)x = en bp.
an bm
amx + bmy cm.
amx -f any = ap.
any bmy ap cm.
(an bm}y ap- cm.
y = W - cm .
an bm
(6)
(7)
EXERCISE 68
-f- 6y = c,
ax + by = 2 a&,
[ nx -f my == m.
-f- 6^ =
.y = 9a + 46.
5.
6.
f x -f my = 1,
I sc ny = 1.
fax -f fy/ = l,
11.
x a
12. Find a and s in terms of n, d, and I if
13. From the same simultaneous equations find d in terms
of a, w, and L
14. From the same equations find s in terms of a, d, and I.
140 ELEMENTS OF ALGEBRA
SIMULTANEOUS EQUATIONS INVOLVING MORE THAS
TWO UNKNOWN QUANTITIES
182. To solve equations containing three unknown quantities
three simultaneous independent equations must be given.
By eliminating one unknown quant iff/ from any pair of equa-
tions, and the same unknown quantity froni another pair, the
problem is reduced to the solution of two simultaneous equations
containing two unknown quantities.
Similarly, four equations containing four unknown quanti-
ties are reduced to three equations containing three unknown
quantities, etc.
Ex. 1. Solve the following system of equations:
= 8, (1)
-4, (2)
l. (3)
Eliminate y.
Multiplying (1) by 4, 8B-12y + 16z = 32
Multiplying (2) by 3, Oa + 12?/- 15z=-12
Adding, 17 x + z = 20 (4)
Multiplying (2) by 3, x + 12 y - lf> z - - 12
Multiplying (3) by 2, 8 x - 12 y + 6 z = 2
Adding, 11 x - 9 z = - 10 (6)
Eliminating x from (4) and (5).
(4) -(5), 100 = 30.
Therefore z = 3. (6)
Substitute this value in (4), 17 x + 3 - 20.
Therefore x = 1. (7)
Substituting the values of x and z in (1),
2 3 y -f 12 =s 8.
3y = 6.
Hence y =* 2. (8)
Check. 2.1-3.2 + 4.3 = 8; 3.1+4.2-5.3=-4;
4.1-6.2 + 3.8 = 1.
SIMULTANEOUS LINEAR EQUATIONS 141
EXERCISE 69
1.
10 x + y -f z = 15, 8.
4.
5.
~6?/ == 6,
15 2 = 45.
7.
9.
10.
11.
12.
13.
14. 2
4- 2/ -f 2
x y -M
k a; -f ?/ 2?
= 4.
2 a? + 2 y -f 2 = 35,
4 = 42,
2z = 40.
a? + 70-9 = 26,
x -f- 2 i/ -f- z = 14,
142
ELEMENTS OF ALGEBRA
15.
16.
17.
18.
19.
20.
21.
22.
60;
= 5,
? = llz,
= 8*.
23.
?/ 3 x = 0,
_.
(3 1510
x 4-
074-2!
_ 2
'
3 J
.2 a; 4- .3 y + .42 = 2,
^ = 2.6
=s 2.
27.
84
; 32.
SIMULTANEOUS LINEAR EQUATIONS
143
29.
x y z
M=i,
y *
30.
31.
x : z = 1 : 2.
# 4- 2/ = 2 m,
2/ + z = 2p,
. z + x = 2 n.
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS
183. Problems involving several unknown quantities must
contain, either directly or implied, as many verbal statements
as there are unknown quantities. Simple examples of this
kind can usually be solved by equations involving only one
unknown quantity. ( 99.)
In complex examples, however, it is advisable to represent
every unknown quantity by a different letter, and to express
every verbal statement as an equation.
Ex. 1. The sum of three digits of a number is 8. The
digit in the tens' place is | of the sum of the other two digits,
and if 396 be added to the number, the first and the last digits
will be interchanged. Find the number.
Obviously it is difficult to express two of the required digits in terms
of the other ; hence we employ 3 letters for the three unknown quantities.
Let x the digit in the hundreds' place,
y = the digit in the tens 1 place,
and z the digit in the units' place.
Then 100 + 10 y + z - the number.
The three statements of the problem can now be readily expressed in
symbols: x + y + z - 8. . (1)
100s + lOy + z + 396 = 100* + 10y + x.
The solution of these equations gives x = l,y = 2, 2 = 6.
Hence the required number is 125.
Check. 1 + 2 + 6 = 8; 2 = 1(1+6); 125 + 396 = 521.
(3)
144
ELEMENTS OF ALGE13KA
Ex. 2. If both numerator and denominator of a fraction be
increased by one, the fraction is reduced to | ; and if both numer-
ator and denominator of the reciprocal of the fraction be dimin-
ished by one, the fraction is reduced to 2. Find the fraction.
Let x = the nurn orator,
and y = the denominator ;
By expressing the two statements in symbols,
x + I 2
then - = the fraction.
y
we obtain,
and
x- 1
These equations give x 3 and y = 5. Hence the fraction is f .
3+1 4_2. 5_ I _4_
(1)
(2)
Check.
5+1
Ex. 3. A, B, and C travel from the same place in the same
direction. B starts 2 hours after A and travels one mile per
hour faster than A. C, who travels 2 miles an hour faster
than B, starts 2 hours after B and overtakes A at the same
instant as B. How many miles has A then traveled?
TlMK
(Hours)
KATE
(Miles per hour)
DlSTANCB
(Miles)
A
X
y
xy
B
X-2
2/4-1
xy + x 2 y
2
C ....
x - 4
V 4- 3
xy -{- ,'} x 4 y
1?,
Since the three men traveled the same distance,
xy = xy + x 2 y 2.
xy = xy -f 3 x 4 y 12.
Or x 2 y = 2.
3x-4y = 12.
(4)-2x(3), a: = 8.
From (3) y = 3.
Hence xy = 24 miles, the distance traveled by A.
Check. 8 x 3 = 24, 6 x 4 = 24, 4 x = 24.
(1)
(2)
(3)
C4)
SIMULTANEOUS LINEAR EQUATIONS 145
EXERCISE 70
1. Four times a certain number increased by three times
another number equals 33, and the second increased by 2
equals three times the first. Find the numbers,
2. Five times a certain number exceeds three times another
number by 11, and the second one increased by 5 equals twice
the first number. Find the numbers.
3. Half the sum of two numbers equals 4, and the fourth
part of their difference equals 3. Find the numbers.
4. If 4 be added to the numerator of a fraction, its value
is J. Tf 3 be added to the denominator, the fraction is reduced
to L Find the fraction.
<>
5. If the numerator and the denominator of a fraction be
increased by 3, the fraction equals .}. If 1 be subtracted from
both terms, the value of the fraction is fa. Find the fraction.
6. If the numerator of a fraction be trebled, and its denomi-
nator diminished by one, it is reduced to J. If the denomi-
nator be doubled, and the numerator increased by 4, the
fraction is reduced to \-. Find the fraction.
7. A fraction is reduced to J, if its numerator and its
denominator are increased by 1, and twice the numerator
increased by the denominator equals 15. What is the frac-
tion ?
8. The sum of the digits of a number of two figures is 6,
and if 18 is added to the number the digits will be interchanged.
What is the number ? (See Ex. 1, 183.)
9. If 27 is added to a number of two digits, the digits will
be interchanged, and four times the first digit exceeds the
second digit by 3. Find the number.
10. The sum of the three digits of a number is 9, and the
sum of the first two digits exceeds the third digit by 3. If
9 be added to the number, the last two digits are interchanged.
Find the number.
146 ELEMENTS OF ALGEBRA
11. Twice A's age exceeds the sum of B's and C's ages by 30,
and B's age is \ the sum of A's and C's ages. Ten years ago
the sum of their ages was 90. Find their present ages.
12. Ten years ago A was as old as B will be 5 years hence ;
and 5 years ago B was as old as is now. If the sum of
their ages is 55, how old is each now ?
13. A man invested $ 5000, a part at 6 % and the remainder
at 5%, bringing a total yearly interest of $260. What was
the amount of each investment ?
14. A man invested $750, partly at 5% and partly at 4%,
and the 5% investment brings $15 more interest than the
4 % investment. What was the amount of each investment ?
15. A sum of $10,000 is partly invested at 6%, partly at
5 %, and partly at 4 %, bringing a total yearly interest of $530.
The 6 % investment brings $ 70 more interest than the 5 % and
4% investments together. How much money is invested at
6 %, 5 %, and 4 %, respectively ?
16. A sum of money at simple interest amounted in 6 years
to $8000, and in 8 years to $8500. What was the sum of
money and the rate of interest ?
17. A sum of money at simple interest amounted in 2 years
to $090, and in 5 years to $1125. What was the sum and
the rate of interest?
18. The sums of $1500 and $2000 are invested at different
rates and their annual interest is $ 190. If the rates of inter-
est were exchanged, the annual interest would be $ 195. Find
the rates of interest.
19. Three cubic centimeters of gold and two cubic centi-
meters of silver weigh together 78 grains. Two cubic centi-
meters of gold and three cubic centimeters of silver weigh
together 69 J- grams. Find the weight of one cubic centimeter
of gold and one cubic centimeter of silver.
SIMULTANEOUS LINEAR EQUATIONS
147
20. A farmer sold a number of horses, cows, and sheep, for
$ 740, receiving $ 100 for each horse, $ 50 for each cow, and
$15 for each sheep. The number of sheep was twice the
number of horses and cows together. How many did he sell
of each if the total number of animals was 24?
21. The sum of the 3 angles of a triangle is 180. If one
angle exceeds the sum of the other two by 20, and their
difference by GO , what are the angles of the triangle ?
22. On the three sides of a triangle ABC, respectively, three
points, /), E, and F, are taken so
that AD = AF, ED = BE, and CE
= OF. If AB = G inches, BC = 7
inches, and AC = 5 inches, what is
the length of AD, BE, and CF?
NOTE. Tf a circle is inscribed in the
triangle An C touch ing the sides in D, 7<7,
and F '(see diagram), then AD = AF,
BD = HE, and GE = CF. A r ^
23. A circle is inscribed in triangle ABC touching the three
sides in D, E, and F. Find the parts of the sides if AB = 9,
BC=7, andCL4 = 8.
24. In the annexed diagram angle a = angle b, angle c =
angle d, and angle e angle/. If angle ABC = GO , angle
B BAG 1 = 50, and angle BCA = 70,
find angles a, c, and e.
NOTE. is the center of the circum-
scribed circle.
25. It takes A two hours longer
than B to travel 24 miles, but if
A would double his pace, he would
walk it in two hours less than
B. Find their rates of walking.
CHAPTER XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND
EQUATIONS
184. Location of a point.
It' two fixed straight lines
XX' and YY' meet in
at right angles, and PJ/_L
XX', and PN _L YY', then
the position of point P is
determined if the lengths
of P3f and PN are given.
185. Coordinates. The
lines PM and P^V are
called the coordinates of
point P. PN, or its equal
OM, is the abscissa; and
PM, or its equal OA r , is
the ordinate of point P. The abscissa is usually denoted by
jr, the ordinate by ?/.
The line XX' is called the jr-axis, YY' they-axis, and point
the origin. Abscissas measured to the riyht of the origin,
and ordinates abore the x-axis are considered positive ; hence
coordinates lying in opposite directions are negative.
186. The point whose abscissa is a;, and whose ordinate is
?/, is usually denoted by (X ?/). Thus the points A, B, (7, and
Dare respectively represented by (3 7 4), (2, 3), (3, 2),
and (2, -3).
* This chapter may be omitted on a first reading.
148
GRAPHIC REPRESENTATION OF FUNCTIONS 149
The process of locating a point whose coordinates are given
is called plotting the point.
NOTE. Graphic constructions are greatly facilitated by the use of
cross-section paper, i.e. paper ruled with two sets of equidistant and
parallel linos intersecting at right angles. (See diagram on page 151.)
EXERCISE 71
1. Plot the points: (4, 3), (4, -2), (-4, 2), (-3, -3).
2. Plot the points: (-4, 2J-), (-5, 1), (4, 0), (-2, 0).
3. Plot the points : (0, 3), (4, 0), (0, 0), (0, - 2).
4. Draw the triangle whose vertices are respectively (4, 1),
(-l,3),and(l, -2).
6. Plot the points (6, 4) and (4, 4), and measure their
distance.
6. What is the distance of the point (3, 4) from the
origin ?
7. Draw the quadrilateral whose vertices are respectively
(4,1), (-1,4), (-4, -!),(!, -4).
8. Where do all points lie whose ordinates tfqual 4 ?
9. Where do all points lie whose abscissas equal zero ?
10. Where do all points lie whose ordinates equal zero?
11. What is the locus of (a?, y) if y = 3 ?
12. If a point lies in the avaxis, which of its coordinates is
known ?
13. What are the coordinates of the origin ?
187. Graphs. If two variable quantities are so related that
changes of the one bring about definite changes of the other,
the mutual dependence of the two quantities may be represented
either by a table or by a diagram.
150
ELEMENTS OF ALGEBRA
Thus the following tables represent the average temperature
of New Y'ork City from January 1 to December 1, and the
volumes of a certain amount of gas subjected to pressures from
1 pound to 8 pounds.
DATE
AVERAGE
TKMPKR.VTURE
January 1 . .
February 1
March 1 . .
- 1C.
- 1 C.
1C.
April 1 ...
Mayl . . .
June 1 ...
C.
12 C.
18 C.
Julyl . . .
August 1 . .
September 1 .
October 1 . .
22 C.
23 C.
20 C.
10 C.
November 1 .
10 C.
December 1
3C.
PKERSITKK
VOLUME
1 lb.
12 c.cin.
21b.
6 c.cm.
31b.
4 c.cm.
41b.
3 c.cm.
61b.
2.4 c.cm.
(5 lb.
2 c.cm.
71b.
1.7 c.cm.
81b.
1.6 c.cm.
188. The same data, however, may be represented graph-
ically by making each number in one column the abscissa, and
the corresponding number in the adjacent column the ordinate
of a point. Thus the first table produces 12 points, A, B, C, D,
etc., each representing a temperature at a certain date.
By representing in like manner the average temperatures for
every value of the time, we obtain an uninterrupted sequence
of points, or the curved line ABCN y the so-called graph of
the temperature.
To find from the diagram the temperature on June 1, we meas-
ure the ordinate of F. Thus the average temperature on May
15 may be found to be 15 ; on April 20, 10 ; on Jan. 15, 1 .
A graphic representation does not allow the same accuracy of results
as a numerical table, but it indicates in a given space a great many more
facts than a table, and it impresses upon the eye all the peculiarities of
the changes better and quicker than any numerical compilations.
GRAPHIC REPRESENTATION OF FUNCTIONS 151
i55$5St5SS
3{utt|s33<0za3
Graphs are possibly the most widely used devices of applied mathe-
matics. The scientist uses them to compile the data found from
experiments, and to deduce general laws therefrom. The engineer, the
physician, the merchant, uses them. Daily papers represent ecpnoniical
facts graphically, as the prices and production of commodities, the rise
and fall of wages, etc. Whenever a clear, concise representation of a
number of numerical data is required, the graph is applied.
EXERCISE 72
From the diagram find approximate answers to the following
questions :
1. Determine the average temperature of New York City
on (a) May 1, (b) July 15, (c) January 15, (d) November 20.
152 ELEMENTS OF ALGEKRA
2. At what date or dates is the average temperature oi
New York (a) G C., (1) 10 C., (c) - 1 C., (d) 9 0. ?
3. At what date is the average temperature highest ? What
is the highest average temperature?
4. At what date is the average temperature lowest? What
is the lowest average temperature ?
5. During what months is the average temperature of
New York above 18 C.?
6. When is the average temperature below C. (freezing
point) ?
7. From what date to what date does the temperature
increase (on the average)?
8. When is the temperature equal to the yearly average of
11 0. ?
9. What is the average temperature from Sept. 1 to Oct. 1?
10. How much, on the average, does the temperature
increase from June 1 to July 1 ?
11. During what month does the temperature increase most
rapidly ?
12. During what month does the temperature decrease most
rapidly ?
13. During what month does the temperature change least?
14. Which month is the coldest of the year?
15. Which month is the hottest of the year?
16. How much warmer on the average is it on July 1 than
on May 1 ?
17. If we would denote the time during which the tempera-
ture is above the yearly average of 11 as the warm season,
from what date to what date would it extend ?
GRAPHIC REPRESENTATION OF FUNCTIONS 153
18. When is the average temperature the same as on April 1 ?
NOTE. Use the graphs of the following examples for the solution
of concrete numerical examples, in a similar manner as the temperature
graph was applied in examples 1-18.
19. From the table on page 150 draw a graph representing
the volumes of a certain body of gas under varying pressures.
20. Construct a diagram containing the graphs of the mean
temperatures of the following three cities (in degrees Fahren-
heit) :
-
-
^
u
y -
-
d
j^
^
^
e;
r .
w
<
W
LJ
o
"
^
"^
^
"* -<
^
<
c/)
v
^
^
*
San Franc-isco
50
52
54
55
57
58
58
59
(>0
59
50
51
50
Tampa
59
00
60
72
70
80
82
81
80
73
05
(53
72
Bismarck
4
10
23
42
54
04
70
08
57
44
20
15
40
21. Represent graphically the populations (in hundred thou-
sands) of the following states :
STATK
1800
1810
1820
.5
1880
1.0
1840
1850
8.5
18(50
1870
1880
1890
1900
48.2
Illinois
4.8
25.3
30.8
38.3
Maryland
New York
3.4
0.9
3.8
9.0
4.0
13.7
4.5
19.2
4.7
24.3
5.8
31.0
(5.9
38.8
7.8
43.8
93
50.8
10.4
00.0
11.9
72.7
Virginia
8.8
9.7
10.7
12.1
12.4
14.2
10.0
12.3
15.1
10.0
18.5
22. One meter equals 1.09 yards. Draw a graph for the
transformation of meters into yards.
23. Draw a temperature chart of a patient.
Hour .
Temperature
p.m.
7 p.m.
8p.m.
9p.m.
10p.m.
llp.m.
m'dn't
100.5
101
101.5
103.2
102.5
102
101.4
154 ELEMENTS OF ALGEBRA
24. If C is the circumference of a circle whose radius is J2,
then C 2 irJl. (Assume ir~ >2 T 2 .) Represent graphically the
circumferences of all circles from R = to R = 8 inches.
25. Represent graphically the weight of iron from to 20
cubic centimeters, if 1 cubic centimeter of iron weighs 7.5
grams.
26. Represent graphically the cost of butter from to
5 pounds if 1 pound cost $.50.
27. Represent graphically the distances traveled by a train in
3 hours at a rate of 20 miles per hour.
28. A dealer in bicycles gains $2 on every wheel he sells.
If the daily average expenses for rent, gas, etc., amount to
$8, represent his daily gain (or loss), if he sells 0, 3, 2 ...
10 wheels a day.
29. The cost of manufacturing a certain book consists of the
initial cost of $800 for making the plates, and $.50 per copy
for printing, binding, etc. Show graphically the cost of the
books from 1 to 1200 copies. (Let 100 copies = about \- inch.)
On the same diagram represent the selling price of the books,
if each copy sells for $1.50.
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE
189. An expression involving one or several letters is called
a function of these letters.
2 x + 7 is a function of x.
2 xy y' 2 + 3 y 8 is a function of x and y.
190. If the value of a quantity changes, the value of a
function of this quantity will change; e.g. if x assumes
successively the values 1, 2, 3, 4, x* x + 7 will respec-
tively assume the values 7, 9, 13, 19. If x increases
gradually from 1 to 2, x 2 x -f 7 will change gradually from
7 to 9.
GRAPHIC REPRESENTATION OF FUNCTIONS 155
191. -A variable is a quantity whose value changes in the
same discussion.
A constant is a quantity whose value does not change in the
same discussion.
In the example of the preceding article, x is supposed to change, hence
it is a variable, while 7 is a constant.
192. Graph of a function. The values of a function for the
various values of x may be given in the form of a numerical
table. Thus the table on page 1G4 gives the values of the func-
tions x 2 , x 3 , and Vsr, for x=l, 2, 3 50. The values of func-
tions may, however, be also represented by a graph.
E.g. to con struct the graph
of x 2 , construct a series of
points whose abscissas rep-
resent X) and whose ordi-
nates are the corresponding
values of x 2 ', i.e. construct
the points (-3, 9), (- 2, 4),
(-1,1), (0,0), (1,1), (2, 4),
and (3, 9), and join the
points in order.
If a more exact diagram
is required, plot points which
lie between those constructed above, as Q-, -J), (1^, 2), etc.
Ex. 1. Draw the graph of x 2 -f- 2 x 4 from x = 4, to x = 4.
To obtain the values of the functions for the various values of a*, the
following arrangement may be found convenient :
x
-3
2
1
1
2
3
X
-4
-3
2
1
1
2
3
4
X 2 =
10
9
4
1
1
4
9
16
x 2 + 2x-
8
3
- 1
3
8
15
11
y, or x 2 -f 2 x 4 =
4
- 1
-4
-5
-4
-1
4
11
20
156
ELEMENTS OF ALGEBRA
Locating the points( 4, 4),
(-3, -1), (-2,4)... (4,20),
and joining in order produces
the graph ABC. (To avoid
very large ordinatcs, the scale
unit of the ordinatcs is taken
smaller than that of the x.)
For brevity, the function
is frequently represented
by a single letter, as y.
Thus in the above example,
y = x 2 -f- 2 x 4 ; and if
r 01 ,, 71 . if /* 1i
*/ +* .,-, rf 4 > > >
?/ = 4J, etc.
193. A function of the
first degree is an integral
rational function involving only
the lirst power of the variable.
Thus 4x + 7, or ax + b -f c are func-
tions of the first degree.
194. It can be proved that the
graph of a function of the first degree
is a straight line, hence two points
are sufficient for the construction
of these graphs.
Ex. 2. Draw the graph of
y = 2x-3.
If z = 0, j/=-3.
If x = 4, y = 6.
Locating (0, 3) and (4, 5), and join-
ing by a straight line produces the required graph.
7
Y'
GRAPHIC REPRESENTATION OF FUNCTIONS 157
EXERCISE 73
Draw the graphs of the following functions:
1. a? + 2. 4. 2x + l. 7. 2-3x. 10. a? 1.
2. x-l. 5. 3x 2. 8. 1 a?. 11. x z + x.
3. 2 a? I. 6. a?. 9. -Jar 8 . 12. 4a? a*
13. a; 2 4 a; + 4. 16. a; 2 x+1. 19. y = 2x 3.
14. a; 2 -3 a -8. 17. 6 -fa- -or. 20. ?/ = a; 2 -4.
15. a; 2 -fa-- 2. 18. 2 a ar.
21. Draw the graph of or 2 from #= 4 to 05 = 4, and from
the diagram find :
(a) (3.5)2; (ft) (_ 1.5)2; ( C ) (-2.8)'; (d) (-If) 2 ;
(e) Va25; (/) Vl2^ ; (0) V5; (^) VlO-'S".
22. Draw the graph of or 4 a? + 2 from x 1 to a; = 4, and
from the diagram determine:
(a) The values of the function if x = \, 1J-, 2J-.
(6) The values of a?, if a; 2 4 # + 2 equals 2, 1, 1-J-.
(c) The smallest value of the function.
(d) The value of x that produces the smallest value of the
function.
(c) The values of x that make it* 2 4 a? + 2 = 0.
(/) The roots of the equation x 2 4 x -f 2 = 0.
(</) The roots of the equation a 2 4 x -f 2 = 1.
(7i) The roots of the equation x 2 4 x -f 2 = 2.
23. Draw the graph ofy=2-j-2# # 2 from # = 2 to a?=4,
and from the diagram determine :
(a) The values of y; i.e. the function, if"a; = -J-, 1-J-, 2J.
(6) The values of a*, if y = 2.
(c) The values of a*, if the function equals zero.
(d) The roots of the equation 2 -f 2 a a* 2 = 0.
(e) The roots of the equation 2 -{-2x a* 2 = l.
158 ELEMENTS OF ALGEBRA
24. Degrees of the Fahrenheit (F.) scale are expressed in
degrees of the Centigrade (C.) scale by the formula
(a) Draw the graph of
C = f (F-32)
from F 4
to F=l.
(b) From the diagram find the number of degrees of centi-
grade equal to -1 F., 9 F., 14 F., 32 F.
(c) Change to Fahrenheit readings : 10 C., C., 1 C.
25. A body moving with a uniform velocity of 3 yards per
second moves in t seconds a distance d = 3 1.
Represent this formula graphically.
26. If two variables x and y are directly proportional, then
y = cXj where c is a constant.
Show that the graph of two variables that are directly pro-
portional is a straight line passing through the origin (assume
for c any convenient number).
27. If two variables x and y are inversely proportional, then
y = - where c is a constant.
x
Draw the locus of this equation if c = 12.
GRAPHIC SOLUTION OF EQUATIONS INVOLVING ONE
UNKNOWN QUANTITY
195. Since we can graphically determine the values of x
that make a function of x equal to zero, it is evidently possible
to find graphically the real roots of an equation. Thus to find
what values of x make the function x 2 + 2x 4 = (see 192),
we have to measure the abscissas of the intersection of the
graph with the o>axis, i.e. the abscissas of P and Q.
Therefore x = 1.24 or x = 3.24.
GRAPHIC REPRESENTATION OF FUNCTIONS 159
196. To solve the equa-
tion x 2 + 2 x 4 = 1, de-
termine the points where
the function is 1. If
cross-section paper is used,
the points may be found
by inspection, otherwise
draw through (0, 1) a
line parallel to the #-axis,
and determine the abscis-
sas of the points of inter-
section with the graph,
viz. 2 and 1.
197. An equation of the
the form ax 2 + bx + c = 0,
where a, 6, and c represent
known quantities, is called
a quadratic equation. Such
equations in general have
two roots.
\-3 -2 --1
Y'
1/2
1
EXERCISE 74
Solve graphically the following equations :
1. 4 x _ 7 0.
2.
3.
4.
6.
6.
7.
8.
9.
10. or 5 -a -5 = 0.
11. a 2 -2a;-7 = 0.
12. (a) a: 2 -6a;-f 9 = 0.
(6) z 2 -
(c) a 2
13. (a) x 2 4x 6 = 0.
14. (a)
(6)
160
ELEMENTS OF ALGEBRA
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO
UNKNOWN QUANTITIES
198. Graph of equations involving two unknown quantities.
Since we can represent graphically equations of the form y =
function of x ( 1D2), we can construct the graph or locus of any
equation involving two unknown quantities, that can be reduced
to the above form.
x \ x
Thus to represent - -L^- = 2 graphically, solve for ?/, i.e.
-
y = -
A
and construct x ( - graphically.
Ex.1. Represent graphically 3x
X'-2
7*
JJ y. _
y ='-"-
Solving for ?/,
Hence if x 2, y = - 4 ;
if x - 2, y = 2.
Locating the points (2, 4) and (2, 2),
and joining by a straight line, produces the
required locus.
199. If the given equation is of the
first degree, we can usually locate two
points without solving the equation for
y. Thus in the preceding example:
3 x - 2 y ~ 2.
If s = 0, y = -l.
Hence we may join (0, 1) and (f , 0).
Ex.2. Draw the locus of 4 x + 3 y = 12.
If x = 0, ?/ = 4 ; if y = 0, fc = 3.
Hence, locate points (0, 4) and (3, 0), and join
them by straight line AB. AB is the required graph.
NOTE. Equations of the first degree are called linear
equations, because their graphs are straight lines.
T
GRAPHIC REPRESENTATION OF FUNCTIONS 161
200. The coordinates of every point of the graph satisfy
the given equation, and every set of real values of x and y
satisfying the given equation is represented by a point in
the locus.
201. Graphical solution of a linear system.
To find the roots of
the system.
By the method of
the preceding article
construct the graphs
AB and CD of (1)
and (2) respectively.
The coordinates of
every point in AB
satisfy the equation
(1), but only one point
in AB also satisfies
equation (2), viz. P, the point of intersection of AB and CD.
By measuring the coordinate of P, we obtain the roots,
x = 3.15, y = .57.
202. The roots of two simultaneous equations are represented
by the coordinates of the point (or points) at which their
graphs intersect.
203. Since two straight lines which are not coincident nor
parallel have only one point of intersection, simultaneous
linear equations have only one pair of roots.
Ex. 3. Solve graphically the equations :
\x-y-\- 1=0.
(1)
(2)
162
ELEMENTS OF ALGEBRA
c \
Y
yj
B
/
V J
f
'\
y
7
\,
X'
/
N
'* X
V
1
1
,Y'
2
\D
Using the method of the preceding para,
graph, construct AB the locus of (1), and
CD the locus of (2) .
Measuring the coordinates of P, the point
of intersection, we obtain
Ex. 4. Solve graphically the fol-
lowing system :
= 25, (1)
= -C. (2)
Solving (1) for y,y~ V25 x 2 .
Therefore, if x equals 5, - 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, y equals
respectively 0, 3, 4, 4.5, 4.0, 5, 4.9, 4.5, 4, 3, 0.
Locating the points (5,0), (-4,
+ 3), (4, 3), etc., and joining, we
obtain the graph (a circle) AB C of the
equation x 2 + y* = 25.
Locating two points of equation (2),
e.g. (-2, 0) and (0, 3), and joining by a
straight line, we obtain DE, the graph of
3 x - 2 y = -6.
Since the two graphs meet in two
points P and $, there are two pairs of
roots. By measuring the coordinates of
P and Q we find :
204. Inconsistent equations. The equations
2 = 0, (1)
4 = 0, (2)
cannot be satisfied by the same values of x and y,
i.e. they are inconsistent. This is clearly shown
by the graphs of (1) arid (2), which consist of a
pair of parallel lines. There can be no point of
intersection, and hence no roots.
In general, parallel graphs indicate inconsistent equations.
GRAPHIC REPRESENTATION OF FUNCTIONS '163
205. Dependent equations, as
2^3 ==l
and 3 x -f 2 y = 6
have identical graphs, and, vice versa, iden-
tical graphs indicate dependent equations.
EXERCISE 75
Construct the loci of the following equations:
1. a+r/=6. 3. 2x 3?/=6. 5. y 4. 7. y=
2. 2x y6. 4. x~y=0. 6. y=x 2 + 5. 8. a 2
Draw the graphs of the following systems, and solve each
system, if possible. If there are no solutions, state reasons.
9.
10.
a; y = 4.
17.
.,
19.
20.
1 6 * + 7 y = 3.
16
16
22.
23.
\ 2 x + 3^
164
24.
25.
28.
U # - 14 y = - 8.
ELEMENTS OF ALGEBRA
"~ 26.
4 a = 3(6 - y).
29 .
3
30.
31. Show that the same values of x and y cannot satisfy the
three equations :
x -f 5 y = 5.
TABLE OF SQUARES, CUBES, AND SQUARE HOOTS
0?
V*
CC3
Vj.
w;
^.2
u;3
Vj
1
1
1
J.OOO
26
6 76
17 576
5.099
2
4
8
1.414
2 7
7 29
1 9 (>3
5.196
3
9
27
1.732
28
7 84
21 952
5.292
4
16
64
2.000
2 9
841
21 .'589
5.385
5
25
125
2.236
30
900
27000
5.477
6
36
216
2.449
3 1
961
29 791
5.568
7
49
343
2.646
3 2
1024
32 768
5.657
8
64
512
2.828
33
10 89
35 937
5.745
1)
81
729
3.000
34
H 56
39 304
5.831
10
1 00
1 000
3.162
35
12 25
42875
5.916
1 1
121
1 331
3.317
3 6
12 96
46 656
6.000
1 2
1 44
1 728
3.464
3 7
13 69
50 653
6.083
13
1 69
2 197
3.606
3 8
1444
54872
6.164
1 4
1 96
2 744
3.742
3 9
15 21
59 319
6.245
15
225
3375
3.873
4
16 00
64 000
(5.325
16
256
4096
4.000
4 1
16 81
68 921
6.403
1 7
289
4 913
4.123
42
1764
74088
6.481
1 8
324
5 832
4.243
43
1849
79 507
6.557
1 9
361
6 859
4.359
4 4
193(5
85 184
6.633
20
400
8 000
4.472
45
2025
91 125
6.708
2 1
441
9261
4.583
4 6
21 16
97 336
6.782
22
484
10648
4.690
47
2209
103 823
(5.856
23
5 29
12 167
4.796
4 8
2304
110592
6.928
24
5 76
13 824
4.899
4 9
24 01
117 649
7.000
2 5
625
15 625
5.000
60
25 00
125 000
7.071
CHAPTER XIII
INVOLUTION
206. Involution is the operation of raising a quantity to a
positive integral power.
To find (#(**&)" is a problem of involution. Since a power
is a special kind of product, involution may be effected by
repeated multiplication.
207. Law of Signs. According to 50,
-fa- -fa- -f a = -f a 3 .
a a = +- a 2 .
a a a = a 3 , etc.
Obviously it follows that
1. ^4/? powers of a positive quantity are positive.
2. All even powers of a negative quantity arc positive.
3. All odd powers of a negative quantity are negative.
( a) is positive, ( aft 2 ) 9 is negative.
INVOLUTION OF MONOMIALS
208. According to 52,
1. (a 2 ) 3 = a 2 a 2 a 2 = 2 + 2 + 2 = a.
2. (5 5 )* = 6 5 b 5 . 6 6 ?> 5 = 6+ 5 + fi +fi = 62.
8. (a n ) m = a n o n to in factors
4. (- 3 2 6 3 )* = (- 3 a 2 6 8 ) . (- 3 a 2 6 8 ) (-
= _ (2m a ) 8 ____
(8 *)"" 27 n 16
165
166 ELEMENTS OF ALGEBRA
To find the exponent of the power of a power, multiply tht
given exponents.
To raise a product to a given power, raise each of its factors to
the required power.
To raise a fraction to a power, raise its terms to the required
power.
EXERCISE 76
Perform the operations indicated :
1. (>y. 2. (-a 2 ) 4 - 3. (-a 2 ) 5 . 4. (afc 2 ) 11 .
5.
6. (-277171 4 )*.
/2mV. 24.
\ 3 J
' -
M-W
10. (-2ar). 27 / _4_V
11. ' ' '
/ _4_
V V/
13. -
/-2?n?A 4
^--- 30.
15. am-Vy) 3 .
16. (-|^^) 2 . ' V 3xy )'
INVOLUTION OF BINOMIALS
209. The square of a binomial was discussed in 63.
210. The cube of a binomial we obtain by multiplying (a + 6) 1
b y + &- ( a _j_ 6) 3 = a 3 + 3a 2 6 + 3a6 2 + * 8 ,
and (a - 6) 8 = a 3 - 3 a 2 6 -f 3 a6 2 - 6 8 .
INVOLUTION 167
Ex. 1. Find the cube of 2 x -f- 3 y.
= (2s) 3 + 3(2aO*(Sy) + 3(2aj)(3y)>
s= 8 a; 3 + 36 z 2 y + 54 xy* + 27 y 3 .
Ex. 2. Find the cube of 3 x* - y n .
(3 x 2 - y) = (3 y?y - 3(3 a*)a(y
= 27 a 6 - 27 ay + 9 x 2 y 2n -
EXERCISE 77
Perform the operations indicated:
1. (a + &) 8 . 7. (5 -a) 3 . 13. (3a-f26) 8 .
2. (a?-?/) 8 . 8. (1 + 2 a;) 3 . 14. (6m+2w) 8 .
3. (a-fl) 3 . 9. (3 a -I) 8 . 15. (3 a- 6 ft) 8 .
4. (m-2) 8 . 10. (l + 4aj) 3 . 16. (3a 2 6 2 -l) 3 .
5. (w+w) 8 - lx - (7 a 2 -I) 3 . 17. (a
6. (a-j-7) 3 . 12. (1 + 5 a 3 ) 8 . 18. (4 or* -
Find the cube root of :
19. a 3 + 3a 2 6 + 3a& 2 -f-& 3 . 22. 1 -f 3 w + 3 w 2 + ra 8 .
20. ^-Sx^ + S^ 2 -^ 3 . 23. 86 3 -126 2 + G6-l.
21. a 8 -3a 2 + 3a-l.
211. The higher powers of binomials, frequently called ex.
pansions, are obtained by multiplication, as follows :
(a + 6) 8 = o 8 + 3 d'b + 3 a6 2 + b*.
(a + 6) 4 = a 4 + 4 a?b + 6 a 2 & 2 + 4 a6 3 + b 4 .
(a + 6) 5 = a 5 + 5 a 4 6 + 10 a*b 2 + 10 a 2 6 3 -f 5 aM + 6 s , etc.
An examination of these results shows that :
1. The number of terms is 1 greater than the exponent of the
binomial.
2. TJie exponent of a in the first term is the same as the expo-
nent of the binomial, and decreases in each succeeding term by L
168 ELEMENTS OF ALGEBRA
3. T7ie exponent ofb is 1 in the second term of the result, and
increases by 1 in each succeeding term.
4. The coefficient of the first term is 1.
5. The coefficient of the second term equals the exponent of the
binomial
6. TJie coefficient of any term of the power multiplied by the
exponent of a, and the result divided by 1 plus the exponent of b,
is the coefficient of the next term.
Ex. 1. Expand (x
= ic 5 -f 5 x*y + 10 ^V + 10 x*y* + 5 xy* + y 5 .
Ex.2. Expand (a??/) 5 .
2 + 10 x' 2 (-
x 5 5 x 4 y +
212. The signs of the last answer arc alternately plus and
minus, since the even powers of y are positive, and the odd
powers negative.
Ex. 3. Expand (2 # 2 - 3 y 3 ) 4 .
<? * 2 ) 4 - 4(2 * 2 )'(3 *f) -f 6(2 ^) 2 (3 y 8 ) 8
- 4(2^(3 ^'+(3 y 8 ) 4
16 ic 8 - 96 ^y -f 216 o?y - 216 a^ 9 4- 81 y 12 .
Expand: EXERCISE 78
1. (p + q) 4 * 7. (1-for) 4 . 13. (2 4- a) s . 19. (mnp I) 5 .
2. (w ?i) 4 . 8. ( &) 5 . 14. (m 3 -fl) 4 . 20. (2w 2 -f-l) 5 .
3. (tf-f-1) 4 . 9. (c-fd) 5 . 15. (l m 2 n 2 ) 5 . 21. (3a 2 -f5) 4 .
4. (1 + ?/) 4 . 10. (?/i-~w) 8 . 16. (m 5 I) 5 . 22. (2 a 2 5) 4 .
5. (m-J) 4 . 11. (a-f 5) : . 17. (m + n) 8 . 23. (2a-5c) 4 .
6. (l-a&) 4 . 12. (a~^) 5 . 18. (?>i?i -f c)*. 24. (1 -f 2 a:) 4 .
25. (l-fa 2 6 2 ) 5 .
CHAPTER XIV
EVOLUTION
213. Evolution is the operation of finding a root of a quan
tity ; it is the inverse of involution.
\/a = x means x n = a.
V 27 = y means y ?> = 27, or y ~ 3.
\/P = x means r' = 6-, or x & 4 .
214. It follows from the law of signs in evolution that :
1. Any even root of a positive, quantity may be either 2wsitive
or negative.
2. Every odd root of a quantity has the same sign as the
quantity.
V9 = + 3, or - 3 (usually written 3) ; for (-f 3) 2 and ( 3) 2 equal 0.
\/"^27=-3, for (_3) = -27.
v/o* = a, for (+ a) 4 = a 4 , and ( a) 4 = a 4 .
\/32 = 2, etc.
215. Since even powers can never be negative, it is evidently
impossible to express an even root of a negative quantity by
the usual system of numbers. Such roots are called imaginary
numbers, and all other numbers are, for distinction, called real
numbers.
Thus V^I is an imaginary number, which can be simplified no further.
109
170 ELEMENTS OF ALGEBRA
EVOLUTION OF MONOMIALS
The following examples are solved by the definition of a
root : ,
Ex.1. v/^i2 = a*, for (a 3 )* = a 12 .
Ex. 2. 3/0** = a m , for (a")" = a mn .
Ex. 3. v^SjW 3 = 2 a a &c*, for (2 a 2 6c 4 ) 8 =
Ex.4. ^/gL^g =
* c*
Ex 5 A 82 a"
.lL,for(*Siy = ?*-
3 6 2 c* \ 3 b' 2 c*J 243 ft^c 20
216. To extract the root of a power, divide the exponent by the
index.
A root of a product equals the product of the roots of the factors.
To extract a root of a fraction, extract the roots of the numerator
and denominator.
Ex. 6. \/18 . 14 63 25 = V2 . 3* 2 . 7 . 7 . 8 2 . 6 2 = V2* . 3i . 6-
= 2 . 3 2 6 . 7 = 030.
Ex. 7. VT8226 = V25 729 = V26TIT81 = 5-3.9 = 136.
Ex. 8. Find (x/19472) 2 .
Since by definition ( v^)" = a, we have (Vl472) 2 = 19472.
Ex. 9.
= 199 + (_ 198) - 200 - (- 201) = 2.
EXERCISE 79
1. -v/2 5 . 3. -fy 5 3 . 5. V5 2 7 2 . 7. V25 9 16.
2. V?. 4. -v/2^. 6. -v/2 3 3 3 5 3 . 8. v- 125- 64
9. V36 9 - 100 a 2 . 10. \/2 4 9 4 5 4 .
EVOLUTION
171
28. -\/d r -\-Vab + b\
29. V8- 75- 98- 3.
30. V20 . 45 9.
31. V5184.
32. V9216.
33. ( VH) 2 + (Vl9) 2 - (V200) 2 -f ( V240) 2 .
34. ( VI5) 2 x ( VT7) 2 -f ( VI5) 2 x ( V3) 3 .
35. (V2441) 2 ~(V2401) 2 .
36. (Vl24) 2 -{
EVOLUTION OF POLYNOMIALS AND ARITHMETICAL
NUMBERS
217. A trinomial is a perfect square if one of its terms is
equal to twice the product of the square roots of the other
terms. ( 116.) In such a case the square root can be found
by inspection.
Ex. 1. Find the square root of a 2 - 6 ofy 2 -f 9 y 4 .
a* _ 6 ary -f 9 y 4 = (s 8 - 3 y 2 ) 2 . ( 116.)
Hence vV - 6 tfif + 9 y 4 = O 3 - 3 ;/).
EXERCISE 80
Extract the square roots of the following expressions :
1. a 2 -f2 + l. 3. ^-40^4- 4/. 5.
2. l 2y-h2/ 2 . 4 - 9^ + 60^ + 2/ 2 . 6.
172 ELEMENTS OF ALGEBEA
7. 4a 2 -44a?> + 121V 2 . 10.
8 . 4a s + 6 2 + 4a&. 11. 49a 8 -
9. mV-14m??2)-f 49;> 2 . 12. 16 a 4 - 72 aW + 81 & 4 .
13. # 2
14.
15. a 2 -
16. a 2 + & 2 + c 2 + 2 a& - 2 ac - 2 &c.
218. In order to find a general method for extracting the
square root of a polynomial, let us consider the relation of a -f- b
to its square, a 2 -f- 2 ab + b 2 .
The first term a of the root is the square root of the first
term a' 2 .
The second term of the root can be obtained by dividing the
second term 2ab by the double of a, the so-called trial divisor;
2ab ,
a-\-b is the root if the given expression is a perfect square.
In most cases, however, it is not known whether the given
expression is a perfect square, and we have then to consider
that 2 ab -f b 2 = b (2 a -f b), i.e. the sum of trial divisor 2 a,
and b, multiplied by b must give the last two terms of the
square.
The work may be arranged as follows :
a 2 + 2 ab + W \a + b
2
EVOLUTION 173
Ex. 1. Extract the square root of 1G x* - 24 afy* -f 9 tf.
16x4 __
10 x*
Explanation. Arrange the expression according to descending powers
of x. The square root of 10 x 4 is 4 # 2 , the lirst term of the root.
Subtracting the square of 4x' 2 from the trinomial gives the remainder
'24 x' 2 */'' + y. By doubling 4x' 2 , we obtain 8x 2 , the trial divisor.
Dividing the first term of the remainder, 24# 2 y 3 , by the trial divisor
8 /-, we obtain the next term of the root 3 y 3 , which has to be added to
the trial divisor. Multiply the complete divisor Sx' 2 3y 3 by Sy 8 , and
subtract the product from the remainder. As there is no remainder,
4 x 2 3 ?/ 8 is the required square foot.
219. The process of the preceding article can be extended to
polynomials of more than three terms. We find the first two
terms of the root by the method used in Ex. 1, and consider
their sum one term, the first term of the answer. Hence the
double of this term is the new trial divisor; by division we
find the next term of the root, and so forth.
Ex. 2. Extract the square root of
16 a 4 - 24 a s + 4 -12 a + 25 a 8 .
Arranging according to descending powers of a.
10 a 4 - 24 a 3 + 25 a 2 - 12 a + 4
Square of 4 a 2 . 10 a 4
First remainder.
First trial divisor, 8 a 2 .
First complete divisor, 8 a 2 8 a. \ 24 a 3 4- a 2
Second remainder.
Second trial divisor, 8 a 2 6 a.
Second complete divisor, 8 a 2 a -f 2.
As there is no remainder, the required root is (4 a' 2 8 a + 2}.
-f 10 a 2 - 12 a + 4
174 ELEMENTS OF ALGEBRA
EXERCISE 81
Extract the square roots of the following expressions :
2. 3 a 2 2a + a 4 2 a 3 + 1.
3. a 4 4- 2 or 3 4-1 x 2 2x.
4. 16 a 4 -|- 24 a 3 + 81 a 2 4-54 a + 81.
5. 25 m 4 20 w 3 + 34 m 2 - 12 m 4- 9.
6. 4-12 a& -f 37 a' J 6 2 - 42 a 3 > 3 -f 49 a 4 6 4 .
7. 25 x 4 -f- 40 afy 4-46 x 2 if 4- 24 a^ 8 4- 9 i/ 4 .
8. 16x 6 4- 73a 4 4-40^4-36^4-60^.
9. l 4.2^4-3^4-2^ 4- a; 4 .
10. 1 4- 4 x 4- 10 x 2 4- 20 o 4- 25 x 4 4- 24 or 5 4- 16 iK 6 .
11. 36a 6 4-60a 5 4-73a 4 4-40a 3 4-16a 2 .
12. 36it- 6 4-36^?/4-69a;V4-30^4-25^ 4 .
13. 4m 6 4- 12m 5 4- 9m 4 20m 3 30m 2 4- 25.
14. 49 a 4 - 42 a*& 4- 37 a 2 ^ 2 - 12 a6 3 4- 4 6 4 .
15. x 6 4- 4 0^4- 20 or 2 16 x 4- 16.
16. 13# 4 4-13ar J 4-4a; 6 - 14^4-4 4^ 12^.
17. ic 4 4-?/ 4 4-2x- 3 j/ 2xif x*y 2 .
18. 729 4- 162 a 2 6 a 5 4- a 6 - 54 a 3 4- 9 a 4 .
19. 60 a 10 4- 73 a 8 4- 36 a 12 4- 40 a 6 4- 16 a 4 .
20. 46 a 4 4- 44 a 8 -f 25 a 2 -h 12 a 4- 4 4- 25 a 6 4- 40 a
22 . 16 _^ + 2 J-
X XT
24. 4-
a? 2 a;
EVOLUTION 175
220. The square root of arithmetical numbers can be found
by a method very similar to the one used for algebraic
expressions.
Since the square root of 100 is 10; of 10,000 is 100; of 1,000,000 is
1000, etc., the integral part of the square root of a number less than 100
has one figure, of a number between 100 and 10,000, two figures, etc.
Hence if we divide the digits of the number into groups, beginning at the
units, and each group contains two digits (except the last, which may
contain one or two), then the number of groups is equal to the number
of digits in the square root, and the square root of the greatest square in
the first group is the first digit in the root. Thus the square root of 96'04'
consists of two digits, the first of which is 9 ; the square root of 21'06'81
has three digits, the first of which is 4.
Ex. 1. Find the square root of 7744.
From the preceding explanation it follows that the root has two digits,
the first of which is 8. Hence the root is 80 plus an unknown number,
and we may apply the method used in algebraic process.
A comparison of the algebraical and arithmetical method given below
will show the identity of the methods.
7744 1 80 + 8
6400
160 + 8 = 168
1344
1344
Explanation. Since a = 80, a 2 = 6400, and the first remainder is- 1344.
The trial divisor 2 a = 160. Therefore 6 = 8, and the complete divisor
is 168.
As 8 x 168 = 1344, the square root of 7744 equals 88.
Ex. 2. Find the square root of 524,176.
a 6 c
f>2'41 '70 [700 + 20 + 4 = 724
a 2 = 41) 00 00
2 a + 6 = 1400 + 20 = 1420
341 76
28400
4 = 1444
57 76
6776
1T6
ELEMENTS OF ALGEKRA
221. In marking off groups in a number which has decimal
places, we must begin at the decimal point, and if the right-
hand group contains only one digit, annex a cipher.
Thus the groups in .0961 are '.GO'61. The groups of 16724.1 are
1'67'24.10.
Ex. 3. Find the square root of 6.7 to three decimal places.
6/.70
4
12.688
45
2 70
2 25
508
4064
6168
41)600
41344
2256
222. Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator,
or by transforming the common fraction into a decimal.
EXERCISE 82
Extract the square roots of :
1.
5625.
11.
95,481.
21.
4,153,444.
2.
4096.
12.
61,009.
22.
57,198,969.
3.
3249.
13.
582,169.
23.
25,836,889.
4.
5041.
14.
956,484.
24.
43,046,721.
5.
7056.
15.
8.0089.
25.
236,144,689.
6.
9604.
16.
42.25.
26.
42^.
7.
9801.
17.
72.25.
27.
IH-
8.
14,161.
18.
.8836.
28.
A*.
9.
10,201.
19.
.001369.
10.
56,169.
20.
1,555,009.
EVOLUTION 177
Find to three decimal places the square roots of the follow-
ing numbers:
29. 5. 31. .22. 33. 1.01. 35. T \.
30. 13. 32. 1.53. 34. J-. 36. J T .
37. Find the side of a square whose area equals 50.58 square
feet.
38. Find the side of a square whose area equals 96 square
yards.
39. Find the radius of a circle whose area equals 48.4 square
feet. (Area of a circle equals irR 1 , when R = radius and
TT = 3.1410.)
40. Find the mean proportional between 2 and 11.
CHAPTER XV
QUADRATIC EQUATIONS INVOLVING ONE UNKNOWN
QUANTITY
223. A quadratic equation, or equation of the second degree,
is an integral rational equation that contains the square of
the unknown number, but no higher power ; e.g. x 2 4 x 7,
6 y 2 = 17, ax 2 + bx + c = Q.
224. A complete, or affected, quadratic equation is one which
contains both the square and the first power of the unknown
quantity.
225. A pure, or incomplete, quadratic equation contains only
the square of the unknown quantity.
axt + bx -f c r= is a complete quadratic equation.
ax 2 = m is a pure quadratic equation.
226. The absolute term of an equation is the terra which
does not contain any unknown quantities. /
In 4 x 2 7 x -f 12 = the absolute term is 12.
PUKE QUADRATIC EQUATIONS
227. A pure quadratic is solved by reducing it to the form
ic 2 = a, and extracting the square root of both members.
Ex. 1. Solve 13 x 2 -19 = 7^ + 5.
Transposing, etc., 6# 2 = 24.
Dividing, x* = 4.
Extracting the square root of each member,
x = + 2 or x =2.
This answer is frequently written x = 2.
Check. 13( 2)2 - 19 = 33 ; 7( 2)* + 5 = 33.
178
QUADRATIC EQUATIONS 179
Ex.2. Solve .=g
Clearing of fractions, ax x 2 4 a 2 + 4 ax = ax + 4 a 2 + x 2 -f 4 ax,
Transposing and combining, 2 x 2 = 8 a 2 .
Dividing by 2, x 2 = 4 a 2 .
Extracting the square root, x = V 4 a 2 ,
or x =
Therefore, x =
EXERCISE 83
Solve the following equations :
1. o; 2 -7 = 162. 4. 16^-393 = 7.
2. 0^ + 1 = 1.25. 5. 15^-5 =
3. 19 a; 2 + 9 = 5500. 6.
7.
8. (a?-
9. 6(--2)=-10(aj-l).
10. -? + x = 4.
s-3 oj + 3
4fc 2 -5'
18. - =:
' y? b* b
180 ELEMENTS OF ALGEBRA
on a; __!_:L . 22 ' 4, 9
& -{- c# a; a x + a 2 a -f- 1
23. If a 2 4- b 2 = c 2 , find a in terms of 6 and c.
24. If s = (' 2 , solve for .
25. If = Trr 2 , solve for r. 26. If s = 4 Trr 2 , solve for r.
27. If 2 a 2 -f 2 b* = 4 w 2 -f c 2 , sol ve for m.
28. If 22 = ~^-, solve for v. 29. If G = m ' g m , solve for d.
EXERCISE 84
1. Find a positive number which is equal to its reciprocal
( 144).
2. A number multiplied by its fifth part equals 45. Find
the number.
3. The ratio of two numbers is 2 : 3, and their product is
150. Find the numbers. (See 108.)
4. Three numbers are to each other as 1 : 2 : 3, and the sum
of their squares is 5(5. Find the numbers.
5. The sides of two square fields are as 3 : 5, and they con-
tain together 30G square feet. Find the side of each field.
6. The sides of two square fields are as 7 : 2, and the first
exceeds the second by 405 square yards. Find the side of each
field.
228. A right triangle is a triangle, one of
whose angles is a right angle. The side
_____ opposite the right angle is called the hypote-
b nuse (c in the diagram). If the hypotenuse
contains c units of length, and the two other sides respectively
a and b units, then c 2 = a 2 -f- b 2 .
Since such a triangle may be considered one half of a rec-
tangle, its area contains square units.
QUADRATIC EQUATIONS 181
7. The hypotenuse of a right triangle is 35 inches, and the
other two sides are as 3 : 4. Find the sides.
8. The hypotenuse of a right triangle is to one side as
13:12, and the third side is 15 inches. Find the unknown
sides and the area.
9. The hypotenuse of a right triangle is 2, and the other
two sides are equal. Find these sides.
10. The area of a right triangle is 24, and the two smaller
sides are as 3 : 4. Find these sides.
11. A body falling from a state of rest, passes in t seconds
over a space s = -J- yt 2 . Assuming g = 32 feet, in how many
seconds will a body fall (a) G4 feet, (b) 100 feet?
12. The area $ of a circle whose radius equals r is found by
the formula /S = Trr 2 . Find the radius of circle whose area S
equals (a) 154 square inches, (b) 44 square feet. (Assume
7r = - 2 7 2 .)
13. Two circles together contain 3850 square feet, and their
radii are as 3 : 4. Find the radii.
14. If the radius of a sphere is r, its surface 8 = 4 wr 2 . Find
the radius of a sphere whose surface equals 440 square yards.
(Assume ir = - 2 7 2 .)
COMPLETE QUADRATIC EQUATIONS
229. Method of completing the square. The following ex-
ample illustrates the method of solving a complete quadratic
equation by completing the square.
Solve or - 7 x -f 10 = 0.
Transposing, x* 7 x = 10.
The left member can be made a complete square by adding
another term. To find this term, let us compare x 2 7 x with
the perfect square x 2 2 mx -f m 2 . Evidently 7 takes the place
of 2m, or m = |. Hence to make x 2 7x a complete square
we have to add (|) 2 , which corresponds to m 2 .
182 ELEMENTS OF ALGEBRA
Adding ( J) 2 to each member,
Or (*-i) 2 = f.
Extracting square roots, x \ = -| .
Hence # = ff.
Therefore x 5 or x = 2.
Check. 6 2 - 7 5 + 10 = 0, 2 2 - 7 2 + 10 =0.
Ex.1. 80^69^-2 =
Transposing, 9 x 2 15 x = 6.
Dividing by 9, sc 2 | x = |.
Completing the square (i.e. adding Q) 2 to each member),
Simplifying, (*~8) a = .
Extracting square roots, at | = \.
Transposing, x - \ J.
Therefore, a; = 2, or J.
230. Hence to solve a complete quadratic :
Reduce the equation to the form x*-\-px==q. Complete the
square by adding the square of one half the coefficient of x.. Ex-
tract the square root and solve the equation of the first degree thus
formed.
Ex.2.
a x
Clearing of fractions, x 2 x + 2 a 2 -f a = 2 ax.
Transposing, x 2 x 2 ax 2 a* a.
Uniting, s a - x(l -f 2 o) = - 2 a 2 - a,
QUADRATIC EQUATIONS 183
Completing the square,
Simplifying,
1 + 2 ?
Extracting square root, x - "*" - = - Vl - 4 a 2
Transposing, x = l+ * a ~ V IT
Therefore * = 1 +2 <* Vl -* <
EXERCISE 85
Solve :
3. or ! -f6tf = 27.
4. 0^ 7 = 60?.
7. a 8 2 2 = 0.
9. 70-faj 2 =-17a?.
11. 4i 2 -23a;=72.
12. 3x> 2x = 65.
13. 3 x 2 + x = 4.
14. x^l-x 2 .
* 25. (o?-3)(
27. (3-4)( 2) = 5.
16. 9aj 18 = 7.
17. 2 a? 8 11 a; = 30.
18. x + = 12.
a;
x
20. + 5 = 4 a;.
a; "" x 2
22 84 -^
/O A. JL
1 11 or 16 ^ r
^ 2) = 3 (5 a? 26).
29. (a + !) + ( + 2)' -61.
184
ELEMENTS OF ALGEBRA
30.
31.
32.
33.
34.
35.
36.
37.
BS~:P
40 9 7
00 ^ U 1 ^' 13
# o #
8# r 2#
#
X 1
39> #4-2 -30
40. a+I-lL
# 4
At X \ ^ "^ ...... 91
#4-60 3# 5
* + 3 #
#-7 1
9 -a a; " 2<
d2. 48 165 5.
#4-9 #
#4-11 2 #4-1
#4-3 #4-5
x+1 2#-l
x + 3 * + 10
A* 10 12 o
#4-4 #4-8
5#-l 3#-1
a; 3 x + 1
[ 2 + r 1
45
46.
4.
2 x 3 x
= 12.
la
48. o^ 3 ax == 4 a 9 .
49. or -}- 7 wr = 8 r/io?.
=0.
231. Solution by formula. Every quadratic equation can be
reduced to the general form,
ao; 2 -\-bx-\- c = 0.
Solving this equation by the method of the preceding
article, we obtain
2a
The roots of any quadratic equation may be obtained by
substituting the values of a, 6, and c in the general answer.
QUADRATIC EQUATIONS
185
Ex. 1. Solve 5 x 2 = 26 x-5.
Transposing,.
Hence
Therefore
5 x 2 20 x 4- 5 = 0.
a = 5, b = - 26, c = 5.
+ 20 V^tT)* - 4 . 6 . 6
10
== 2024 =6or l.
10 6
Ex. 2. Solve j>o? 2 p*x x p.
Reducing to general form, px*
Hence
Therefore
a = p t b = (p 2 + 1), c p.
P 2 + 1 VQ^+ T? ^4^ -
EXERCISE 86
Solve by the above formula :
1. 2or } -5o; + 2 = 0.
2. 3 x 2 -11 a? + 10 = 0.
3. 2# 2 11 a; + 15 = 0.
4.
6.
6.
7.
8.
9.
10.
= 64-120?.
11. 21 = 44 x - 15 x 9 .
12. 25x* =
13.
14. 7 x 2 = 12 -
15. 6^+5^
16. 7^ + 9 x
17. a; 2 6m
18. a; 2 + 12 TIO; = 64 ?i 2 .
19.
20.
25 x.
56.
90.
= 7 m 2 .
21. o; 2 -
186 ELEMENTS OF ALGEBRA
Find the roots of the following equations to two decimal
places :
22. x 2 = 1 - x. 26. x(x - 4) = - 2.
23. 3x?+x = 7. 27 <2 x== 2S-3x 2 .
24. ar>-8o; = 14. 1
4-2a; 28 - 7a-l=--
25. a=:i^-^. 7s
a?
232. Solution by factoring. Let it be required to solve the
e( l uation: 5^ + 5=26*;
or, transposing all terms to one member,
Eesolving into factors,
(5 a? -!)(- 5) =0.
Now, if either of the factors Bx 1, or # 5 is zero, the prod-
uct is zero. Therefore the equation will be satisfied if x has
such a value that either _,
5 x 1 = 0, (1)
or a?- 5 = 0. (2)
Solving (1) and (2), we obtain the roots
x = ^ or x = 5.
233. Evidently this method can be applied to equations of
any degree, if one member of the equation is zero and the other
member can be factored.
Ex. 1. Solve a *= 7a ? + 15x .
Clearing for fractions, 2 2* = 7 se 2 + 16 x.
Transposing, 2a^7x 2 --16rc = 0.
Factoring, sc(2 x + 3) (x 5) = 0.
Therefore a = 0, 2x-f3=0, orz 5 = 0.
Hence the equation has three roots, 0, }, and 6.
QUADRATIC EQUATIONS 187
Ex. 2. Solve x? - 3 x 2 4 x + 12 = 0.
Factoring, x*(x - 3) - 4 (x 3) = 0.
O 2 -4)(z-3) = 0.
Or (*-2)(x + 2)(a-3)=0.
Hence the roots are 2, 2, 3.
234. If both members of an equation are divided by an
expression involving the unknown quantity, the resulting
equation contains fewer roots than the original one. In order
to obtain all roots of the original equation, such a common
divisor must be made equal to zero, and the equation thus
formed be solved. E.g. let it be required to solve
If we divide both members by x 3, we obtain x 4- 3 = 5
or x = 2. But evidently the value x = 3 obtained from the
equation x 3 = is also a root, for a: 2 9 5 (x 3) = 0, or
(x - 3)(x + 3 5) = 0. Therefore x = 3 or x = 2.
Ex. 3. Form an equation whose roots are 4 and 6.
The equation is evidently (x - 4)(x - (- 6)) = 0.
I.e. (aj-4)(a; + 6)=0.
Or x 2 -f 2 x - 24 = 0.
EXERCISE 87
Solve by factoring :
-|-6 = 0. 9. 2o 3 -f9a; 5 = 0.
2. 0^ + 21 = 10 a?. 10. 3^ 25^ + 28 = 0.
3. ar'-Sa^ -12. 11. or 3 + 9 a; 2 -f 20 x = 0.
4. a* 10a=24. 12. 4or } + 18a 2 -f 8a;:=0.
5. 0^ + 100;= 24. 13. 3# 2 y 5 = 0.
6. ar> + 10 a = 24. 14. 3^ = 0(110-6).
7. a?-10a=:-24. 15. 0(0-2) = 7(0-2).
8. aj( + 8=s: 7. 16. (5
188 ELEMENTS OF ALGEKRA
17. tt(3tt f + 7tt)=6tt. 19. w(w 2 w)=6tt.
18. u z + u 2. 20. x 2 a 2 =(x a)b.
21. (a + 1) (a- 3) = (s + l) (3 -a).
22. (2a? 3) (a + 2)= (+ 3)(a?+2). 23. or 3 - a 2 - 2 a?
24. (y-
25. ( ? j_
26. ara + ft + c*. 27.
Solve :
28. (3 a 11) (2 x 10) = 2
oq ^ |. 3
39 f> 5 6
4 or
OA />. 1 .
x 6 07 4 a; 3
1 9 Q
/I A * 1 w
31 * 21 1
V " "^ it
x 1 x 2 x 6
11 1 i 2 c -
' 100 25 x 4
_ 15 72-6* 9
,. i .,
a; 2 cc 1 a;
40 3 a + 4 ^ ^ j 3
32. . j.
a; 2 or
33 * 4._?1-_1.
' 100 ^25 a? 4
o/i ^ i #+ 1 91
43 tta/ 2 1 fti; ex
AA ^ , ft
"' a, -6 ' x - a ~"'
A 5 ?/ + a ,.V-a -01
a; + l a;
36 5 1 6 o
ie , ft i
3B - -,. + 3 ' ,, + 4 -
,. 4 5 ,
ID. r -i..
O3~a a; ft
3a; 5 2 + l
3 L 2 K
37 *-5 + *-3 5 "
_ 2-6 . 24 x+2
ft + a; a + a; 2
50. '-3a! J -
QUADRATIC EQUATIONS 189
Form the equations whose roots are :
51. 3,1. 53. -2, -5. 55. 1, -2,3. 57. 1,2,3.
52. 3, -4. 54. 0,9. 56. -2,3,0. 58. 2,0, -2.
PROBLEMS INVOLVING QUADRATICS
235. Problems involving quadratics have in general two
answers, but frequently the conditions of the problem exclude
negative or fractional answers, and consequently many prob-
lems of this type have only one solution.
EXERCISE 88
1. A number increased by three times its reciprocal equals
6J. Find the number.
2. Divide CO into two parts whose product is 875.
3. The difference of two numbers is 4, and the difference
of their reciprocals is |. Find the numbers.
4. Find two numbers whose product is 288, and whose sum
is 36.
5. The sum of the squares of two consecutive numbers is
85. What are the numbers ?
6. The product of two consecutive numbers is 210. Find
the numbers.
7. Find a number which exceeds its square by -|.
8. Find two numbers whose difference is G, and whose
product is 40.
9. Twenty-nine times a number exceeds the square of the
number by 190. Find the number.
10. The sides of a rectangle differ by 9 inches, and its area
equals 190 square inches. Find the sides.
11. A rectangular field has an area of 8400 square feet and
a perimeter of 380 feet. Find the dimensions of the field.
190 ELEMENTS OF ALGEBRA
12. The length AB of a rectangle, ABCD, exceeds its widtK
1 B AD by 119 feet, and the line BD joining
two opposite vertices (called "diagonal")
. c equals 221 feet. Find AB and AD.
13. The diagonal of a rectangle is to the length of the rec-
tangle as 5 : 4, and the area of the figure is 96 square inches.
Find the sides of the rectangle.
14. A man sold a watch for $ 24, and lost as many per cent
as the watch cost dollars. Find the cost of the watch.
15. A man sold a watch for $ 21, and lost as many per cent
as the watch cost dollars. Find the cost of the watch.
16. A man sold a horse for $144, and gained as many per
cent as the horse cost dollars. Find the cost of the horse.
17. Two steamers ply between the same two ports, a distance
of 420 miles. One steamer travels half a mile faster than the
other, and is two hours less on the journey. At what rates do
the steamers travel ?
18. If a train had traveled 10 miles an hour faster, it would
have needed two hours less to travel 120 miles. Find the rate
of the train.
19. Two vessels, one of which sails two miles per hour faster
than the other, start together on voyages of 1152 and 720 miles
respectively, and the slower reaches its destination one day
before the other. How many miles per hour did the faster
vessel sail ?
20. A man bought a certain number of apples for $ 2.10. If
he had paid 2 ^ more for each apple, he would have received
12 apples less for the same money. What did he pay for each
apple ?
21. A man bought a certain number of horses for $1200.
If he had paid $ 20 less for each horse, he would have received
two horses more for the same money. What did he pay for
each horse ?
QUADRATIC EQUATIONS 191
22. On the prolongation of a line AC, 23 inches long, a point
B is taken, so that the rectangle, con-
structed with AB and CB as sides, contains - - B
78 square inches. Find AB and CB.
23. A rectangular grass plot, 30 feet long and 20 feet wide,
is surrounded by a walk of uniform width. If the area of the
walk is equal to the area of the plot, how wide is the walk ?
24. A circular basin is surrounded by a path 5 feet wide,
and the area of the path is - of the area of the basin. Find
the radius of the basin. (Area of a circle = TT r 2 .)
25. A needs 8 days more than B to do a certain piece of
work, and working together, the two men can do it in 3 days.
In how many days can B do the work ?
26. Find the side of an equilateral triangle whose altitude
equals 3 inches.
27. The number of eggs which can be bought for $ 1 is
equal to the number of cents which 4 eggs cost. How many
eggs can be bought for $ 1 ?
EQUATIONS IN THE QUADRATIC FORM
236. An equation is said to be in the quadratic form if it
contains only two unknown terms, and the unknown factor of
one of these terms is the square of the unknown factor of the
other, as
0, ^-3^ = 7, (tf- I) 2 -4(aj*-l) = 9.
237. Equations in the quadratic form can be solved by the
methods used for quadratics.
Ex. 1. Solve ^-9^ + 8 = 0.
By formula, ** = 9
Therefore x = \/8 = 2, or x = \/l = 1.
192
ELEMENTS OF ALGEBEA
238. In more complex examples it is advantageous to sub
stitute a letter for an expression involving a?.
Ex. 2.
Let
Then
x J
+ 15 = <
or
or
Hence
Le.
Solving,
r-f 15 = 0,
y-8)=0.
>, or y = 8.
= 1,
6.
6.
4 -8 = 2 a*
EXERCISE 89
Solve the following equations :
1. a; 4 -10a; 2 -h9:=0. 3. a 4 -5o; 2 =-4.
2. a; 4 4-36 = 13.T 2 . 4. a 4 -21or=100.
7. 3 a 4 -44s 2 + 121=0. 9. 4 4 -37aj 2 = -9.
8. 16 a^-40 aV+9o 4 =0. 10. (a: 2 +aj) 2 -18(x 2 +a;)+72=0,
11. (^-Z) 2 -
12.
3 15
"
14.
1=2*.
a? T
16. ^^
~ 28
17. (a?-
18.
19.
2:=Q>
^
QUADRATIC EQUATIONS 193
CHARACTER OF THE ROOTS
239. The quadratic equation oa/* 2 -f- bx -f- c = has two roots,
(
2a 2a
Hence it follows :
1. If b 2 4 ac is a positive or equal to zero, the roots are real.
Ifb* 4c is negative, the roots are imaginary.
2. If b 2 4 ac is a perfect square, the roots are rational.
Iflr kac is 'not a perfect square, the roots are irrational.
3. Ifb 2 4 ac is zero, the roots are equal.
Jfb 2 4ac is not zero, the roots are unequal.
240. The expression b 2 4 ac is called the discriminant of
the equation ay? 4- bx 4- c = 0.
Ex. 1. Determine the character of the roots of the equation
3 a 2 - 2 z - f> = 0.
The discriminant =(- 2) 2 4 . 3 (- 5) = 04.
Hence the roots are real, rational, and unequal.
Ex. 2. Determine the character of the roots of the equation
4 x 2 - 12 x + 9 = 0.
Since ( 12) 2 4 4 9 = 0, the roots are real, rational, and equal.
241. Relations between roots and coefficients. If the roots of
the equation ax 2 4- bx 4- c are denoted by i\ and r 2 , then
__ b 4- Vfr 2 4 ac
Tl T* '
b Vi 2 4 ac
2a
Hence / 1 4-r 2 = ,
a
Or
194 ELEMENTS OF ALGEBRA
If the given equation is written in the form a? 2 + -x + - = 0,
these results may be expressed as follows : a a
If the coefficient ofx 2 in a quadratic equation is unity,
(a) The sum of the roots is equal to the coefficient of x with the
sign changed.
(b) The product of the roots is equal to theubsolute term,
E.g. the sain of the roots of 4 x 2 -f 5 x 3 =: is j, their product
is-f.
EXERCISE 89 a
Determine without solution the character of the roots of the
following equations :
1. o; 2 -lla; + 18 = 0. 8. 5aj 2 + 2-a;.
2. 5a 2 -26a? + 5 = 0. 9. x 2 -7 = 5x.
3. 2x* + 6x + 3 = 0. 10. 12~x = x 2 .
4. or + 10 a; + 4520 = 0. n a?-3 == l
5. ^-12. ' ~
6. 3a; 2 + 4a: + 240 = 0. 12. 10 x = 25 x 2 + 1.
7. 9x 2 ~
In each of the following equations determine by inspection
the sum and the product of the roots:
13. x 2 -!i>x + 2 = Q. 16. Sa^ + Ooj 2 = 0.
14. z 2 -9a-3 = 0. 17. tfmx+p^Q.
15. 2a 2 -4z-5 = 0. 18. 5oj 2 -aj + l = 0.
Solve the following equations and check the answers by
forming the sum and the product of the roots :
19. a 2 - 19 # + 60 = 0. 22. x 2 -4 x 12 = 0.
20. ^ + 2^-2 = 0. 23. 0^ + 205 + 2 = 0.
21. ar 2 + 2a-15 = 0. 24. or j -
CHAPTER XVI
THE THEORY OF EXPONENTS
242. The following four fundamental laws for positive integral
exponents have been developed in preceding chapters :
I. a m a" = a m+t1 .
II. a m -f- a" = a m ~ n , provided w > n.*
III. (a m ) a s=a mn .
IV. (ab) m = a w b m .
The first of these laws is the direct consequence of the defi-
nition of power, while the second and third are consequences
of the first.
FRACTIONAL AND NEGATIVE EXPONENTS
243. Fractional and negative exponents, such as 2*, 4~ 3 , have
no meaning according to the original definition of power, and
we may choose for such symbols any definition that is con-
venient for other work.
It is, however, very important that all exponents should be
governed by the same laws; hence, instead of giving a formal
definition of fractional and negative exponents, we let these
quantities be what they must be if the exponent law of mul-
tiplication is generally true.
244. We assume, therefore, that a m a n = a m+n , for all values
of m and n. Then the law of involution, (a m ) w = a"" 1 , must be
*The symbol > means "is greater than" ; similarly < means "is
smaller than."
195
196 ELEMENTS OF ALGEBRA
true for positive integral values of n, since the raising to a
positive integral power is only a repeated multiplication.
Assuming these two laws, we try to discover the meaning of
8*, a, 4~ 2 , a n , etc. In every case we let the unknown quantity
equal x, and apply to both members of the equation that opera-
tion which makes the negative, fractional, or zero exponent
disappear.
245. To find the meaning of a fractional exponent; e.g. at.
Let x a*.
The operation which makes the fractional exponent disappear
is evidently the raising of both members to the third power.
Hence ^=(a^) 3 .
Or 3* = a.
Therefore 0?=-^.
-
Similarly, we find a?
Hence we define a* to be the qth root of of.
EXERCISE 90
Find the values of:
1. 9*.
8. 8l
15. 16*.
2. 4*.
9. 4*.
16. O^r.
3. 16*.
10. 27*.
17. ft).
4. 8*.
11. 125*.
18. ()*.
5. 27*.
12. 32*.
19. (J)f.
6. 64*.
13. 1.
20. (w a -
7. 32*.
14. (-8)*.
21. (0*4-
Write the following expressions as radicals :
22. m$. 24. a\ 26. (xy$. 28. (bed)*. 30. '&M
23. a?*. 25. A 27. 3*. 29. as. 31. ml.
THE THEORY OF EXPONENTS 197
Express with fractional exponents :
32. -\fi?. 33. -v/o&cT 34. ty?. 35. v'mT
36. Vo 5 . 37. -\/xy- 38. -\/m.\/n.
Solve the following equations :
39. 2' = 4. 41. a* = 3. 43. 3* = 27. 45. 5 a* = 10.
40. 4* = 2. 42. * = 2. 44. 27* = 3. 46. 7z* = 49.
Find the values of :
47. 4* + 9* + 16* + 25* + 36*.
48.
49. 64* + 9* + 16* + (-32)*.
50.
246. To find the meaning of zero exponent, e.g. a.
Let a = a.
The operation which makes the zero exponent disappear is
evidently a multiplication by any power of a, e.g. a 2 *
a 2
Or a=l.
Therefore the zero power of any number is equal to unity.
NOTE. If, however, the base is zero, 5L is indeterminate ; hence is
Indeterminate. a
198 ELEMENTS OF ALGEBRA
247. To find the meaning of a negative exponent, e.g. cr n .
Let x = or".
Multiplying both members by a", a n x = a.
Or a"# = l.
Hence
x = .
a"
Therefore
--=!.
a"
248. Factors may be transferred from the numerator to the
denominator of a fraction, or vice versa, by changing the sign of
the exponent.
NOTE. The fact that a = 1 sometimes appears peculiar to beginners.
It loses its singularity if we consider the following equations, in which
each is obtained from the preceding one by dividing both members by a.
a 8 = 1 a a a
a 2 = 1 . a a
a 1 = 1 a
a- 2 = , etc.
a 2
Find the values of:
THE THEORY OF EXPONENTS 199
EXERCISE 91
1.
7- 2 .
8.
15- 2 .
13.
(I)' 2 -
2.
fi- 1 .
9.
217- 1 .
14,
, 4-*.
3.
3~ 3 .
10.
1
15.
, 8-*
4.
I- 9 .
4-1'
16.
64-*.
5.
199.
1
17.
0)~ 5 .
11.
\ 2/
6.
2~ 5 .
2~s*
1
18.
7.
(TT)-
12.
(I)- 1 -
7~ 2
19.
625"^.
20
>. 12 -5- 1
-25*.
Express with positive exponents :
21. or 5 .
25. ^-^ ^. 27.
22. 6 or 2 . a;- 3
24. 7~ l a 2 b 2 . * ""^T"*'
Write without denominators :
29. * 31 <W*
l> ' arV 8
30. ^L. 32. ^?- 34.
c y' 2
Write with radical signs and positive exponents :
35. mi 40. (2w)~i a^
44.
36. m~^. f 1 66
41. -.
37. 3 cci 2 a;"* 1
, . 45. -L
38. 3 a? * . 42. m 2 . ?>i""i
39. 2m~i 43. rfS.
200 ELEMENTS OF ALGEBRA
Solve the equations :
46. x~ l = l. 50. 17' = 1. 54. 10* = .1.
47. ar 2 = i. 51. z* = 5. 55. 5* = -^.
48. 2 z = f 52. 5or*=10.
49. 3* = f 53. 10* = .001.
Find the values of:
56.
57. 3-ll-
58. 4~* + 1~* - (I-) 1 -f 21 - 9*.
59. (81)* + (3f)*-(5 T V)*-32- 2 .
60. 49 5 + 16 * - 81 - 75 -f (a - 6).
61. (.343)* + (.26)* - (.008)* + A. + A_.
USE OF NEGATIVE AND FRACTIONAL EXPONENTS
249. It can be demonstrated that the last three laws for any
exponents are consequences of the first law, and we shall hence
assume that all four laws are generally true. It then follows
that:
Fractional and negative exponents may be treated by the same
methods as positive integral exponents.
250. Examples relating to roots can be reduced to examples con-
taining fractional exponents.
Ex. 1. (a*&~*)* + (aVM = a*&~* + V ' = '*&*
Ex 2
THE THEORY OF EXPONENTS
201
251. Expressions containing radicals should be simplified as
follows :
(a) Write all radical signs as fractional exponents.
(6) Perform the operation indicated.
(c) Remove the negative exponents.
(d) If required, remove the fractional exponents.
NOTE. Negative exponents should not be removed until all operations
of multiplication^ division, etc., are performed.
Simplify :
EXERCISE 92
2. &.&.&.$-".$-*.
3. 7 2 . 7 9 . 7~ 5 . 7~ 6 .
4 2 5 2 6 2 7 2~ 8 2~ 9
5. a- 3 - a- 4 - a 8 .
6. aj" 3 a- 4 2 a? 2 ar 1 .
7. 6a-.5a.
8 ' 6 *- 6 *' 6 *-
9. 7*.7i.7*.7W.
. , 4
10. #* a; ' x^.
11. V5.^/5-^5.
12. 9 5 -^9i
13. 5-*-*- 5.
14. S-'-s-S- 8 .
16. a 9 -i-a- 4 .
16. 14an-
17. (4**-
18. (Va) 4 .
OA
20.
22. 3-s-VS.
23. 7-f--v / 7.
25.
26.
27 -
28.
/-
__ V m
29- '-= --
202
ELEMENTS OF ALGEBRA
32.
33.
34.
35.
4/ - 3
V ra
-\/m 6
40.
252. If we wish to arrange terms according to descending
powers of a?, we have to remember that, the term which does
not contain x may be considered as a term containing #. The
powers of x arranged are :
Ex. 1. Multiply 3 or 1 + x 5 by 2 x 1.
Arrange in descending powers of x.
Check. lix =
2x-l
=+1
Ex. 2. Divide
2 a
by
^ 3 qfo 4- 2 d
THE THEORY OF EXPONENTS 203
EXERCISE 93
Perform the operations indicated:
2.
3.
4. (7r-8Vr + r>)(9 Vr-7).
5. (a- 2 + a 2 -f 1) (a~ 2 + a 2 - 1 ).
6.
7.
8.
9.
10.
11. (4 a- 3 - 24 a- 1 - 9 - 3 a~ 2 ) -r- (a" 1 - 3).
12. (13Vp + l + 47i) + 35V5?)-*-(5Vp + l).
13. (Va^-f aV^-&Va VS" 3 ) ^- ( Vo Vft)
14. (3 a~ 5 + 1C a-*b~ l - 33 a- 3 6~ 2 + 14 a- 2 ^>~ 3 ) H- (a~ 2 -f 7 a- 1 ^- 1
15. ( a _&)-*. (-^? + ^/-^ + */fr^
16. (a-6 + 2V6c c)-^-(Va+V6 Vc).
17. -y^TTOa; -f 13 - 12 *- 1 + 4 aF*.
18. Vor 2 2 x -h or 2 or 1 -f- 3.
19. V25 # 2 - 2()"ar r + 34 - 12 x -f 9 x*.
20. ^^
21. l + 2
22. (l+4^-flO^ + 20oT-f 25^ T -f-24-\/i?-f 16 a? 8 )*.
23. (1+V2)V2. 26. (1-3VS)(2 + V5).
24. (2+V2)(V2-2). 27. (VU - V2)(Vn~3V2)
25. (5+V3)(5-2V3).
204 ELEMENTS OF ALGEBRA
Find by inspection :
28. (x* + 3)(tf*-f 2). 35. (a;* yi) 8 .
29. a* + 3l-5. 36. (5*-2* 2 .
30.
31.
V- 38.
32. (3^ 2*) 2 .
39.
33. (#* -f- 5) (x* 5). 11
^ 40. (m n) -f- (m* -f- n 5 ).
34. (fl
CHAPTER XVII
RADICALS
253. A radical is the root of a quantity, indicated by a
radical sign.
254. The radical is rational, if the root can be extracted
exactly; irrational, if the root cannot be exactly obtained.
Irrational quantities are frequently called surds.
^9 (* + V) * are radicals.
4^ = 2, V(a + 6) 2 are rational.
\/2, V4a-f b are irrational.
255. The order of a surd is indicated by the index of the
root. /- .
v a is of the second order, or quadratic.
\/2 is of the third order, or cubic.
Vc is of the fourth order, or biquadratic.
256. A mixed surd is the product of a rational factor and a
surd factor; as 3Va, a;V3. The rational factor of a mixed
surd is called the coefficient of the surd.
An entire surd is one whose coefficient is unity; as Va,
257. Similar surds are surds which contain the same irrational
factor. 3v/ 2 and 6 av ^ ar e similar.
3V2 and 3 V8 are dissimilar.
206
206 ELEMENTS OF ALGEBRA
258. Conventional restriction of the signs of roots.
All even roots may be positive or negative,
e.g. VI = + 2 or 2.
Hence
which results in four values, viz. 14, 6, 14, or 6. To avoid
this ambiguity, it is customary in elementary algebra to restrict
the sign of a root to the prefixed sign.
Thus 5 V4 4- 2 V4 = 7 VI = 14.
If the object of an example, however, is merely an evolution,
the complete answer is usually given ; thus
=- (oj- 2).
259. Since radicals can be written as powers with fractional
exponents, all examines relating to radicals may be solved by the
methods employed for fractional exponents.
Thus, to find the nth root of a product ab we have
T 1 1
(a6)"==a"6" (242).
I.e. to extract the root of a product, multiply the roots of the
factors.
TRANSFORMATION OF RADICALS
260. Simplification of surds. A radical is simplified when the
expression under the radical sign is integral, and contains no
factor whose power is equal to the index.
Ex. 1. Simplify
= \/25~a~ 4 Vb = 6 a*VS.
Ex. 2. Simplify -v/16.
-J/lB^^. 4/2 = 2^.
RADICALS
207
261 . When the quantity under the radical sign is a fraction, we
multiply both numerator and denominator by such a quantity
as will make the denominator a perfect power of the same
degree as the surd.
Ex. 3. Simplify V|.
Ex. 4. Simplify
EXERCISE 94
Simplify :
1.
V27.
9.
vW. 17.
|V45?.
2.
V45.
10.
VcW. 18.
-\/'l6.
3.
V32.
11.
V28 al 19.
A/54.
4.
V28.
12.
A/320 a 2 6 2 . 20.
A/IOOO a 4 b.
5.
V24.
13.
V8 a :J 6 2 . 21.
A/250 7 6 fi .
6.
V243-
14.
6V80P. 22.
^48^5*.
7.
V363.
15.
5 V40 a 2 c. 23.
S^-108^
8.
Vor\
16.
7v / 48a li ar 3 . 24.
V2(a + &) :
25.
V3a 2 + 6a?
> + 3 6*.
26. (125a;y)i
27
/f
/o"2l2
/iy"2
A 1 *
v 2".
31.
\/ . 34.
^ 2 -y/ w t
28.
vT-
c
i 8
fo
(x 35.
^"i"
29.
\5T
32.
oc
3 L
nf\
to a 2 i>*
oo
/ 2 00.
\T"*
208
ELEMENTS OF ALGEBRA
37.
38.
39.
/s j
*x+y
n \ 2m
262. An imaginary surd can be simplified in precisely the
same manner as a real surd ; thus,
,
42. V-16a 2 . 44. 2\-
Simplify and find to three decimal places the numerical
values of :
47. VJ.* 49. Vf.
48. VJ. 50. VA
263. Reduction of a surd to an entire surd.
Ex. Express 4 a V& as an entire surd.
EXERCISE 95
Express as entire surds :
1. 4V5. 3. 2-\/lL 5.
2. 3V7. 4. 3^5. 6. a VS.
7.
8.
* See table of square roots on page 164.
RADICALS 209
264. Transformation of surds to surds of different order.
Ex. 1. Transform -\/uW into a surd of the 20th order.
Ex. 2. Transform \/2, V3, and \/5 into surds of the same
lowest order. V 2 = 2* = a* = '#64.
|^ = 8* = 3A= 1 ^gi.
^5 = 6* = 6* =^125.
Ex. 3. Reduce the order of the surd tyaP.
Exponent and index bear the same relation as numerator and
denominator of a fraction ; and hence both may be multiplied by
the same number, or both divided by the same number, without
changing the value of the radical.
EXERCISE 96
Reduce to surds of the 6th order :
1. Va?. 2. -fymn. 3. \/ 4. v'c?. 5. \| ^- 6. mn.
v z \ 3
Reduce to surds of the 12th order :
7. V2~a. 9. \/a 4 6 2 c. 11. -\/oP6. 13.
8. ^v/mV 10. -\/3ax. 12. \/5a5V. 14. a.
Express as surds of lowest order with integral exponents
and indices :
15. -v/o 5 . 16. \/oW. 17. -v/IaT 2 . 18. -\/
20. A/^ 22. VSlmV. 24.
210 ELEMENTS OF ALGEBRA
Express as surds of the same lowest order :
25. V3, </2. 29. 2\ 3*. 32.
26. A/2, s!/3. 30. V2, A/3, ^5. 33. V3, </3, -^4.
27. -v/3, ^2. 31. -v^S, -\/5, -^7. 34. ^2, ^4, </20.
28. -\/7, V2.
Arrange in order of magnitude :
35. -v/3, V2. 37. \/7, VS. 39. 5V2, 4^/4.
36. -v/4, -^6. 38. V5, ^/IT, ^126. 40. -^2, ^3, ^30.
ADDITION AND SUBTRACTION OF RADICALS
265. To add or subtract surds, reduce them to their simplest
form. If the resulting surds are similar, add them like similar
terms (i.e. add their coefficients) ; if dissimilar, connect them by
their proper signs.
Ex. 1. Simplify V| + 3 VlS- 2 V50.
VJ + 3VT8 - 2 V50 = I V2 + 9 V2 - 10 V2 = - V2.
Ex.2. Simplify/a 3 5
~ v~ 8ft 2 s/- 3 s/- / 3ft 2 3
- - -
o , 3:
- 3-\| . + \/=^-
^y 8 a;
Ex. 3. Simplify V|~
RADICALS
211
EXERCISE 97
Simplify the following expressions :
2.
3.
4.
6.
6.
7.
8.
9.
10.
11.
12.
13.
14.
2V8-7Vl8-f5V72-V50.
VT2 + 2V27 + 3V75-9V48.
V18+V32-VT28+V2.
V175-V28+V63-4V7.
VJ+V8-V1 + V50.
4V80-5V45-.3V20 + 6V5.
8VT8-J-2V32 + 3V8-35V2.
V45c 3 V80~c~ 3 -f V5a 2 c + c
3 abv'ab + 3 aVo^ 3 Va^ ;J a6 V4 aft.
212
ELEMENTS OF ALGEBRA
.
23.
98 ab
fab .fab jab FW
^" 1 " V \~\
MULTIPLICATION QEJRABIQALS
266. Surds of the same order are multiplied by multiplying
the product of the coefficients by the product of the irrational
factors, for a~\/x b~\/y ab^/xy.
Dissimilar surds are reduced to surds of the same order, and
then multiplied.
Ex. 1. Multiply 3-\/25^ by 5\/50Y 2 .
3 v / 26^ . 5 4/6072 = 16^6272. 6*. y* =
Ex. 2. Multiply V2 by 3\/l.
Ex. 3. Multiply 5 V7 - 2 VS by 3 Vf + 10 VB.
6\/7- 2v/6
8\/7 + IPV6
105- 6V35
4-60V35-100
106 -f 44 VS6 - 100 = 6 + 44\/36.
RADICALS
213
1. V3
2. V2
3. V3
4. V
5. Vr
16.
17.
18.
19.
20.
21.
Vl2.
-V50.
V6.
-VTO.
V42.
EXERCISE 98
6. VlO V15.
7. Vll.VSS.
8. V20 V30.
9. -v/4.^/2.
10. -\/3 -\^).
aVa; 6 V4 a?.
V2a-V8^.
11. -v/18 -v"3.
12. V5
13. Va
14. Va-
15. V?/
fWa
25. (V2+V3+V4)V3.
27. (5V2-2V3-CVS)V3.
28 . (3 + VB)(2-V5).
40
10
30. (Vm-Vn)(Vm+Vn>
33. ( Vm -\- 1 Vm) (Vm-f 1 -{- Vm).
34. (Va Va 6(Va-f Va 6.
36. (6V2-3V3)(6V2-|-3V3).
37. (5V5-8V2)(5V5 + 8V2).
38. (Vm-Vn) 8 . 40. (V6 + 1) 1 .
39. (V3-V2) 2 . 41. (2-V3) 8 .
214 ELEMENTS OF ALGEHRA
42. (3V5-5V3) S . 43.
44. (3V3-2Vo)(2V3+V5).
45. (2 V3 - V5) ( V3 + 2 VS).
46. (5V7-2V2)(2VT-7V2).
47. (5V2+V10)(2V5-1).
48.
49. (3V5-2V3)(2V3-V3).
60.
51.
52. Va -v/a. 53. -v/a -
DIVISION OF RADICALS
267. Monomial surdn of the same order may be divided by
multiplying the quotient of the coefficients by the quotient of the
surd factors. E.y. a VS -f- a?Vy = -\/ -
x*y
Since surds of different orders can be reduced to surds of
the same order, all monomial surds may be divided by this
method.
Ex. 1
Ex. 2. (V50-f 3Vl2)-4-V2==
268. If, however, the quotient of the surds is a fraction, it
is more convenient to multiply dividend and divisor by a
factor which makes the divisor rational.
RADICALS 215
This method, called rationalizing the divisor, is illustrated by
the following examples :
Ex. 1. Divide VII by v7.
In order to make the divisor (V?) rational, we have to multiply
by V7.
VTL_Vll \/7_V77 , /~
' ~~" }
Ex. 2. Divide 4 v^a by
The rationalizing factor is evidently \/Tb ; hence,
4\/3~a'
36
Ex. 3. Divide 12 V5 + 4V5 by V.
Since \/8 = 2 V*2, the rationalizing factor is \/2,
12 Vil + 4\/5 _ 12v x 3 + 4\/5 g V2
V8 V8 ' V2
269. To show that expressions with rational denominators
are simpler than those with irrational denominators, arith-
metical problems afford the best illustrations. To find, e.g.,
- by the usual arithmetical method, we have
V3 1.73205
But if we simplify JL-V^l
V3 *> ^>
Either quotient equals .57735. Evidently, however, the
division by 3 is much easier to perform than the division by
1.73205. Hence in arithmetical work it is always best to
rationalize the denominators before dividing.
216 ELEMENTS OF ALGEBRA
EXERCISE 99
Simplify :
1. ^ /H . 7. V8?^ T 13 11 n
V7 V7 xy VH
5
~ - 8. Vf-f-V?. 14. -2-.
Vn 2V5
V7 * ' 2 V3
o
' vfi* ^
Vll ' Va
212*. 12. --.
Given V2 = 1.4142, V3 = 1.7320, and V5 = 2.2361, find to
four decimal places the numerical values of:
19. -i. 20. A. 21.
V2 V3 V8
22 . 12.. 23 . A. 24 . JL. 25. 4=-
V5 V8 V48 V50
270. Two binomial quadratic surds are said to be conjugate,
if they differ only in the sign which connects their terms.
Va + Vb and Va Vb are conjugate surds.
271. The product of two conjugate binomial surds is rational
, 272. To rationalize the denominator of a fraction whose denom-
inator is a binomial quadratic surd, multiply numerator and
denominator by the conjugate surd of the denominator.
Ex. 1. Simplify
RADICALS 217
2V3-V2
V3-V2 '
~ = 4 + V5.
Ex.2. Simplify s
a; - vffi^T _ a; - Vs 2 - 1 x-Vtf
Ex. 3. Find the numerical value of :
V2 + 2 ,
2V2-1
V2+2 _ V2+2 e 2\/2+l_6 + 6\/2. = 18.07105 =
2V2-1 2V2-1 2V2 + 1 7 7
EXERCISE 100
Eationalize the denominators of :
x 1
B ^
B 14
2+V3
3
2V5-3V2
fl 1
8-5V2
10 12
1+V2
3
V2 + V3
7
7-3V5
~' 5-V21
A
" V7-V2
' 2-V2
. .
V8-2 2-V3 1-fVS
218 ELEMENTS OF ALGEBRA
13 . 6 ~ 3 A 16. 6V7-.W3. 19.
14
V5-1 V5-2 1-Va?
Vg+v/2 17 5V7-7V5
V3-V2 ' V5-V7
15. ^-SVg. 18>
2V5-V18 m-Vm
Va
22.
Given V2 = 1.4142, V3 = 1.7320, and V5 = 2.2361; find to
four places of decimals :
23 . _!_. 25 . -J?_. 27.
V2-1 Vo-1 2-V3
24. -= 26. v _ 28.
V3 + 1 1+V5 3-V5
V5+2 ' V3-2*
31. Find the third proportional to 1 + V2 and 3 -f- 2V2.
INVOLUTION AND EVOLUTION OF RADICALS
273. By the use of fractional exponents it can easily be
shown that VcT = ( V) w .
Hence V25~ 3 = ( V25) 3 - 5 3 = 125.
RADICALS 219
274. In other examples of involution and evolution, intro-
duce fractional exponents :
Ex. 1. Simplify
Ex. 2. Find the square of
EXERCISE 101
Simplify :
1. (3Vmw) 2 . 5. V64 3 . 9.
2.
3. (V2~u-) 3 - 7. -\/l6*. 11.
4. V25 5 . 8. \/125" 2 . 12.
SQUARE ROOTS OF QUADRATIC SURDS
275. To find the square root of a binomial square by inspection.
According to G3,
( V5 + V3) 2 = 5 + 2 V5~^3 + 3
= 8 + 2 VIS.
If, on the other hand, we had to find v8-f 2\/15, the
problem would be quite simple if presented in the form
v5-|-2V3 5 + 3. To reduce it to this form, we must find
two numbers whose sum is 8 and whose product is 15, viz.
5 and 3.
220 ELEMENTS OF ALGEBRA
Ex. l. Find Vl2 4- 2 \/20.
Find two numbers whose sum is 12 and whose product is 20. These
numbers are 10 and 2.
Ex. 2. Find Vll - 6 V2.
Write the binomial so that the coefficient of the Irrational term is 2.
^TT- 6 A/2 = Vll - 2 \/18.
Find two numbers whose sum is 11, and whose product is 18. The
numbers are 9 and 2.
Hence ^11 - 6\/2 = ^9 - 2 A/2 + 2
= V9-A/2
= 3 - A/2.
Ex. 3. Find V4 + VJ8.
EXERCISE 102
Extract the square roots of the following binomials :
1. 8 + 2V15.
6. 7-V40.
11. 6-V32.
2. 3-2V2.
7. 2 + VS.
12. 9 + 4V5.
3. 6-2V5.
8. 4 + 2V3.
13. 15-4VU.
4. 11 + 2VI8.
9. 7-V48.
14. 8-T-V55.
'5. 5-V24.
10. 3-V5.
15. 14 + 8V3.
16. a + 6 + 2Va
I. 17. :
2 a 4- b 2 Va(a -f- &)
RADICALS 221
Simplify the following expressions :
18. Vl3-2V22.
19. -+=. 22. * 4-- *
r - 2 V6 VT 4. V48 VT - V48
20. 4 . 23.
V4 + V12
RADICAL EQUATIONS
276. A radical equation is an equation involving an irrational
root of an unknown number.
Vx = 5, -\/x + 3 = 7, (2x xrf 1, are radical equations.
277. Radical equations are rationalized, i.e. they are trans-
formed into rational equations, by raising both members to
equal powers.
Before performing the involution, it is necessary in most
examples to simplify the equation as much as possible, and to
transpose the terms so that one radical stands alone in one
member.
If all radicals do not disappear through the first involution,
the process must be repeated.
Ex.1. Solve vV-f!2-a = 2.
Transposing a;, Vsc 2 + 12 = x -f 2.
Squaring both members, x 2 -f 12 = x a + 4 x -f 4.
Transposing and uniting, 4 x 8.
Dividing by 4, x = 2.
Check. The value x = 2 reduces each member to 2.
222 ELEMENTS OF ALGEBltA
Ex. 2. Solve V4 x + 1 -f V4 a; -f- 25 = 12.
Transpose V4 x -f 1 , Vitf -f 25 = 12 V4afT~l.
Squaring both members, 4 x -f- 25 = 144 24 \/4 #-|-
Transposing and uniting, 24\/4# + 1 = 120.
Dividing by 24, \/4 jc~+~l = 5.
Squaring both members, 4 x -f- 1 25.
Therefore x = 0.
CftecAr. V24~+~l -f V/2TT25 = 5 + 7 = 12.
278. Extraneous roots. Squaring both members of an equa-
tion usually introduces a new root. Thus x 2 = 3 has only
one root, viz. 5.
Squaring both members we obtain or 4#-f 4 = 9, an equa-
tion which has two roots, viz. 5 and 1.
The squaring of both members of the given equation intro-
duced the new root 1, a so-called extraneous root. Since
radical equations require for their solution the squaring of
both members, the roots found are not necessarily roots of
the given equation ; they may be extraneous roots.
279. The results of the solution of radical equations must be
substituted in the (jlren equation to determine ivhether the roots are
true roots or extraneous roots.
Ex. 3. Solve -Vx -f-
Squaring both members,
x + 1 + 2 Vx' 2 + 1 x + (.
Transposing and uniting, 2 Vx^
Dividing by 2, VzM- 7 x -f = 3 x - 3.
Squaring both members, x 2 -f 7 x + = 9 x 2 18 x + 9.
Transposing, 8x 2 25x-f3 = 0.
Factoring, (x 3) (8 x - 1) = 0.
Therefore x = 3, or at = .
Check. It x = J, the first member =|\/2 + -jV2=|v^;
member = V2.
RADICALS 223
Hence x \ does not satisfy the given, equation ; it is an extraneous root.
If x = 3, both members reduce to 5. Hence there is only one root, viz.
a; = 3.
. NOTE. If the signs of the roots were not restricted, x = } would be a
root of the preceding equation, for it satisfies the equation
VaT+T 4- VxT~0 = \/8 x -f 1.
Ex. 4. Solve Vz+T + V2aT+3 = A 5 _
Clearing of fractions, V2x' 2 + "b"x -f 8 4-2x4-3 15.
Transposing, ViTie- + 6~ieT~3 - 12 - 2 r.
Squaring, 2 z 2 4 6 x 4 3 = 144 - 48 x + 4 z 2 .
Transposing, 2 x 2 53 -f 141 = 0.
Factoring, (x - 3) (2 x - 47) = 0.
Therefore, x = 3, or x *j-.
Check. If x = 3, both members reduce to 5.
If x V, tlie Jeft member = 1 2 T V2, and the right member = |V2.
Hence x = 3 is the only root.
= G.
Solve the following equations :
* Exclude all solutions which do not satisfy the equation or which make
the given radicals imaginary.
224
ELEMENTS OF ALGEBRA
280. Many radical equations may be solved by the method
of 238.
Ex. 1. Solve af*- 33 af* + 32=0.
Factoring,
Therefore
RADICALS
225
Raising both members to the | power,
x = 32~* or 1"* = ^ or 1.
Ex. 2. Solve x* 8 x
Adding 40 to both members, x* 8 x -f 40
Let Vz 2 $x + 40 = y, then x 2 - 8 x + 40 =
Hence y' 2 2 y = 35.
y _ 2 y - 35 = 0.
Therefore
- 8 x + 40 = 36,
y = 7, or y = 5.
2 - 8 z-|-40 = 7,
2_8z 4-40 = 49,
x = 9 or 1.
or Vi 2 -8a;-f40= 5.
= 26.
x = 6 or 3.
Since both members of the equation were squared, some of the roots
may be extraneous. Substituting, it will be found that 9 and 1 satisfy
the equation, while 6 and 3 are extraneous roots.
This can be seen without substituting, for 6 and 3 are the roots of the
equation Vx' 2 8 x + 40 = 6. But as the square root is restricted to its
positive values, it cannot be equal to a negative quantity.
EXERCISE 104*
Solve the following equations:
1. x + Vx = 6. 6.
2. a? 2Va; 3 = 0.
3. 4-12a* = 16.
4. 45 14VJB = .
5. o;*-2a;i~24 = 0.
* Exclude extraneous roots and roots which make the given radicals
imaginaries.
Q
226 ELEMENTS OF ALGEBRA
11. or 2 8 a -f- 40 2 V* 2 8. a 4-40 = 35.
12. a^-
13. x 2 + x 4 V SB* 4- a; + 3 = 6.
14. 5 ar ; -fll x 12 V5l? +1 1^7-^30 =
15. 2 ^ 3 x + G V2^"-^I + 2 = 1 4.
16.
17.
18.
19. a; 2 7a?H-V^ 7a;-f 18 = 24.
20. 6 Va?~3o~ 3 = y? 3 x -f 2.
CHAPTER XVIII
THE FACTOR THEOREM
281. If x* - 3 x~ + 4 x + 8 is divided by x - 2 and there is a
remainder (which does not contain a?), then
or* 3 x 2 -f- 4 a; -f 8 = (a? 2) x Quotient -f Remainder.
Or, substituting Q ani ^ ^ respectively for " Quotient " and
" Remainder," and transposing,
R = x* - 3 x 2 + 4 a? + 8 - (a? - 2) Q .
As 72 does not contain a?, we could, if Q was known, assign
to x any value whatsoever and would always obtain the same
answer for R.
If, however, we make a? = 2, then (x 2)Q 0, no matter
what the value of Q. Hence, even if Q is unknown, we can
find the value of R by making x = 2.
# = 2 3 -3- 2 2 + 4- 2 + 8-0 = 12.
Ex. 1. Without actual division, find the remainder obtained
by dividing 3 x* -f- 2 x 5 by x 3.
Let z = 3,
then ^ = 3-81+2.3-6-0 = 244.
Ex. 2. Without actual division, find the remainder when
ax 4 4- bx? + ex 2 4- <fo -f e is divided by x m.
E = ax 4 + &z 8 + ca: 2 -f (to + e (x m) Q.
Let = w,
then R = am* + 6m 3 + cm 2 + tZw + e.
227
228 ELEMENTS OF ALGEBRA
282. The Remainder Theorem. If an integral rational expres-
sion involving x is divided by x m, the remainder is obtained
by substituting in the given expression m in place of x.
E.g. The remainder of the division
(4x 6 - 4x4-11)^0 + 3) is 4 (- 3) 5 - 4(- 3)-f 11 =- 949.
The remainder obtained by dividing
(x + 4)4 _ (3 + 2) ( X - 1) + 7 by x - 1 is 6* - 3 . + 7 = 632.
EXERCISE 105
Without actual division find the remainder obtained by
dividing :
2. x* + 3x 3 -2x* 32x12 by a?-3.
3. x*-x s + 4x 2 -Tx + 2\)y x + 2.
4. a 100 -50 a 47 4- 48 a 2 -}- 2 by a-1.
5. x 5 - b 5 by x - b.
6. a^ + ^by x + b.
7. a 7 -f b 7 by a + 6.
8. ^-14y j ~132/ 2 --
283. If the remainder is zero, the divisor is a factor of the
dividend.
The Factor Theorem. If a rational integral expression involv-
ing x becomes zero when m is written in place of x, x m is a
factor of the expression.
E.g. if x 8 3 x 2 2 x 8 is divided by x 4, the remainder equals
43 - 3 42 - 2 4 - 8'= 0, hence (x - 4) is a factor of x 8 - 3 x 2 - 2 x - 8.
00 *. Only factors of the absolute term need be substituted
fora?.
TEE FACTOR THEOREM 229
Ex. 1. Factor a? - 7 a? 2 -f 7a?-f 15.
The factors of the absolute term, i.e. 15, are -f 1, 1, -f- 3, 8, -f 5,
5, + 15, _ 15.
Let x = 1 , then x 8 7 x 2 + 7 a; -f 15 does not vanish.
Let x = - 1, then x 8 7 x' 2 4- 7 x + 15 = 0.
Therefore x ( 1), or x -4- 1, is a factor.
By dividing by x -f 1, we obtain
a?8 - 7 x 2 + 7 x + 16 = (x + l)(x 2 - 8 a; -f 16)
EXERCISE 106
Without actual division, show that
1. 4 x j + 3 x 2 2 as 5 is divisible by a; 1.
2. 2 or 5 2 a? 4 + 3^ - 7 or 2 - 5 a 18 is divisible by x 2.
3. x * 34 ar 5 -f 225 is divisible by x 5.
Resolve into factors :
4. m 3 -6ra 2 -fllm 6. 8. a 8 -}- 2o? 5x 6.
5. a 3 -2a + 4. 9. 2m 3 -5m 2 - 13m + 30
6. p 3 -5^ + 8p 4. 10. a s -8a 2 -f 19 a -12.
7. p*- 9^ + 23^-15. 11. & 3
12. m 4 4m 8 p~m 2 p 2 + 16m^
13. m 4 -f m s n 25 mV + 19 ran 3 -t- 4 n 4 .
14. a 5 + 32.
Solve the following equations by factoring :
15. ar*-f 6aj 2 + lla;- r -6 = 0. 21. or* -f 9 or* + 27 a? + 27.
1ft o?-5ar l + tt-t-15 = 0. 2 2. aj? - 7 a? 2 + 16 a? - 12.
17. ^-10^4-29^-20=0. 23. ^ + 7y 2 + 2y-40 = 0.
18. oj 3 5x 2 -f3a;4-9 = 0. 24. x 4 -4o 8 + 2a^ + 4a?~3 =0
19. a^-8^ + 19a;-12 = 0. 25. 4^
20. a; 3 7 a? 4-6 = 0.
230 ELEMENTS OF ALGEBRA
285. If n is a positive integer, it follows from the Factoi
Theorem that
1. x n y n is always divisible by x y.
For substituting y for x, x n y n y n y n = 0.
2. x n -f- y n is divisible by x -f ?/, if n is odd.
For ( y) n -f y n = 0, if w is odd.
By actual division we obtain the other factors, and have for
any positive integral value of n,
If n is odd,
ar + p =
e.g. z 6 - y 5 = (x -
286. It can readily be seen that # n -f y is not divisible by
either x + y or x y, if n is even.
287. Two special cases of the preceding propositions are of
importance, viz. :
x* -f-/ = (x +/)O 2 - xy +/),
Ex. 1. Factor 27 a* -f 8.
27 a 6 + 8 = (3 a 2 ) 8 +
288. The difference of two even powers should always be
considered as a difference of two squares.
Ex. 2. Factor m 6 n 9 .
We may consider m 6 n 6 either a difference of two squares or a dif-
* The symbol means " and so forth to."
THE FACTOR THEOREM
231
ference of two cubes. The first method, however, is preferable, since it
leads more directly to the prime factors. Hence
= (m -f n)(m 2 mn -f w 2 )(wi ;i
Ex. 3. Factor a 12
mn -f w 2 ).
EXERCISE 107
Resolve into prime factors :
1.
tt 3 -
1
9.
216 a 3 + 27 & 3 .
17.
a'-fl
.
2.
x* +
1
10.
1000 - x fi .
18.
m 12 +
3.
1-
a 3
b 3 .
11.
X 4 - X.
19.
l + e
; 6 6 .
4.
'-
8
.
12.
a -f a 4 .
20.
r>i2 -
a 3 .
5.
0^4-
8
13.
a-Z> 6 .
21.
ary-h
125.
6.
+
1.
14.
a-h& 6 .
22.
64 m 3 n 3 y 6 .
7.
8 a 3
1.
15.
afy 8 64.
23.
27m 9
- 343.
8.
a 3 -
64 b 3 .
16.
a 5 -!.
24.
r,\Q J
u c
Solve the following equations:
25. x 3 -8=0. 26. y 3 +8=0. 27. a s -27=0. 28. a;=
CHAPTER XIX
SIMULTANEOUS QUADRATIC EQUATIONS
289. The degree of an equation involving several unknown
quantities is equal to the greatest sum of the exponents of the
unknown quantities contained in any term.
xy -f y = 4 is of the second degree.
x*y + 6 a?V - y 4 is of the fifth degree.
290. Simultaneous quadratic equations involving two un-
known quantities lead, in general, to equations of the fourth
degree. A few cases, however, can be solved by the methods
of quadratics. *
I. EQUATIONS SOLVED BY FINDING x + y AND x-y
291. If two of the quantities x -f y, x y, xy are given, the
third one can be found by means of the relation (oj-j-y) 2 4 xy
Ex.1. Solve ==5 > (1)
1^ = 4. (2)
Squaring (1), & + 2 xy + 2/ 2 = 25. (3)
(2) x 4, 4 xy = 16. (4)
Hence, x-y- 3. (5)
Combining (5) with (1), we have
= 6,
Hence
/ X = }
| y = 4.
" *The graphic solution of simultaneous quadratic equations has been
treated in Chapter XII.
232
SIMULTANEOUS QUADRATIC EQUATIONS 233
292. In many cases two of the quantities x -f y, x y, and
xy are not given, but can be found.
(1)
(2)
(3)
(4)
F* 2
Lx ' 2 '
Square (2),
a 2 2 xy + y 2 = 1.
(3) x 2,
2 a 2 - 4 ay -f 2 y 2 = 2.
Hence
4 ay = 24.
(3) + (5),
a 2 -f 2 xy + y 2 = 25.
Therefore
+ y = + 6,
or x -f y = 6.
But
x - y - 1,
x - y = 1.
Hence
x = 3, y = 2,
j. 2 v
f 2
3 2 - 3 - 3 - 2 + 2 2 2 = 8,
f2.2 2 -3-2.3-f
Check. \
1
3 2 = 1.
1
-2 + 3 = 1.
293. The roots of simultaneous quadratic equations must be
arranged in pairs, e.g. the answers of the last example are :
r*=-2,
b=-3.
Solve:
1.
EXERCISE 108
2.
3.
'
10.
r - (" 1 = 876. "' {
8. I " "' 12.
I x + y=7.
r ^, =
4 ELEMENTS OF ALGEBRA
[ x -4- i/ =
13.
14.
6 r
"I
,o
18.
I x + y = a.
19. I* Jj ^
^ = 18*
[. ONE EQUATION LINEAR, THE OTHER QUADRATIC
294. A system of simultaneous equations, one linear and
ne quadratic, can be solved by eliminating one of the unknown
uantities by means of substitution.
Ex. Solve 2 x + 3 y = 7, (1)
From (1) we have, x - ~ " (3)
Substituting in (2) , ( 7 ~^V\ 2 + 2 y y = 5.
\ 2 /
Simplifying, 49 - 42 y + 9 y 2 + 8 ?/' 2 - 4 y = 20.
Transposing, etc., 17 y 2 40 y + 29 = 0.
Factoring, (y - 1 ) (17 y - 20) = 0.
Hence y = 1 , or f J.
Substituting in (3), aj = 2, or JJ.
EXERCISE 109
Solve :
] as 2 -f- a;?/ = 6, ^ f or* -f 4 xy = 28,
-47/ = 0.
3. r ^ - . - 5.
la; i
' '
SIMULTANEOUS QUADRATIC EQUATIONS
y - > lla 1 '
13.
235
12~ 10
7.
8-
9.
10.
III. HOMOGENEOUS EQUATIONS
295. A homogeneous equation is an equation all of whose
terms are of the same degree with respect to the unknown
quantities.
4^ 3 x 2 y 3 y 3 and # 2 2 xy 5 y 2 are homogeneous equations.
296. If one equation of two simultaneous quadratics is
homogeneous, the example can always be reduced to an example
of the preceding type.
' x *- 3 y* + 2y = 3, (1)
. 2 x 2 7 xy + G if = 0. (2)
Ex. 1. Solve
Factor (2), (x 2t/)(2 x 3y) = (
Hence we have to solve the two systems :
From (3), x-2y.
Substituting in (1),
4 f- 3 y 2 + 2 y = 3,
Hence y = 1 , 1 3,
':il -e 3 :)
(3)
(1)
8 V-~80
y =
236
ELEMENTS OF ALGEBRA
297. If both equations are homogeneous with exception oi
the absolute terra, the problem can be reduced to the preceding
case by eliminating the absolute term.
= 2, (1)
Ex. 2 . Solve
Eliminate 2 and 6 by subtraction.
(1) x 5,
(2) x 2,
Subtracting,
Factoring,
Hence solve :
15 x 2 - 20 xy + 15 y 2 = 2 x 5.
11 a 2 16 xy -f 5 y 2 = 0.
(rc-2/)(llx-5y) = 0.
From (3), j
Substituting y in (2),
109 a;2
^ VI09, y =
(3)
(4)
(3)
(2)
Solve:
6ar } --7aK/4-27/ 2 ==0,
EXERCISE 110
f 10^-370^ + 7^ =
16^-7^
SIMULTANEOUS QUADRATIC EQUATIONS 237
m <"" ' -=m 14 ' &-
3^4-2^=43.
U. ^ ' _ 15.
f 150 a?- 125 ay = - 6,
1 150 */ 2 - 175 ay = 12.
"
IV. SPECIAL DEVICES
298. Many examples belonging to the preceding types, and
others not belonging to them, can be solved by special devices,
which in most cases must be left to the ingenuity of the
student.
Some of the more frequently used devices are the following:
299. A. Division of one equation by the other. Equations of
higher degree can sometimes be reduced to equations of the
second degree by dividing member by member.
E,!. Solve { * + '-*
Dividing (1) by (2), y? - xy 4- y 2 = 7. (3)
Squaring (2), a? -f 2 xy + y 2 = 10. (4)
(4) -(3), Bxy-9,
238 ELEMENTS OF ALGEBRA
EXERCISE 111
Solve :
faj-y=152, f^ + 7/ 3 = 133,
* i ^ *>.
= 189,
* '
300. B. Some simultaneous quadratics can be solved by
considering not x or ?/, but expressions involving x and ?/, as
-, xy, x 2 , x + y y etc., at first as the unknown quantities. In
x
more complex examples it is advisable to substitute another
letter for such expressions.
Ex. 1. Solve i" <--- -' (1 >
(2)
Considering V# + y and Vx y as unknown quantities and solving,
we have
from (1), Vx -f y 4 or 6,
from (2), V^^y = 3 or 2.
But the negative roots being extraneous, we obtain by squaring,
x 4- y = 16,
jc~ y = 9.
Therefore x = 12 J, y = 3|.
SIMULTANEOUS QUADRATIC EQUATIONS 239
,
Ex. 2. Solve
Let
Then
r
Hence
I e.
V x 4 7
Hence we have to solve the two systems :
__ 17^ + 4-0.
= |, or = 4.
x U)
! + */ = 17. [2x + y= 17.
The solution produces the roots :
(1)
(2)
Solve :
2.
4.
EXERCISE 112
5.
6.
36*
M-6.
7.
F + y
+
240 ELEMENTS OF ALGEBRA
Solve by any method :
far' + a^lSG,
9 ' **
= 198.
5x+ 7y =
'
'
1 13 f- 21 ^ =
15.
18.
19
'
'
16.
2 or 5 CCT/ + 3 f + 3 a; - 4 y = 47 .
=34,
1 6 xy = 15.
( xy m 2 n*.
f (7 + o5)(6-hy) = 80,
25.
26.
27.
*
, = .
x y 20'
1 1 = 41
x 2 y* 400'
30.
31.
SIMULTANEOUS QUADRATIC EQUATIONS 241
i y . Q OK
~\ 7, OO.
y %
9 f*K 36.
32.
33.
34. .
3 a 2
25
7'
38.
39.
j/ = 48-
Solve graphically (see 201, 203):
40.
ix y
INTERPRETATION OF NEGATIVE RESULTS AND THE
FORMS OF 5 , .
oo
301. The results of problems and other examples appear
sometimes in forms which require a special interpretation, as
a ^ etc
--, -, , etc.
oo
302. Interpretation .of - According to the definition of
division, - = x y if = x. But this equation is satisfied by any
finite value of a?, hence -- may be any finite number, or ~ is
indeterminate.
242 ELEMENTS OF ALGEBRA
303. Interpretation of ? The fraction - increases if x de-
x
creases; e.g. ^-100 a, ~~f = 10,000 a. By making x
ToU" TO^UU"
sufficiently small, - can be made larger than any * assigned
number, however great. If x approaches the value zero, be-
comes infinitely large. It is customary to represent this result
by the equation ~ = QQ.
The symbol oo is called infinity.
304. Interpretation of The fraction - decreases if x in-
QO X
creases, and becomes infinitely small, or infinitesimal) if x is
infinitely large. This result is usually written :
305. I,i solving a problem the result or oo indicates that the
problem has no solution. If in an equation all terms containing
the unknown quantity cancel, while the remaining terms do not
cancel j the root is infinity.
306. The solution x = - indicates that the problem is indeter-
minate, or that x may equal any finite number. If all terms of an
equation, without exception, cancel, the answer is indeterminate.
Hence such an equation is satisfied by any number, i.e. it is an
identity.
Ex. 1. Find three consecutive numbers such that the square
of the second exceeds the product of the first and third by 1.
Let 2, as + l, x -f 2, be the numbers.
Then (a: + I) 2 -x(x + 2)= 1. (1)
Simplifying, x 2 -f 2 x + 1 - x' 2 2 x = 1.
Or, ' = 0.
Hence any number will satisfy equation (1), i.e. (1) is an identity, and
the given problem is indeterminate.
SIMULTANEOUS QUADRATIC EQUATIONS 243
Ex. 2. Solve the system :
(1)
(2)
From (2), z = 1
Substituting,
Or, 1=0.
Hence y QO, and a; = oo.
/.e. no finite numbers can satisfy the given system.
EXERCISE 113
1. One half of a certain number is equal to the sum of its
third and sixth parts. Find the number.
2. Find three consecutive numbers such that the square of
the second exceeds the product of the first and third by 2.
v o r K
3. Solve ~ ~
4. Solve
x - 3 x - 6
x 4 x 6
a; - 3 x - 5 a 2 - 8 x + 15
6. Solve
x - 2 y = 4.
6. Solve | 9 *
7. Solve (aj + 1) : (x + 2) = ( + 3) : (a? + 4).
EXERCISE 114
PROBLEMS
1. The sum of two numbers is 76, and the sum of their
squares is 2890. Find the numbers.
2. The sum of two numbers is 42 and' their product is 377.
Find the numbers.
244 ELEMENTS OF ALGEBRA
3. The difference between two numbers is 17 and the sum
of their squares is 325. Find the numbers.
4. Find two numbers whose product is 255 and the sum of
whose squares is 514.
5. The sum of the areas of two squares is 208 square feet,
and the side of one increased by the side of the other e.quals
20 feet. Find the side of each square.
6. The hypotenuse of a right triangle is 73, and the sum of
the other two sides is 103. Find these sides. ( 228.)
7. The area of a right triangle is 210 square feet, and the
hypotenuse is 37. Find the other two sides.
8. To inclose a rectangular field 1225 square feet in area,
148 feet of fence are required. Find the dimensions of the
field.
9. The area of a rectangle is 360 square feet, and the diago-
nal 41 feet. Find the lengths of the sides. (Ex. 12. p. 190.)
10. The diagonal of a rectangular field is 53 yards, and its
perimeter is 146 yards. Find the sides.
11. The mean proportional between two numbers is 6, and
the sum of their squares is 328. Find the numbers.
12. The area of a rectangle remains unaltered if its length
is increased by 20 inches while its breadth is diminished by
10 inches. But if the length is increased by 10 inches and
the breadth is diminished by 20 inches, the area becomes -f% of
the original area. Find the sides of the rectangle.
13. Two cubes together contain 30| cubic inches, and the
edge of one, increased by the edge of the other, equals
4 inches. Find the edge of each cube.
14. The volumes of two cubes differ by 98 cubic centimeters,
and the edge of one exceeds the edge of the other by 2 centi-
meters. Find the edges.
SIMULTANEOUS QUADRATIC EQUATIONS 245
15. The sum of the radii of two circles is equal to 47 inches,
and their areas are together equal to the area of a circle whose
radius is 37 inches. Find the radii. (Area of circle = irR 1 *.)
16. The radii of two spheres differ by 8 inches, and the
difference of their surfaces is equal to the surface of a sphere
whose radius is 20 inches. Find the radii. (Surface of sphere
= 47T# 2 .)
17. If a number of two digits be divided by the product of
its digits, the quotient is 2, and if 27 be added to the number,
the digits will be interchanged. Find the number.
CHAPTER XX
PROGRESSIONS
307. A series is a succession of numbers formed according
to some fixed law.
The terms of a series are its successive numbers.
ARITHMETIC PROGRESSION
308. An arithmetic progression (A. P.) is a series, each term
of which, except the first, is derived from the preceding by
the addition of a constant number.
The common difference is the number which added to each
term produces the next term.
Thus each of the following series is an A. P. :
3, 7, 11, 16, 19, ....
17, 10, 3, -4, - 11, ....
a, a + d, a + 2 d, a -f 3d, ....
The common differences are respectively 4, - 7, and d.
The first is an ascending, the second a descending, progression.
309. To find the nth term / of an A. P., the first term a and
the common difference d being given.
The progression is a, a -f d, a + 2 d, a -f 3 d.
Since d is added to each term to obtain the next one,
2 d must be added to a, to produce the 3d term,
3 d must be added to a, to produce the 4th term,
(n 1) d must be added to a, to produce the nth term.
Hence / = a + (n - 1) d. (I)
Thus the 12th term of the series 9, 12, 15 is 9 -f- 11 3 or 42.
246
PROGRESSIONS 247
310. To find the sum s of the first n terms of an A. P., the first
term a, the last term 1 9 and the common difference d being given.
= a + (a
Reversing the order,
Adding, 2*=(a + Z) + (a + l) + (a + l) .- (a + l) + (a + l).
Or 2s = n
Hence * = (+/). (II)
2
Thus to find the sum of the first 60 odd numbers, 1, 3, 6 we have
from (I) ' ' I = I + 49 . 2 = 99.
Hence . = *({ + 99) = 2600.
EXERCISE 115.
1. Which of the following series are in A. P. ?
(a) 1, 3, 5, 7, .-;
(6) 2,4,8,16,...;
(c) -3, 1, 5, 9,.-.;
(d) 1J, -|, -24, -4^....
2 Write down the first 6 terms of an A. P., if
(a) a = 5, d = 3;
(6) a = 2,' cZ == - 3 ;
(c) a = -l, d = -2.
3. Find the 5th term of the series 2, 5, 8, .
4. Find the 10th term of the series 17, 19, 21, ....
5. Find the 7th term of the series 1-J, 2, 2J, .
6. Find the 21st term of the series 10, 8, 6, .
7. Find the 12th term of the series -4, -7, -10,
8. Find the 101th term of the series 1, 3, 5, ....
9. Find the nth term of the series 2, 4, 6, .
248 ELEMENTS OF ALGEBRA
Find the last term and the sum of the following series :
10. 3, 7, 11, , to 8 terms.
11. 2, 4, 6, ', to 7 terms.
12. 8, 12, 16, , to 20 terms.
13. 3, 2J, 1|, , to 10 terms.
Sum the following series :
14. 7, 11, 15, , to 20 terms.
15. 33, 31, 29, , to 16 terms.
16. 15, 11, 7, -, to 20 terms.
17. 1, 1, 1J, , to 15 terms.
18. 2-f H + i-f > to 10 terms.
19. 2.5 + 3.1 -f 3.7 -f , to 12 terms.
20. (x +"l) 4- (# -f- 2) -f (x -f 3) H , to a terms.
21. 1 + 2-f-3 + 4 H hlOO.
22. 1+2+3+4H \-n.
23. Find the sum of the first n odd numbers.
Q^) How many times does a clock, striking hours only, strike
in 12 hours ?
(&fi) For boring a well 60 yards deep a contractor receives
$1 for the first yard, and for each yard thereafter 10^ more
than for the preceding one. How much does he receive all
together ?
^S5 A bookkeeper accepts a position at a yearly salary of
$ 1000, and a yearly increase of $ 120. How much does he
receive (a) in the 21st year j (6) during the first 21 years ?
311. In most problems relating to A. P., Jive quantities are
involved; hence if any three of them are given, the other two may
be found by the solution of the simultaneous equations .
rf. (i)
(ii)
PROGRESSIONS 24ft
Ex. 1. The first term of an A. P. is 12, the last term 144,
and the sum of all terms 1014. Find the series.
s = 1014, a = 12, I = 144.
Substituting in (I) and (II),
l)e?. (1)
1014 = ^(12 + 144). (2)
2
From (2), 78 n = 1014, or n = 13.
Substituting in (1), 144 = 12 + 12 . d.
Hence d=ll.
The series is, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122, 133, 144.
Ex. 2. Findn, if s = 204, d = 6, J = 49.
Substituting, 49 = a + (w- 1) .6. (1)
204 = ^ (a + 49). (2)
From (1), a = 49 -6(71 - 1).
Substituting in (2), 204 = ^ (98 - ~n~\ 6).
408 = n(104 - 6 n).
6 n 2 - 104 w + 408 = 0.
3 n 2 52 n + 204 = 0.
Solving, n = 6, or 11 J.
But evidently n cannot be fractional, hence n = 6.
312. When three numbers are in A. P., the second one is
called the arithmetic mean between the other two.
Thus x is the arithmetic mean between a and 6, if a, #, and
b form an A. P., or if
x a = b x.
Solving, x = -
4
I.e. the arithmetical mean between two numbers is equal to
half their sum.
250 ELEMENTS OF ALGEBRA
EXERCISE 116
Find the arithmetic means between :
1. a -f- b and a b. 3. and
m n
2. x y and #-f-5y. 4. and -
a + b a b
5. Between 4 and 8 insert 3 terms (arithmetic means) so
that an A. P. of 5 terms is produced.
6. Between 10 and 6 insert 7 arithmetic means
7. How many terms has the series ^ , T j ? , ^, , f ?
8. How many terms has the series 82, 78, 74, , 6?
9. Given d = 3, n = 16, s == 440. Find a and I.
10. Given s = 44, n = 4, f = 17. Find a
11. Given a = 7, J = 83, n = 20. Find d.
12. Given a = - 3, n = 13, 1 = 45. Find d.
13. Given a = 4, n = 17, 1 = 52. Find d and .
14. Given a = 1700, d = 5, / = 1870. Find w.
15. Given a = |, I = ^ 3 = 1. Find n.
16. Given a = 1, n = 16, s = 70. Find?.
17. Find I in terms of a, n, and s.
18. A man saved each month $2 more than in the pre
ceding one, and all his savings in 5 years amounted to $ 6540.
How much did he save the first month?
19. $300 is divided among 6 persons in such a way that each
person receives $ 10 more than the preceding one. How much
did each receive ?
PROGRESSIONS 251
GEOMETRIC PROGRESSION
313. A geometric progression (G. P.) is a series each term of
which, except the first, is derived from the preceding one by
multiplying it by a constant number, called the ratio.
E.g. 4, 12, 36, 108, ....
4, -2, +1, -I,....
a, or, <zr 2 , ar 8 ,
The ratios are respectively 3, |, and r.
314. To find the nth term / of a G. P., the first term a and
the ratios r being given.
The progression is a, ar, a?* 2 , .
To obtain the nth term a must evidently be multiplied by r n ~ l .
Hence l = ar n ~ l . (I)
Thus the 6th term of the series 16, 24, 36, ., is 16(f) 4 , or 81
315. To find the sum s of the first n terms of a G. P., the first
term a and the ratio r being given.
s = a + ar -for 2 -- ar n ~ l . (1)
Multiplying by r, rs = ar -f ar 2 4- ar n , (2)
Subtracting (1) from (2),
s(r 1) = ar" a.
Therefore 8 = ^ZlD. (II)
7* JL
Thus the sum of the first 6 terms of the series 16, 24, 36, .
8 =s lg[(i) fl -l] == 32(W - 1) = 332 J.
NOTE. If n is less than unity, it is convenient to write formula' (II) in
the following form : nf + *.
8 g== q(l-r")
1 r
252 ELEMENTS OF ALGEBRA
316. In most problems relating to G. P. Jive quantities are in.
volved ; hence, if any three of them are given, the other two may
be found by the solution of the simultaneous equations :
/=<!/-', (I)
,_!=!>. (it,
Ex. l. To insert 5 geometric means between 9 and 576.
Evidently the total number of terms is 5 + 2, or 7.
Hence n = 7, a = 9, I = 670.
Substituting in I, 676 = r 6 .
t = 64.
r^2.
Hence the series is 0, 18, 36, 72, 144, 288, 676,
or 0, - 18, 36, - 72, 144, - 288, 576.
And the required means are 18, 36, 72, 144, i 288.
EXERCISE 117
1. Which of the following series are in G. P. ?
(a) 2,6,18,54,-.; (c) f,l,,4, ....
(b) 1, 4, 9, 25, ... ; (d) 5, - 5, + 5,*- 5, ....
2 Write down the first 5 terms of a G. P. whose first
term is 3, and whose common ratio is 4.
3. Write down the first 6 terms of a G. P. whose first
term is 16, and whose second term is 8.
4. Find the 6th term of the series J, f, 1, .
5. Find the 7th term of the series ^, -fa, +-f% 9
6. Find the 6th term of the series 6, 4, 3, %
7. Find the 9th term of the series 5, 20, 80, ....
8. Find the llth term of the series ^, -fa, \ t .
9. Find the 7th term of the series |, , , .
10. Find the 5th term of a G. P. whose first term is 125 and
whose common ratio is .
PROGRESSIONS 25S
Find the sum of the following series :
11. 32, 48, 72, .-., to G terms.
12. 243, 81, 27, -, to 6 terms.
13. 14, 42, 126, .-., to 8 terms.
14. 1, 2, 4, , to 7 terms.
15. 81, 54, 36, ..-, to 6 terms.
16 - nV> i*> >"> to 6 terms.
!7- M,i -, to 12 terms.
18. a 9 , a^, a; 7 , , to 5 terms.
19. Given r = 4, n = 3, Z == 160. Find a and s.
20. Given r = -J-, n = 4, Z = 3. Find a and 5.
21. Given r = 2, n = 5, s = 310. Find a and J.
22. Given r = 3, n = 5, s = 605. Find a and I.
23. Find the geometric mean between 7,J- and 270.
24. Prove that the geometric mean between a and b equals Vo6.
INFINITE GP:OMETRIC PROGRESSION
317. If the value of r of a G. P. is less than unity, the value
of r n decreases, if n increases* The formula for the sum may
be written
= fl _ flf
By taking n sufficiently large, r n , and hence , may be
made less than any assignable number. ~ r
Consequently the sum of an infinite decreasing series is
-r^-
Ex. 1. Find the sum to infinity of the series 1, J, -J,
i 1
Therefore 8^ = = '- .
1 + 1 4
254 ELEMENTS OF ALGEBRA
Ex. 2. Find the value of .3727272 ....
.;)7?7272 ... = .3 + .072 + .00072 -f ....
The terms afteAhe first form an infinite G. P.
a = .072, r = .Ql.
Hence .= _4Z* - = ^ = .72. = .
1 - .01 .99 990 66
Therefore .37272 . . . = A + i. = 1L .
10 65 110
EXERCISE 118
Find the sum to infinity of the following series :
1. 1, i J, -. 3. 16, 12, 9, -.. 5. 5, 1, I, ....
2. 1, i 1, .... 4. 3, - 1, i, -. 6. 250, 100, 40, ...
7. 9, 6, 4, ....
8. If a = 40, r = j. Find the sum to infinity.
Find the value of:
9. .555.... 11. .191919-... 13. .27777 ....
10. .717171-... 12. .272727-.. 14. .3121212-..
15. The sum of an infinite G. P. is 9, and the common ratio
is J. Find the first term.
16. The sum of an infinite G. P. is 16, and the first term is
8. Find r.
17. Given an infinite series of squares, the diagonal of each
equal to the side of the preceding one. If the side of the first
square is 2 inches, what is (a) the sum of the areas, (6) the sum
of the perimeters, of all squares ?
BINOMIAL THEOREM 257
EXERCISE 119
Expand the following :
3. (1 + xy. 5. (s + i). 7. l
2. (x-y) 6 . 4. (a-2) 7 . 6. /2a+|Y- 8. (z 2 -^
\ 4
Simplify :
9. (1+V#) 4 + (1 Va) 4 . 10. (\
11. Find the 5th term of (a + b) 9 .
12. Find the 3d term of (a - b) w .
13. Find the 4th term of (w -f ri) 12 .
14. Find the 5th term of (1 + a) 11 .
15. Find the 4th term of (a -f 2 b) 7 .
16. Find the 6th term of (x - a 2 ) 25 .
17. Find the 5th term of f Vx + -^r
18. Find the 3d term of fa -f -^Y -
V Va/
19. Find the coefficient of a?b 13 in (a -f 5) u .
20. Find the coefficient of a 4 b 12 in (a -f 6) 16 .
21. Find the coefficient of a 5 b 15 in (a - 6) 20 .
22. Find the coefficient of a?V" in (a - 6) 100 .
23. Find the coefficient of a 8 6 16 in (a 2 - b 2 ).
24. Find the middle term of (x + y) 4 .
25. Find the middle term of (a b)\
f }\8
26. Find the middle term of f x : )
27. Find the middle term of (m ri) 16 .
28. Find the 99th term of (a + b) m .
29. Find the 1000th term of (a + b) im .
258 ELEMENTS OF ALGEBRA
REVIEW EXERCISE
Find the numerical values of :
1. 27 x* ~ 27 x-y -f 9 xy~ - # 8 , if
*=M or M 2 1 2 ] 2 ] 2 1 3 1 3 1 M.
y=2j 3J lj 2j 3} 4j 2J 4J 5J
2. 16 x* - 32 afy -f 24 afya - 8 ^ 8 + y 8 , if
x^l, 1, 2, 2, 2, 3, 3, 4.
y = 2, 3, 2, 3, 4, 3, 4, 5.
3. 4 * 2 - 4 xy - 4 ^ + ?/ 2 + 2 ^ + ^ 2 , if
a: = 2, 2, 3, 3, 3, 4, 4, 5.
?/ = 1, 2, 4, 2, 3, 4, 3, 2.
= 2, 1, 1, 2, 2, 1, 3, 6.
4. (2 a 4 - 13 a a b + 31 a 2 ft 2 - 38 aft 3 + 24 ft 4 ) - (2 a 2 - 3 aft -f- 4 ft 2 ), if
a = 2, 2, 3, 3, 4, 4, 5, 6.
ft = 1, 2, 1, 2, 2, 4, 2, 3.
a 8 + 3 + C 3 + ;] a 2^ + 3 a l } 2
5. ~T -r T -r ,f
a 2 + ft' 2 + c 2 + 2 aft ac be
a - 1, 2, 4, 4, 5, 5, 5, 5.
ft = 2, 2, 1, 5, 1, 3, 5, 6.
c = 3, 3, 2, 1, 2, 4, 2, 7.
6. (ft c)(c 4- ) - (c - a)(a + ft) - a(a 4- ft - c) + c(a -|- c), if
a = 3, 3, 4, 1, 2, 4, 2, 2.
ft = 2, 1, 2, 2, 1, 1, 3, 3.
c = l, 2, 1, 3, 3, -3, -5, -6.
7 , i c if
(a-ft)(a-c) (ft- c )(ft-a) (c-a)(c-ft)'
a = 2, -2, 1, -1, 3, 4, -4 - 1.
6 = 1, 1, - 2, - 2, 2, 2, 2, + 1.
c = 3, -3, 4, -3, -1, -1, -3, 2.
x c) . b(x c)(x a} , c(x g)(x
- 6-) (b ~c)(b- a) (c - a) (c - b) '
259
a = 1, 1, - 1, - 2, 3, - 3, 2,
/> = 2, 3, + 1, -f 8, 4, - 4, 6, - 2.
c = 3, 2, - 2, - 4, 5, - 5, 1, 2.
x = 4, 5, + 2, + 4, (5, + 2, 3, 1.
9. The radius r of a circle inscribed in a triangle whose sides are
a, by and c is represented by the formula
Find r, if
a = 3, 10, 8, 25, 29, 41.
6 = 4, 21, 17, 24, 21, 9.
c = 5, 26, 15, 7, 20, 40.
Add the following expressions and check the answers :
10. x 3 - 2 ax* -f a z x + a 8 , x 3 -f 3 ax' 2 , 2 a 3 - ax' 2 - 2 x 8 .
11. x 2 + y 4 + z 8 ,' - 4 x 2 - 5 z 3 , 8 .c' 2 - 7 y 4 -f 10 z 8 , 6 y 4 - 6 2 8 .
12. x 4 - 4 x 3 y + 6 x 2 ?/ 2 - 4 xy 8 + ?y 4 , 4 x 3 // - 12 zy + 12 xy* - 4 y 4 ,
C xy - 12 xy* + G y 4 , 4 xy* - 4 / 4 , y 4 .
13. x 3 + x/y 2 + .r
x' 2 z + y' 2 z +
14. 3 x 2 - 10 y' 2 + 5 z 2 - 7 ys, - x 2 + 4 7/ 2 ~ 10 z 2 + 3 ary,
z 2 + 11 yz + 8 2:2 - 2 x?/, 4 z 2 - \ yz + xz,
and 2 x 2 + y' 2 9 2:2 xy.
15. 1 + 3 x + 2 x 8 - x 5 , 4 - 2 x 2 - 8 a: 3 + 7 x 4 ,
12 a; - 4 x' 2 -f 12 x r> , and 5 or 2 + 7 x 8 - 11 x 5 .
16. 11 z 4 - 2 */, 7 xy 3 - 2 a?y + 3 aty - 8 y 4 , y 4 + 8 x 4
-12 x 4 + 5 *y + 4 * 8 y, 7y 4 - 3 a?y - 5 xy 3 .
17. 4 a 5 - a + 12 a 8 - 10, 6 a 4 - a 8 - 7 + 2 a 2 , 4 a + 9 a 2 - 3 a 5 ,
9 - 2 a 4 + 11 a - a 5 , 4 a 8 - a 4 - 5.
18. 11 x 8 + 14 x^ij -7xy* + z 3 , x 3 - 2 x 2 // + 3 x?/ 2 - 7 y 3
+ 3 y 2 * - 2 z 8 , 4 x- 8 + 2 // - 11 z 3 4 4 ?p 2 - 3 xyz,
and 3 y 8 -f 12 z 8 - 7 y 2 * 4- 4 xyz + 4 xy' 2 - 4 yz\
260
ELEMENTS OF ALGEBRA
19. 6 VI 4-X-5V14- #4-8, - 4\/i + x - 2VT+7 - ?>
3Vl 4- x 4- 4 Vl 4- */ 4- 3, and 2 Vl 4- x 4- 4vTT~y 3.
20. Take the sum of 2 x 8 4- 4 x 2 4- 9 and 4 x - 2 - x 2 4- 4 x 8 from
6 x 8 4- 4 x 4- 7.
21. Take the sum of G a 8 4- 4 a 2 x 4- 3 ax 2 , x 8 - ax 2 4- 2 a' 2 x, and
a 2 x -2 ax' 2 4- 3 x 3 from G a 8 4- 2 a 2 x - 4 x 8 .
22. Take the sum of 3 x 4- 8 2, 2 x 8 x 4- x 2 4- 5, and 4 7 x
4- 2 x 2 4- 4 x 3 from the sum of 9 x 4- 2 x s and 5 x 3 4- 3 x' 2 .
23. From the sum of 12 x 5 4- 4 x?y 4 4- y 5 , 2 x 6 4 x 4 ?/ x// 4 4- 3 y/ 5 ,
and G x 4 y 4- 2 x 2 // 3 - 3 .n/ 4 take the sum of G x 5 -f 2 x 2 / - ?/ 5 , x 5 - 2 x 4 #
+ 2 y 5 4- xy 2 , 4 ^V 4- G x 5 - 2 x 8 y 2 4- 3 y 5 .
24. From the sum of 4 - x - x 2 , G x - 4 - 5 x*, 54-2 x 2 , and 7 x
-x 2 take 4 - 4 x 2 4- 11 x.
25. From the sum of ft 4- c 4- 3 a, c 4- a 3 ft, and a 4- ft 3 c take
the sum of 2 ft 2 c - a, 2 c 4- 2 a ft, and 2 a 2 ft c.
26. Find what expression added to 3 x 2 5 x 2 x ;] 4- 3 will give
27. From the sum of ft 4- c 3 a, c 4- a 3 ft, and a 4- ft 3 c take
the sum of 2 ft - 2 c - a, 2 c - 2 a - ft, and 2 a - 2 ft - c.
28. Subtract the difference of x 8 4- 3 x 2 - 3 x - 1 and x 8 - 3 x 2 +
3 x - 1 from G x 2 4- 6 x.
29. Add 2 5 10 4- G 11 4- 3 . 7 12 , 3 5 10 - 5 G 11 4- 4 7 12 , - 4 . 5 10 +
7 . 0" - 9 7 12 , and - 5 10 - 3 G 11 4- 2 - 7 12 .
30. Ifcc = x4-y4-2, b = x -\- y z, c = x y -}~ z, and d= x4-#4-z
find (a) a -f ft, (c) a 4- ft 4- c, (*) a + ft 4- c 4- </,
(ft) ft - c, (</) a - ft + c, (/) a - 6 + c - rf.
Simplify :
31. 2 x - [3 if - (3 x - (5 y - 6T
_^
32. a _[5ft-{^ - (5 c - 2 6- - ft - 4 ft) 4- 2 a - (a - 2 ft -f- c)}] .
33. x* - [4 z 8 - {G * 2 - (4 * - 1)}] - (x 4 4- [4 x 8 4- 6 x 2 ] 4- 4 a: 4- 1).
34. 13-3ft-[l7a-5ft^[7fl-3ft-{4fl~4ft-(2a-3ft)}]].
35. 3 x 2 - (x* - 4) - [4 x - 5 - {2 x 2 - (7 x 4- 2) - (4 x 2 - 27~-~7)}].
36. 2a4-7c-(7ft4-4c)-[6a-3ft4 I 2~c4-4c-{2a-(ft-2T-2)}].
REVIEW EXERCISE 261
37. 7 a 2 -{5 a 2 -2 a + (2 a 2 - 7 a - 5)} + (3 a 2 - 4 a - 12).
38. (5 a 2 - 2 a - i j- - {3 2 - (2 a + 5 a - 0)} - (2 a 2 - 7).
39. 2 x - [3 y - 2 z + {4 ar - (5 y - 3T~2~s)} + 5 2].
40. a - [2 ft -f [3 c - (3 a - JT^T+1)} + (2 ft - 3 c)].
41. 3 x 2 - [4 x - 5 - (2 x 2 - (7 a; + 2) - (4 x 2 - 2 x -7)}].
42. 5a-(7ft+4c) + [6 a- 3~ft -f 2 c + 4 ^ - {2 a -(ft - 'J a~^~c)K].
43. a -{- b -(c - rf)} + a -[- & + {- 2c-(V/ - e -'/)}]
-(2a + 2b - 3c).
44. 5 a - (7 i + 4 r:) - [0 a - 13 ft + 2 c + 4 c - {2 a - (6 - 2a - 6-)}].
45. 13 a - 96 -[17 a- 56- [7 a - 36 -{4 a - 46 - (2 a - 3 ft)}]].
46. a - [2 ft - {3 c - (4 d - 5 )}] + {4 c - (2 ,Z - 2 <?)}
_[&-{2c-(3d + 7e)-a}].
Perform the operations indicated :
47. (* 2 + *+!){> + 2). 51. (1 -ar+a; a )(l-z a ).
48. (;r 2 -2:c+ l)(ar- 1). 52. (.r 2 + !>ar + 3)(^ 2 - 2x + 3).
49. (x 2 + 4x + 5)(j; - 3). 53. (2 x 2 -3 ar+ 1)(3 z 2 -2 x+ 1).
50. (1 - 6 x + 5 x' 2 ) (2 - 3 *). 54. (/> 4 - 2 ft 2 + 1)(7, 4 + 2 ft 2 + 1),
55. (4 + 3a 2 - 2)(1 - 4 a 2 + a 4 ).
56. (a 2 + ft 2 + c 2 - ab - ac - 6c) (a + ft -f c).
57. (a 2 + ft 2 + c 2 + aft + ac - be) (a - ft - c).
58. (x 2 + 4 y 2 + 3 z 2 ) (.c 2 - 2 ?/ 2 - 3 z 2 ).
59. (a 2 + ft 2 + 9 - 3 a + 3 ft + aft)(a - ft + 3).
60. (4 z 2 + 9 ?/ 2 + ^ 2 - 6 xy - 2 zz - 3 yz)(2 a:
61. (ar + 7)(ar + 5)(a: + 3).
62. (x- 3)(*-5)(* -7).
63. (a:-2)(r-4)(a:-9).
64. (x 2 + x + l)(a: 2 - x + !)(* - * 2 + 1).
65. (z - a) (2: + a)(x 2 + a 2 )(a: 4 + a 4 ).
66. (r 2 + ^ + 7/ 2 ) (x 2 - ary + ?/ 2 ) (^ 4 - *V +
67. (1 -*)(! + ar)(l + ^ 2 )(1 + **).
68. (a; - y)(x
262 ELEMENTS OF ALGEBRA
69. (a 8 - z 3 ) (a 8 + z 3 )(a 6 + a: 6 ) (a 12 + a: 12 ).
70. (a 2 - 2 a + l)(a 2 -f 2 a + l)(a a + 1).
71. (a - b)*(a 4- ft) 2 . 73. ( - 2ft) 8 ( + 2 ft).
72. (2 ar - y)\x 4- y). 74. (x - 3y) a (* 2 4- 6*y - 9y 2 ).
75. (a a -2a + l)(u 2 -f 2 a -f I)(a 2 + 1).
76. (:r 2 + a;y'*4-y 2m )OK n --y m ).
77. (rtP+i 4- 6)(a^+ 2 - am&t 4- A 2 *).
78. (a 2 " 4- ft 2n 4- c 2 4- 2 a m ft n - ac - ft n c)(a"* + ft" + c).
79. (??2 + n 6 -f- p c ) (w 20 + n~ b + /? 2c - ?n a n b - m n p c - n l p c ).
Simplify :
80. 4 (a + ft)(a + 4 A) (a - ft) + 4(2 ft - a) (2 /; 2 + a 2 ).
81. p(p + ? ) 2 + 7(7> ~ 'y) 2 4- <f(p - q).
82. a(2 + 3ft) 2 -(2a 4- ^) 8 .
83. (x 4- 2 y) (2 ^ 4- y) (^ - ,v) 4- (a? 4- y)*(x - y).
84. (p 4- 3 y)2(/ - 3 V ) - (/> - 3 v)^(;> + 3 9).
85. O 2 + y 2 J a - (x - y) O 4- y) (^ 2 4- y 2 ).
86. (x 4- y) 4 - (^ 2 4- y 2 ) 2 - 4 ^/(.r 2 4- xy 4- y 2 ).
87. (a 4- b 4- c) 2 - (a 4- ft- c) 2 .
88. (x 4- y) (y 4- 2) (s + *) - j; 2 (y 4- z) - z\x 4- y).
89. (x 4- y + z)(x + y - z)(x -y + z)(- x + y + z).
90. ( _ ft) (a: + a)(x + b) + (b-c)(x + ft) (a: 4- c)
--(c - a) (a? 4- c)(ar + a).
91. ( a 4- ft 4- c) 2 - (ft 4- c) 2 - (c 4- a) 2 - (a 4- ^>) 2 4- a 2 4- ft 2 4- c 2 .
92. (a 4- ft 4- c) 8 - (ft 4- O 8 - 4- ) 8 - (a 4- ft) 8 4- a 8 4- ft 8 4- c 8 .
93. 3 a - (4 ft - {3 a - 2 ft}) -f (3 a - 5 ft - ft - a}.
94. 3[a{2 a - 3 (ft - c)} - 2 ft (a - c)].
95. Prove the following identities, by multiplying out each side
of the equality.
(a) (a 4- b 4- c) 8 rr a 8 4- ft 8 4- c 8 4- 3(6 4- c)(c 4- a)(n + ft).
(ft) (.:-y)( a ;-2y)(.r-3y)4-l)y( a :-y)^-2y)4-18 // 2 (2r-y)4-6 // 8
REVIEW EXERCISE 263
Simplify :
96. [10( 4- &) 8 - 5(a 4- 6) 2 ]- 5(a 4- &)
6 (a 4- b) n
98. (10 a 3 *- 5 a 21 4- 5 a*) -=- 5 a*.
99 O3a O2a i O4a Qa-f-l^ 2 a
100. (3' n 4~ 3 m ~*~ n 3 3n 3") -T- 3".
102. (6 x 4 4- 23 x s 4- 42 a; 2 4- 41 x 4- 20) -*- (3 a* 4- 4 a? + 5).
103. (20 x* - 33 z 3 4- 72 x 2 - 35 a; 4- '30) ~ (4 ^ - 5 x 4- 10).
1O4.
105.
106. (2<
107.
108.
109. (x* - 9 ax 8 4- 12 .
110. (2 y 4 4- 2 y 2 4- 02 y - 16 y a 4- 48) - (y 2 - 5 y - 12).
111. (80 a 4- 3 a 4 - 23 a 3 4- 50 - 5 a 2 ) ~ (a 2 - 6 a - 10).
112. (4 4 - 4 a 2 // 2 4- 25 /> 4 - 16 a 3 6 4- 40 />) - (2 a 2 - 4 aft - 5 b*).
113. (25 a,v/ 8 ~ 3 xy - 6 y 4 4- 27 x* - 35 x 2 // 2 ) - (7 xi/ - 9 x 2 - 2 y 2 )
114. (8 x* - 2 2% 4- y 4 - 2 xy 8 - 21 x*if) -=- (4 ^ 2 - y 2 4- 5 xy).
115. ( y 8_o7)^^2 + 3 y + 0).
116. (.r 4 4- a: 2 y 2 4- y 4 ) - (x 2 - xy 4- y 2 ) .
117. (a 4 4- 16 a 2 4- 256) ~- (a 2 4- 4 a 4- 16).
118. (a 8 4- 4 ^ 4 + & 8 ) -s- ( 4 - 2 ^ 2 + ^ 4 ).
119. (a 8 - 8 6 8 4- c 8 4- 6 afo) -f- (a - 2 6 4- c).
120. Cr 8 -27y 8 -l-9a:y) -(a:-3y- 1).
121. (z y 6 ) -r- (a: 2 4- xy 4- y 2 ).
122. (x 10 - *) -(x 8 - 1).
123. (a+ 3 - 3 a"+ J 4- a"- 1 ) - (a 2 - a - 1).
124. (a** 1 - a*- 2 6 8 ) -r- (a - &).
264 ELEMENTS OF ALGEBRA
125. (1 - a 8 - G ax - 8 z 8 ) -5- (1 - a - 2 x).
126. (* + y 3 + z 8 - 3 a#z) *- (ar + y + s).
127. (*+ l + x- n - x+ l + x a ) ~ (x a + l + x).
128. (1 - 27 a 3 " 1 - 8 6 3 - 18 *&) -f- (1 - 3 a - 2 &).
129. What is the remainder when a 4 3 a B b + 12 a 2 6' 2 - b* is divided
by a*-ab + 26 2 ?
130. By what expression must a: -f 3 be multiplied to give x* 7 + 2187?
131. By what expression must 3 a 4 - 8 a*b + 4 a 2 6 2 - 8 ttfc 8 - 12 M
be divided to give the quotient 3 a 2 2 ab + & 2 ?
132. By what expression must x* + G x 2 - 4 a: 1 be divided to give
x 2 + 5 # 9 as quotient, with 8 as remainder?
Solve the following equations and check the answers:
133. 3(2 x - 1) - 4(0 x - 5) = 12(4 x - . r >) - 22.
134. 4(ar - 3) - 5(2 x - 3) = 12 - (x -f 9).
135. 7(2 x - 3)- 2(j: - 2) = 3 a: - 2(5 a: - 9) + 3.
136. 10(2 x - 9) 4- 7(4 * - 19) + 5 = 4 a: - 3(2 z - 3).
137. 5 x + 3(7 x - 4) - 2(10 x - 7) = 4 - (x - 5).
138. + ? + = 13. 139. 1 + 5 + 1=15.
2 o 4 o o o
140. 10(2 x - 9) - 7(0 x - 32) + 5 = 4 x - 3(2 j - 3).
141. 4(j - G) - 2 {3 a? - (j - 8)} ^ 5(13 - 3 a:).
142. 2(3 x + 4) = 5{2 x - 3(* + 4) + 9} - (1 - 3 x).
143. 8 [2 (a: - 1) - (x + 3) ] - 5{.r + 7[or - 2(4 - a:)]}.
144. 4-2(3ar
145. 5 a? - (3 a? - 2 [2 x - 2 7^~5] + 4} = 2(3 x - 1).
146. x 2 + (x 4- 1) (a? -- 1) = 2(* - 2) (a: + 3).
147. (ar - l)(ar + 2) (a: - 3) = x\x - 2) + 2(ar + 4).
148. (2ar-
149. (5a:
150. (4 x - 3) (3 x 4- 7) = (7 x - 1 1) (3 x - 4) - (9 x + 10) (a: - 3) .
REVIEW EXERCISE 265
151. (x + 4) (2 x + 5)- (* + 2)(7 z + 1) = (* - 3) (3 - 5*) + 47.
152. (x - 2) (j? + 1) + (x - 1) O + 4) = (2 * - 1) (s + 3).
153. (a - 3) (a: - 4) (a - 5) = (3 - l)(z - 14) (a: + 3)- 24.
154. (a; - 2) (7 -*) + (*- 5)(.r + 3) - 2(x ~ 1) + 12 = 0.
155. (2 a; - 7) (a; + 5) = (9 - 2 x) (4 - a:) + 229.
156. (7 - 6 x) (3 - 2 x) = (1 a: - 3) (3 ar - 2).
157. 14 - a; - 5(x - 3) (j; + 2) + (5 - z) (4 - 5 x) = 45 x - 76.
158. (ar + 5) 2 -(4-a:) 2 =r21a:.
159. 5(ar - 2) a + 7(x - 3) 2 = (3 x - 7) (1 x - 19) + 42.
160. (3 x - 17) 2 + (4 x - 25) 2 - (5 x - 29) 2 = 1.
161. O + ;T)O - 9) + (a; + 10) (ar - 8) = (2 x 4- 3)(* - 7) - 113.
162. ^ + ?=13 + . 163. f-^ + ^s-O.
10 o 2o 2 ;j 4
164. Write down four consecutive numbers of which y is the
greatest.
165. By how much does 15 exceed a ?
166. How much must be added to k to make 23?
167. A man is 30 years old ; how old will he be in x years?
168. Find five consecutive numbers whose sum equals 100.
169. There are 63 sheep in three flocks. The second contains 3
sheep more than the first, and the third twice as many as the first.
How many sheep are there in eacli flock Y
170. The sum of the three angles of a triangle is 180. The second
angle of a triangle is twice as large as the first, and if 15 were taken
from the third and added to the first, these two angles would be equal.
What are the three angles?
171. A picture which is 3 inches longer than wide is surrounded
by a frame 2 inches wide. If the area of the frame is 108 square
inches, how wide is the picture ?
172. The formula which transforms Fahrenheit (F.) readings of a
thermometer into Centigrade readings is C. = | (F 32).
(a) If C. = 15, find the value of F.
(b) At what temperature do the Centigrade scale and the Fahren-
heit scale indicate equal numbers?
(c) How many degrees C. transformed into F. will produce F. = 2 C.?
266 ELEMENTS OF ALGEBRA
173. A number increased by 3 gives the same result as the numbet
multiplied by 3. Find the number.
174. A number divided by 3 gives the same result as the number
diminished by 3. Find the number.
175. An express train runs 7 miles an hour faster than an ordinary
train. The two trains run a certain distance in 4 h. 12 m. and 5 h. 15 m.
respectively. What is the distance?
176. A square grass plot would contain 73 square feet more if each
side were one foot longer. Find the side of the plot.
177. A boy is as old as his father and 3 years younger than his
sister ; the sum of the ages of the three is 57 years. Find the age of
the father.
178. A house has 3 rows of windows, 6 in each row ; the lowest
row has 2 panes of glass in each window more than the middle row,
and the middle row has 4 panes in each window more than the upper
row ; there are in all 168 panes of glass. How many are there in each
window ?
179. Four years ago a father was three times as old as his son is
now, and the father's present age is twice what the son will be 8 years
hence. What are their ages ?
180. Two engines are together of 80 horse power ; one of the two
is 16 horse power more than the other. Find the power of each.
181. The length of a floor exceeds its width by 2 feet; if each
dimension is increased 2 feet, the ana of the floor will be increased
48 square feet. Find the dimensions of the floor.
182. The age of the elder of two boys is twice that of the younger;
three years ago it was three times that of the younger. Find the age
of each.
183. A boy is 5 years older than his sister and | as old as his
father; the sum of the ages of all three is 51. Find the age of the
father.
Resolve into prime factors :
184. x* + x - 2. 187. 2 + a _ no. 190. 4 a 2 + 13 a + 3.
185. y-y -42. 188. 7/ 2 -ll?/-102. 191. 10x 2 + 11 a; ~ 6.
186. z 2 -92;-36. 189. aW + llab-2&. 192. z 2 + x - 56.
REVIEW EXERCISE
267
193. y 2 - 77 y + 150. 196.
194. 2 a 2 - 19 a - 10. 197.
195. a 2 + 3 a - 28. 198.
202. z 2 -2;r?/-f y 2 -9.
203. x 5 - 19 z 4 ?/ +
204. 5
205. 14x 2 -25ary + Gy 2 .
206. 3 x* -x 2 - 12 x +4.
207. 16x 4 -81.
208. 2 a 8 - 8 6 2 .
209. if-W-y+b.
210. a: 8 -11 a: 2 + 10 ar.
211. 20 x 4 - 20 z 8 - 5 x 2 .
212. 3 x 2 - 21 a: - 54.
224. 7x 2
225. a^a
226. a; 4 -f yx* + z*x + z*y.
227. 7a
228. a: 2
229. * 2
234.
235. a; 8 -a; a + 2 a: - 1.
236. 24 a: 2 -23 a: -12.
239. (13z 2
240. 4a 2 & 2
241. (a +
242. x m+l - x m y + xy m -
243. 4
244. 2a
248. te
6 a; 2 - 5 xy - 6 y 2 . 199. 2 a; 2 + ary - 10 y a .
3y 2 - 13 y + 4. 200. x* - 12 * - 64.
x* + 8 a; 2 + 15. 201. # 2 - 29 y + 120.
213. 60 a 2 + 1 1 a*b - a 2 /A
214 - 12 x*y - 14 a: 2 // - 10 xy.
215. z + 5x 2 - 6s.
216. 2 x 2 - 22 z + 48.
217. 2 afy - 28 ary + 66 y.
218. x + 13 a: 2 + 30 x.
219. (a 2 + x 2 ) 2 - (a + z 2 ) 2 .
220. (r + y + a;y) 2 - (x y) 3 .
221. 6 a 2 + 5 a - 6.
222. x*y + G * 2 # 2 + 9 x*y\
223. 6 x* + 5 a:y - 6 ?/ 2 .
230. 15 x 2 + 26 x + 8.
231. 9a-4a6 a .
232. (a + b - c) 2 - (a - c) 2 .
233.
237. a: 4 - a: 2 - x + 1.
238. a: V - a: 2 - y 2 -f 1.
5# 2 ) 2 - (12 a: 2 + 4y 2 ) 2 .
(a 2 6 2 - c 2 ) 2 .
(a + c) 2 - (c + rf) 2 - (b + rf) 2 .
m +^. 245. 3 ap - 6 aq - 3 c/> + 6 cq.
3y 2 . 246. 3 x V - 3 xf + 3 * 2 y - 3 xy.
3% 247. a+a* + o a +l.
ly + la mx + wiy aw.
268 ELEMENTS OF ALGEBRA
249. 7 ax 4- 14 bx 3 ay 4- a + 2 ft - 6 by.
250. a%% 8 - 3 abc - a 2 />c 2 -f 3.
251. 2 a; 8 - 2 ax 2 + 2 for 2 - 2 aft*.
252. 2 a.r + a# + az -f 2 6z -I- fry 4- &z.
Find the II. C. F. of:
253. G(x+ l)'\ 9(x 2 - 1).
254. 3 #2 -|- 10 a; + 8, a; 2 + 23 x -f 20.
255. 5 x 2 + 7 r -f 2, 15 # 2 + 20 x 4- 8.
256. x*y* 4- 18 xy + 5, 18 x 2 z/ 2 + 39 xy 4- 15.
257. a 2 - 11 a 4- 10, a 2 - 10 a 4- 9.
258. 3 a 2 /; 2 - 5 ab -f 2, 3 a% 2 - 4 ab + 1.
259. 10 x 2 - 23 a? + 12, 30 ^ - G7 x -f 33.
260. 7 x 2 4- 16 x - 13, 28 a: 2 -f 71 x - (55.
261. 12 a: 2 // 2 - 1 9 ;ry -21,48 afy 2 - 73 xy - 91.
262. x 2 + 8 x + 1 5, x 2 -f 9j: + 20.
263. * 2 - 9 x + 14, x- 2 - 11 x -f 28.
264. x 2 + 2 x - 120, x* - 2 z - 80.
265. x* - 15 ar + 30, * 2 - 9 x - 36.
Find the L.C.M. of:
266. z 2 + 3 x + 2, x 2 + 4 ar + 3, x 2 + 5 ar -f 6.
267. z 2 - 3 x - 4, * 2 - x - 12.
268. 2 x 2 - 7 a: -f 5, * a - 23 x -f 20.
269. 2 z 2 -f 13 x + 1 5, 8 ;r 2 + 10 x - 3.
270. x 2 - 18 ry + 32 y 2 , a: 2 - 9 xy + 14 y 2 .
Reduce to lowest terms :
271. 2-2-
~ 8 x < f +
_
x 2 + 3 xy - 40 y 2
272 x 2 -f 4 a: - 77 2?5 5 a; 2 - 7 a; - 6
' ' *
a; 2 -f- !8a: + 77 ' 5 a: 2 - 17 x + 14
273 P a -5y>+4. 2?6 12 J r 2 __7^/_ J / 2
^2-7/7 + 12 28 x 2 + 3 .ry - 2/ 2
REVIEW EXERCISE 269
277 8 agg - 6 a; - 9 2Q4 4 * 2 - 8 x + 8
' ' '
278. _ m ~ n . 285
w 4 + 2 7w% 2 -f n* z 4 + a; 8 - ar - 1
279.
sa + Og-e. 286 1 - 2 * -f * 2 - 2*
* 2 - 9 1 + 3 x + ar a + 3 x*
280. "* " w - )P 287
m 2 4- J' 2 4- 2 mp - n 2 * "
281 2 q^( 2 - a: 2 ) m 288
'
- 2c
282. - 289 a: 2 - (y - z) * t
z 2 (a + c)a; + ac ' (j; y)' 2 z 2
283 t- . 290
'
x' 2 - y* + z 2 + 2 0:2
291 * 2 + y 2 + 2 2 + 2 yz 4- 2 zx + 2 ary
0;2 _ 2 _ 2 2 _ 2
292 ?/.rL.!/... 293
^ - a; - y '
294 2 fr 2 <? 2 + 2 cV + 2 a 2 ^ 2 - 4 - ft* ~ c 4
295
296
'
297
'
270 ELEMENTS OF ALGEBRA
Find the value of :
298 23. 19 -23. 2 99 i
* " '
23 19(23 + 19) ' x + 4 *-3 a + 7
300. Lnl + 2Lz| + ^.
ar 2 x 3 a; 4
1 3
301.
(a: + l)(ar + 2) (x + l)(ar + 2)(* + 3)
302. __ + -*_ + *
303.
a? a -la?-la? + l
1 1
(a + c) (a + ct) (a + c)(a -f e)
304. x ~~ a + ^ ^ ^ a ~ ''^ .
x b x a (: a) (x 6)
~ i 7. 2. i _
305.
2 + 7 3. _ 44 ^ _ 2 * -
308. ^-
n m + n (m + n) 2
309. "*" x + ^ "" g
310 x 3 a: -2 a: 2 - 2 a: - 17
a:-2 ar-3 x 2 -5a:-i-6"
O(c-a) (c-a)(a-i) (a-
306. i_ _L 1
X 2 + 9 a . + 20 x 2 4- 12 x + 35
1 1
307.
311.
312.
313.
BE VIEW EXERCISE
i + b a 2 + ft 2
_ _.
x 8 -.- 1 x 2 + a; + 1 a: - 1
2ft 2
a 8 -
314. . } .,+ 2
3
315.
a * 2 - 9 * + 20 * 2 - 8 * + 15
10 z 2 ,2 1
(1 + :
1 -f * 2 1 - 2 x'
316.
(a? -
2)
317. nl g(jL+ 2 ) 4- ^^^_
318. 1-
a: + y
a; 2 + y 2
319.
D x x(l - *)
* 8(1-*) 4(1 +*) 2 8(1 + *) 4(1 + * 2 )
271
321
322.
ft-c
( 6_ c) 2_ (a ._ (
- c ) 2 - (a - ft) 2
g~ft
'(a-6)*-(a:-r)a
323.
272 ELEMENTS OF ALGEBRA
Simplify:
324. 2 x * ~ 8 x2 -i- 4 r8 + 2 ^ "" 12 *.
z 2 - 4 x - 5 10 a; 2 - 250
325. -*
3a; 2 -lOx 5x- 2 +lOar 2 a; 2
a 2 - 2 a - 03 i^+^T- 42 ' ^2 _ l5rt~+~54*
327 8 ^ - 28 J?_ x fl^-^ffjje _ 27^-12^7 t
' 2 8 - 11 2 + 12 a 4 a 2 + a - 4 6~7** - 5 a - 6*
328. 2 ^ "" 1B x + 40 y * 2 + 5 x - 3;B ~ 1037 -
329
4g~0y g x 6 a* - 9 *// + 27 .y x (a?-4y) _
4 a; 2 - 10 o# ~ 24 y 2 x' 2 - 7 xy + 12 ^/ 2 3(2 x - 3 y) 2 '
330 .V 8 - ?/ 2 + y - 1 x 3y a - ll.y-20 x 2 y + 4 ;/ 2 - 3 y - 6 (
331 g gy -f 3 y ~ 6 q - 9 G y/ 2 + 5 ?/ - 6 G fl y~4-y+ 15 ^e - 10
' 6y a +lly-10
332 3 a: 2 - 4 xy - 4 y 2 5 x 8 + 8 a: 2 ?/ - 4 x?/ 2 ^_ G x 2 + 13 gy_+ 6 y
2 z 6 - 8 a; 3 y 2 10 a: 2 - 19 xy + 6 y 2 ' 8 x* - ~"
333. a: 2 - 7 acy + 12 y 2 . a: 2 - 5 sy + 4 y*
x* + 5 a:y + y 2 ' a: 2 -f zy - 2 y 2 '
334 * 2 + 2 y - 15 .a: 2 + 0^ + 20
'
8 a; -33 ' ^2^7 a... 44
333.
REVIEW EXERCISE 278
336. ~ c * \ c * ~ a2 h q2 - 5 *| : f 5 ~ c | C - q | q - 6 1 -
a b c \ { a b c \
337. (a-f2/,) 2 . 338. (a + lV. 339. fl-.1V. 340. f V.
\ aj \x yj \5yl
341. + ni + 1 -. 345. -
342. (a-Wi + iJ. 346. ( 5 -
343. (ar + l + IV. 347.
\ xi
344. ( 7 -?-f!?.y. 348.
w
349 _ ** -i. ~ ~ \.
o; 2 -"l ' x 2 + 8 a: + 2"
350. .;r a>74 ; 5 '
7 6 -5 2 .13 13 s 11
Find the numerical values of :
351. 1+ i--, if a = 3.
352.
Simplify :
353 ^-3 pE+1 /2x~l 5a:~2
* 2 L a V 4 10
354. - + .
?_2 r + ^
^ a: y x
274
ELEMENTS OF ALGEBRA
355 f 5 ___ _/| x
U<>-3) 2(*-l)J
356 fl 1-* 1-10*W*-1
V YTx 1-** JUa-l
357.
ar 2 xy + y a x l + xy + y l
x*
358.
_
+ l a
359.
y-
y-x y-
360. (a
a b c
abc
361.
1+2 i+5 1+1
362.
9 x 2 -f
363. (~
364. (* + 1 + W?* _ 1 +
\3a 2x) \3a 2x
365.
366.
,
a -f -
1 + x x
I I
a 2 4-
367.
368.
369.
REVIEW EXERCISE 275
f 2 + ^\ (b* -f c*)
\b* + c* b*-c*)^ }
b c
b c b 4- c
a b , b c
(1 +ab)(l+bc)
a + 6 a b
a b . b c
1 _ ^
~
370 -
' 1 (-/')(&-o) ' 1 .
a
372. 2 -
m
373
374.
"1*7
276
375.
ELEMENTS OF ALGEBRA
-3 or
Solve the equations :
376. 5*-8,*-2
= 15.
a: 2 (a; 8 -
377. iLf-5 + !*=! = 2 J.
45 <3
378. 2(3 x + 1) - |(x + 4) + 20 + |(x + 7) = 0,
379.
. 4(* + 6)- -(* + 10) = j(* + 5)-
O J 1 v/
^ _j_ O 10
380. 5 {2 x + 1 - 3(* + 1)} - ^-^ + ^-\:) x -f 51) +2J = 0,
o /
381.
383.
J r !__7. - r ~ x-f ! .
a: 2 + 5"^ - 10 x - 1 '
__4 3_ = !.
2 x 5 3 vC 7 a:
382.
# _j- k j a: 2 a: 3
385
10
17
387 L*J> _ -7ar = 10ar + 15 , 2J
* 14 (5 ar + 2 28 """ 7 '
ear-7 1 , , 1 + 16ar_63 -24 g 12f - 8 a
i3JTo^ + 2T~~~ia a '
389
390.
-
5 14(ar-l) 7 18 105
391.
REVIEW EXERCISE 277
x 4 _ x 5 _ a: 7 _ x 8
ar- 5 a;- 6~a: -8~a; - 9*
a: 37-0 _ x + 1 a? - R
- *
a; u
a: 37- _ x
^ ~ ~r; "i ^ 7 ~r
x 2 x i x 1
'2 a; - 7 7 - 1(5 a; + 4 a;*
2 + 6 * 2 " 2 ^ 2 ~ ^ H- 1 .
^ (a . - l)(x - a) (a: - 3) 4- 3(4 * - 2)(ar + 1).
396. (8 x - 3) 2 (x - 1) = (4 x - 1) 2 (4 x - 5).
397.
m
x 1 a: -f 1
398. .5 x - 2 = .25 x + .2 a: - 1.
399. .5 x -f .6 x - .8 = .75 x -f .25.
400. 3* =r 177,147.
401. y ~ rt ^= & ~ c . 402.
a; -f 1 1 + a
403. JLg:== c -q. 404.
7wa: n c -}- a b
4O5
b b x
f -(a: - a) + a-(a; - ft) = 2(ar - ) (a; - J).
40,. a:- a)(a: - &)(>: + 2a +2&) = (a: + 2 a)
408. (x ~ a)(x -f 6) -f c = (z + a)(a: - i).
^ a: 2 a: 5
____-_
278 ELEMENTS OF ALGEBRA
410. (x -f a)(z - b) - 2 alb = (x -f b)(x - a) - 2 6 2 a.
411. (x - a)(x - b) = (x - a ~ ) 2 .
a b a b
412.
414.
415.
x a x b x c
b _ a -f b
x -f a x -f b x -f c
1 1 a-b
x a x b x 1 ab
1 1 1
x a x + c x b ~ c x b
416 mx ~ a ~ b mx a c
nx c d nx b d
417. (a - b)(x - c) - (5 - c)(:r - a) - (c - a)(x - b) = 0.
418 a I2x - lfi:r l rt
~j-o. ~r
419.
a 2 ar a IJ a; 4 x a a Qx
x 2 a x ~ 2 b x 2 c 3 x
) -f c a c + a 6 a + 6 c a + 6 -f
420. Tn 6 hours A walks 2 miles more than B walks in 7 hours ;
in 9 hours B walks 11 miles more than A walks in 5 hours. Find
the number of miles an hour that A and B each walk.
421. In a number of two digits the first digit is twice the second,
and if 18 be subtracted from the number, the order of the digits will
be inverted. Find the number.
422. A man drives to a certain place at the rate of 8 miles an
hour. Returning by a road 3 miles longer at the rate of 9 miles an
hour, he takes 7 minutes longer than in going. How long is each
road ?
423. A person walks up a hill at the rate of 2 miles an hour, and
down again at the rate of 3^ miles an hour, and was out 5 hours.
How far did he walk all together ?
REVIEW EXERCISE 279
424. A steamer which goes at the rate of 264 miles a day is fol-
lowed in 2 days by another which goes 286 miles a day. When will
the second steamer overtake the first?
425. Find two consecutive numbers such that the sum of the fifth
and eleventh parts of the greater may exceed by 1 the sum. of the
sixth and ninth parts of the less.
Find the fourth proportional :
426. x - y, z 2 - y\ x* -xy + y*.
427. , i, |. 428. a + 5, a - t>, a 8 -f 2 ab -f 6 2 .
Find the mean proportional to
429. 3 and 1J. 430. z 2 - iand 2 2 - I .
2 2 a
431. Find the ratio x : y, if
5 x = 7 y ; wi* + y= ny; ax -\- by - ex + dy.
432. A line 10 inches long is divided in the ratio m:n. Find
the length of the parts.
433. The sum of the three angles of any triangle is 180. If one
angle of a triangle is to another as 4 : 5 and the third angle is equal
to the sum of the first two, find the angles of the triangle.
434. If a : b = 5 : 7, and b : c = 14 : 15, find a : c.
435. Solve : n m : n(n x) = p m : n(p x).
436. Which ratio is greater, 5 : 7 or 151 : 208?
437. Prove that the number of miles one can see from an elevation
of h feet is very nearly equal to ^- - miles.
438. Which of the following proportions are true?
a. (3a + 4ft):(Oo + 86)= (a-26):(3o-46).
b. (9 2 - 46 2 ): (15a 2 - 31 afc + UV 2 )
= (15 a 2 + 31 ab + H 6) : (25 a 2 - 49 6 3 ),
c. (a 8 + &*) : (a 2 -h & 2 ) = (a 2 - fc 2 ) : (a -6).
d. (a 8 + 6 8 ) : (a + ft) = (a 5 ~ a*b + a*b* - a 2 ^ 8 + aft* - & 5 ) : (a 8 - 6 8 ),
280 ELEMENTS OF ALGEBRA
439. Find the value of x, if
a. 29(a + &) : x = 551 (a 3 - ft 2 ) : 19(a - &).
c. (3 a 2 + 2 ab - 8 ft) : (5 a 2 -f 4 ai - 12 ft 2 ) = a? : (5 a - 6 ft).
440. The volumes of two spheres are to each other as the cubos of
their diameters. If a sphere 2 inches in diameter weighs 1:2 ounces,
what is the weight of a sphere of the same material having a diam-
eter of 3 inches ?
Solve the following systems:
441. 7 a: - 2 y = 1 ; 3 a; + 5 y = 59.
442. x + 17 # = 53; 8 x + y = 19.
443. 33 x + 35 y = 4 ; 55 * - 55 y = - 16.
444. 7jr-9y = 17; 9ar-7# = 71.
445. 7a?-y = 3; 5x+4y=lQ.
446. 7 a: - 3 y = 3 ; 5 a; -f 7 ?/ = 25.
447. x + 5 y = 49 ; 3 x - 11 y = 95.
448. ax + ly = 2 ; a*x + & 2 # = a + b.
449. 5z-4:# = 3;r-f-2# = l.
450. ox -f &// = 2 ; />(.*; + y) = a + ft.
451. 28 = 5 a - 4 ft; 8a + 21+3ft = 0.
452. 5j + 7 7 = ; 12 /) - 89 = q.
453. 20y + 21 = 2; 42 = 15y + 137.
454. 18a = 50 + 25y; 5#+ 10 = -27 a.
455. 56 + 10y = 7a;; 15ar = 20 + 8y.
456. 9/> = 2 - 11 7; 21 7 = 27 + Op.
457. 3 ar - 7 y = 25; 4 a; = 5 y + 29.
458. 8 a: - 59 = 3 z; 5 2 = 7 a: - 35.
459. |-l(*-2y)=0; 1(3 a; + 5y)- | (or
460.
3 x + 7
REVIEW EXERCISE
~~~^ = 5;7;c=56-3y.
8
28i
461.
3 a? _ y 7 a? 3 y _ 1
12
__
15 ~~10 4 10 "10
463.
465.
4 g ~
2
7 g + 3 .?/ 1 . a: 2 g + 3 y _
+ ' +
467. -
= 2; (or - 2y)- (2
= 2J.
468. ^ + i^ = 7;- + -=2. 469. --i = 5; i-
47O
_ _
3~12 4*
471. ax } by = c \ 472. ax by = m; 473. car = rf
cte - ey =/. cx + ey-n. x 4- y
474.
475.
-_
& + y 3
dx+frj- c\
282 ELEMENTS OF ALGEBRA
476. In a certain proper fraction the difference between the nu
merator and the denominator is 12, and if each be increased by 5 the
fraction becomes equal to |. Find the fraction.
477. What is that fraction which becomes f when its numerator is
doubled and its denominator is increased by 1, and becomes j| when
its denominator is doubled and its numerator increased by 4 ?
478. If 1 be added to the numerator of a fraction it becomes equal
to , if 1 be added to the denominator it becomes equal to ^. Find the
fraction.
479. The sum of three numbers is 21. The greatest exceeds the
least by 4, and the other number is half the sum of the greatest and
least. Find the numbers.
480. There are two numbers the half of the greater of which ex-
ceeds the less by 2, also a third of the greater exceeds half the less by
2. Find the numbers.
481. Of the ages of two brothers one exceeds half the other by 4
years, and a fifth part of one brother's age is equal to an eighth of
that of the other. Find their ages.
482. If 31 years were added to the age of a father it would be
thrice that of his son ; also if one year were taken from the son's age
and added to the father's, the latter would then be twice the son's
age. Find their ages.
483. A and B together have $6000. A spends } of his money and
B spends \ of his. B then has J as much as A. How much money
had each at first?
484. Find two numbers such that twice the greater exceeds the
less by 30, and 5 times the less exceeds the greater by 3.
485. A sum of money at simple interest amounted in 10 months to
$2100, and in 18 months to $2180. Find the sum and the rate of
interest.
486. A sum of money at simple interest amounts in 8 months to
$260, and in 20 months to $275. Find the principal and the rate of
interest.
487. A number consists of two digits whose difference is 4; if the
sum of the digits be multiplied by 4, the digits will be inverted. Find
the number.
REVIEW EXERCISE 283
488. There is a number of two digits which is equal to seven times
the sum of the digits ; also if the digits be transposed the new number
will exceed 10 times the difference of the digits by 6. Find the
number.
489. Find two numbers whose sum equals s and whose difference
equals d.
490. The sum of two numbers is , and the difference of their
squares is b. Find the numbers.
Solve the following systems :
491. x -f y -f z 29| ; x + y z = 18J ; x y -f z = 13|.
492. , + 2 = 41; * + s - = *i, , + 5=84.
2 425
493. 5 a; + 3 y 62 4 ; 3 a: y -f- 2 z = 8 ; a; 2y + 2z = 2.
494. 4a;-5#+2z = G; 2 ar + 3 y - z = 20; 7 a: - 4 # + 3 z = 35.
495. $x
496. 4 a:
497. y x + z = 5; z y x 25 ; a; + # + z = 35.
Solve :
498. a: + // = 11; ;/ -f z = - 1; a: - z = 12.
499. 3 x + 5 y = 101 ; 7 a; + 2 z = 209; 2 a: -f z = 79.
500. i-f-i = a; 1+1 = 6; i-fi = e.
x y y z z x
502. 3ar + 2y = 8; 4z+3z = 20;
503. 2a:-f 7;/ = 15; 2y + 3a = ll;
504. 7;? -2^ = 20; 2/>-3r = 4; 30 4r=-9.
506. 2 a; 3^ = 8; 5^ 9z = 10; a: + 4 ?/ 2 z = 15.
507. --\ ' ' ~
284 ELEMENTS OF ALGEBRA
516.
517.
523.
511. x + '2z = 4:x +1
512 x + ?/ + 2 *
!y + l_6j, + 3 e 2-
13.
z 1
514.1 + 1=1.1 + 1
y ' y *
*2 z x
515. 1-1 = 1; 1 +
# y 6 y
1 434
2 x y z
(a b c
fl 1 1
x y z
_ + = &, 518. ,
y z x
a b c _
x y ^ 520. <
434
g + ar-Sy^'J/)
x + y-3z^c
(x y z
[a: y z 521 ' J
ay + bx^ c.
x y
1 5,1 o
+ _- _g ?
x y z
y-\-z 6+ c
1 + 1 = 6, 519. <
y z
Z X
5 + * _ 2 _ 5 522 ^
a? ^ *
5,7,6 1R
- H h - = lo.
37 ^ S
zx __ ra
2! + X C + rt
y?/ ^>
+ 2+?. + !f == 2800,
3579
9 + ll"
: = 1472.
ra? + y + z = 3a-f& + r,
524 \ x +y + t = + 36 + c,
jx 2 i=a + 6 c,
[y + z- =3a-&-c.
REVIEW EXERCISE 285
525. When weighed in water, 37 pounds of tin lose 5 pounds,
and 23 pounds of lead lose 2 pounds.
(a) How many pounds of tin and lead are in a mixture weighing
120 pounds in air, and losing 14 pounds when weighed in water?
(b) How many pounds of tin and lead are in an alloy weighing
220 pounds in air and 201 pounds in water ?
526. A and B together can do a piece of work in 2 days, B and C
in 3 days, and C and A in 4 days. In how many days can each
alone do the same work?
527. Throe numbers are such that the sum of the reciprocals of
the first and second equals ; the sum of the reciprocals of the first
and third equals \\ the sum of the reciprocals of the second and
third equals \. Find the numbers.
528. A vessel can be filled by three pipes, L, M, N. Tf M and N
run together, it is filled in 35 minutes; if N and L, in 28 minutes;
if L and Af t in 20 minutes. Tu what time will it be filled if all run
together?
529. A boy is a years old ; his mother was I years old when he was
born; his father is half as old again as his mother was c years ago.
Find the present ages of his father and mother.
530. A can do a piece of work in 12 days ; B and C together can
do the same piece of work in 4 days ; A and C can do it in half the
time in which B alone can do it. How long will B and C take to do
it separately ?
531. A number of three digits whose first and last digits are the
same has 7 for the sum of its digits; if the number be increased by
90, the first and second digits will change places. Find the number.
532. In A ABC, AB=6, BC = 5, and CA=7. An (escribed)
circle touches A C in /), and the prolongations of BA and BC in E
and F respectively. Find AD, CD, and BE.
533. Two persons start to travel from two stations 24 miles apart,
and one overtakes the other in 6 hours. If they had walked toward
each other, they would have met in 2 hours. What are their rates of
travel?
286
ELEMENTS OF ALGEBRA
534. Represent the following table graphically :
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES, FRANCE,
GERMANY, AND BRITISH ISLES
Year
1800
1810
IHL'O
180
1840
1850
1800
1870
1880
1890
1!>00
U.S.
5.3
7.2
0.0
12.1)
17.0
23.2
31.4
38.0
50.2
02.0
70.3
France
27.2
28.8
30.5
32.4
34.0
35.0
37 .3
30. I
37.0
38.0
38.9
Germany
22.0
23.4
20.2
29.7
32.4
35.2
38.1
40.5
45.2
49.4
50.4
British
Isles
10.0
17.0
20.5
21.0
20.4
27.2
28.7
31.2
34.5
37.5
11.2
535. One dollar equals 4.10 marks. Draw a graph for the trans-
formation of dollars into marks.
536. The number of workmen required to finish a certain piece
of work in D days is Draw the graph from D 1 to D = 12.
How long will it take 11 men to do the work?
537. If I feet is the length of a pendulum, the time of whose swing
is t seconds, then / = 3.3 t' 2 . Draw a graph for the formula from / =0
to / = 3 and write down the time of swing for a pendulum of length
8 feet.
Draw the graphs of the following functions :
538. 3 x + 5. 542. x 2 + x. 546. 2 - x - x 2 .
539. 2 x - 7. 543. x 2 - x - 5. 547. x*.
540. 2 - 3 x. 544. z 2 + x - 3. 548. x 8 - 2 x.
541. - 3 x. 545. x *-x + 2. 549. x* - x + 1.
550. Draw the graph of y 2 + 2 x x*, from x = 2 to x = 4,
and from the diagram determine :
a. The values of y, i.e. the function, if x = f , 1, 2|.
b. The values of x if y = 2.
c. The greatest value of the function.
d. The value of x that produces the greatest value of y.
e. The roots of the equation 2 + 2 x x z = 1.
REVIEW EXERCISE 287
551. The formula for the distance traveled by a falling body is
a. Represent ] f/f 2 graphically from t = to t = 5. (Assume g = 10
meters, and make the scale unit of the t equal to 10 times the scale
unit of the \ ^ 2 .)
b. How far does a body fall in 2^ seconds?
c. In how many seconds does a body fall 25 meters?
Solve graphically the following equations :
552. x*-"2x-7 = Q. 559. 2 x 2 - 4 x - 15 = 0.
553. a: 2 ~0a: + 9 = 0. 560. 2 j; 2 + 10 x - 7 =
554. a; 2 + 5 .r - -1 = 0. 561. 3 x 1 - G a: - 13 = 0.
555. x* - 5 x - 3 = 0. 562. a; 3 - 3 x - 1 = 0.
556. z 2 - 3 x - = 0. 563. .r 3 + 3 z - 11 = 0.
557. x 2 ~ 2 a; - 9 = 0. 564. 2 .r 8 - 6 a: + 3 - 0.
558. 3 x* - 3 a? - 17 = 0. 565. z 4 - 10 x 2 + 8 = 0.
566. x 4 - 4 x 2 + 4 a: - 4 = 0.
567. x 5 - x- 4 - 11 x* + a; 2 + 18 x - 4 = 0. 568.' 2* + Z - 4 = 0.
569. If y - 8 + 5 a; 2 - 10,
a. Solve// = 0. c. Solve y = 5.
J. Solve // = 5. r?. Solve y 15.
e. Determine the number of real roots of the equation y 2.
f. Determine the limits between which m must lie, if y = m has
three real roots.
g. Find the value of m that will make two roots equal if y = m.
h. Find the greatest value which ?/ may assume for a negative x.
i. Which negative value of x produces the greatest value of y ?
Solve graphically :
570
570 '
571.
572. . '
= 8.
288 ELEMENTS OF ALGEBRA
|4,-5 y =10, j^ + ,-4,
[ xy = 0. [ x -f ?/ = 3.
577. (f-, 3 ^ 4 ' 581. {f_7l = j?
578. \*> = : or 582. |''- o 2* + 3' > -
[ a: 2 + y* = 25. [ ?/ 2 a:.
579. jj+;frf 583. jf:ji + f-
Perform the operations indicated :
584. f-MV 586 ' ^ + 6)T ' 590> (2 + <r)3 -
587. (a-iy. 591. (3 - 2 a:) 3 ,
588. (a -f ?>) 3 (a - ft) 8 . 592. (1 + xY.
585 --
589. (1-a:) 3 . 593. (# - 2) 4 .
594. (a; + ^) + (air-%)8. 597. (1 + x) 4 (l - ^) 4 .
595; (aa; + %) 4 -f- (a* - %) 4 . 598. (1 + x + x' 2 )' 2 .
596. (1 + *) 5 - (1 - a:) 6 . 599. (1 - x + a; 2 )' 2 .
600. (2 + 3 x + 4 a:' 2 ) 2 -f (2 -3 x + 4 a; 2 ) 2 . 601. (1 + x + z 2 ) 8 .
Extract the square roots of the following expressions:
602. 64 a 12 - 128 a 10 6 + 100 a 8 /; 2 - 100 aW + 100 aW- 48 a*h 6 + 10 6.
603. 4-8 xf + 30 a: 2 ?/ 4 - 04 aty 6 + 9(5 x^f - 128 a*^ 10 + 04 aty 1 2 .
604. a: 16 - 2 a: 14 ?/ + 3 a: 1 ' 2 ?y 2 - 4 a: 10 // 3 -f 5 zy - 4 x*y* + 3 a: 4 ?/ 6 - 2 a: 2 // 7 + y.
605. a 8 - 4 fSb + 4 a 6 & 2 + ^s_ 14 a 4/,4 + 4 a 8^6 + 9 a a^e _ 6 aW + fe 8 .
606. 9 - x -f 13 a; 2 - 4 a: 8 -f 4 a: 4 .
607. (2 a 2 ft + i 2 ) 2 -f (a 1 - 2 & 2 ) (4 6 + 1) .
608. a: 6 -f ~ + 10 ^i - 1 V 6x 4 - - + 5 a*.
x [
609.
610. a 2 a; 2 -f 2 aAa: + ?/ 2 - 2 6a: + a; 2 - 2 aa: 2 .
611. a 2 + 9 6 2 + 25 c 2 - 30 &c + 10 ac - a6.
612. a: 2 -f + - + -~-bx.
943 3
REVIEW EXERCISE 289
Find the fourth root of :
613. 4 4- 4 a*b + a 2 /; 2 -f 4 aft 8 + ft 4 .
614. 10:r 4 + 9G* 3 + 21G.*; 2 + 21Ga; + 81.
615. HI x s - 108 afy + 54 'x*y* - 12 a?y + y*.
616. 10 a 4 - 32 fe 2 + 24 a 2 /; 4 - 8 aft 6 + ft 8 .
Find the eighth root of:
617. a 8 - 8 tvb + 28 a 6 // 2 - 50 a c ft 8 + 70 a 4 ft 4 - 50 a 8 ft 6 + 28 a 2 ft
- 8 aft 7 + ft 8 ,
618. a 8 - 10 a* + 112 a 8 - 448 z + 1120 a: 4 - 1792 x* + 1792 a: 2
- 1024 x + 256.
Find the square root of :
619. 942841. 621. 0090.2410. 623. 49042009.
620. 25023844. 622. 4370404. 624. 44352.30.
625- VOIOOD + V582T09. 626. V950484 - V250 - \/4090.
Find to three decimal places the square roots of the following
numbers :
627. 49.871844. 629. 035.191209. 631. 494210400001.
628. 371240.49. 630. 210.15174441. 632. 2.
633. 21. 634. 3 2 V 635. 4J. 636. 9g. 637. 40. 638. GGff.
639. According to Kepler's law, the cubes of the distances of the
planets from the sun have the same ratio as the squares of their
periods of revolution about the sun. If the distances of Earth and
Jupiter from the sun are at 1 : 5.2, and the Earth's period equals 3G5J
days, find Jupiter's period.
Solve the following equations :
640. x 2 + 9 x = 70. 646. x 2 -f x ~ 16 = 0.
641. a- 2 -21 x = 100. 647t x 2 + 9 x - 22 = 0.
642. *+* = 156. 648 x2 _ 5x _ 66 = .
643. x 2 - 53 x ~ - 150. 9
644. 8*' + 24* = 32. 649. * + ? , = 87.
645. 9a; 2 + 189 z = 900. 650. 3a; 2 +x = 14.
651. (x - 2) 2 -f (x + 5) 2 = (x + 7) 2 .
290
ELEMENTS
OF ALGEBRA
652. 12 a: 2 + 7;
653. -a; 2 4- 5 a
654. 7 a: 4- 3 a; 2
655. 6 a- 2 4- 7*
656. 3 a: 2 -28.
C= 02.
:= G.
= 20.
= 17 a:,
o.
- _ 3
+ 2 5*
2 j: 22
2 = "
8 32
666.
667.
668.
669.
671.
672.
673.
674.
1
a:-l 5 2
x 4-1 G 7(a:-l)
4 5 _ 12
a?-l ar-2 2 x 4- 13
658.
x 4- 1 x-{- 2 a: 4- 16
a: 4- 1 , *4- 2 2.r 4- 13
659 . io_iir
X X
66O i
a; - 1 a; - 2 x 4- 1
2s -1 } 3aj-l 5a:-ll
a:4-l a;4-2 a: 1
a;_! 4a: .=l n _*L= : \
8 x- - 3 x 4- 1
a 2 a; 2 - 2 a*x 4- a 4 - 1 rr 0.
4a 2 j; = ( 2 - //-'4- a-) 2 .
* + -?+*.
uov. -|- _ ^ -
5 x 4 - x x 4- 2
661. x 2 4- O 4- b)x 4- 6 = 0.
662. a: 2 - 2 <u" 4- ^ - // 2 = 0.
663. z 2 4- - x - < ^~ = 0.
6 6 2
664 *+l + *-l_2ar-l.
6 ' *4-2 + a:
665 x - 2 a:
- 2 ar - 1
4-2 x 4- 3
a y: ^ a:
1 , 1 _1 , 1
a; 4- 2 x
676.
677 1 1 1
- 2 a: - 3
1 1
a:'a:4-& a'a4-&
^ 4- a a: 4- 6
4. ^
682.
683.
684
685.
686
fift?
a m + b
a + 1:r+1 |all-0
a a 4- a:
fi7fl *-l-" + 2ft
a; a
a ar ar A 13
678. -
a* 4- ( i ~ - o
fi7q a:4--ft_
^J~~7)
x b ' a ~ x 6
1 ,1 a 4- &
ar-4-A
___ 1 ,
2 3
680. 4- -
681 x
-2 a:-3
1
a 4- a: /> 4- x ab
1,5 2
681 l
1 1 _ +ft
24--
x
ar a """ x b ab
REVIEW EXERCISE
291
X + 1>
~
x a + c c
" ~T~ ~ i~ ~ *
- ~ " i ~
x + a + b x a b
690.
691.
692.
693.
694.
= 4 x +
rj* 2 4(5 + Ox + 4
. + a 2 + 2 ax 1 + V* -2bx
695. (1 - a a ft 2 )jr 2 - 2 a(l + & 2 )z -fa 2 - fi 2 = 0.
696. ax 2 + to 2 -f ru: 2 - ax - bx - c = 0.
697. ax 2 + fa + ex - a - b - c = 0.
698. a; 2 - 2V3:r -f 1 = 0. f x 4
699. a: 2 2 V5 a: + 4 = 0. ^ ^
701. (x 2 +3a:) 2 -2a; 2 -6a:- 8 = 0.
702. 2(4 :r 2 - 3 a:) 2 - 28 x 2 + 21 a: + 5 = 0.
7r\O /'r'S __ ' : )'*' _i. 7^^ ^3" ^^ ^T '^ 1
704. (:r 2 + :r)O 2 + :c-f 1) = 42.
706.
707.
708.
292 ELEMENTS OF ALGEBRA
709. **-13a: 2 +36 = 0. * 2 + 16 | 25
710. 16 x* - 40 a 2 * 2 + 9 a 4 = 0. ' 25 a 2 -16
711. a: 2n -f-2aar n + a 2 -5 2 = 0. 713. 3or 4 - 44# 2 + 121 = 0.
714 ___ _ i
2* 2 -5 3*2-7
715.
716. Find two consecutive numbers whose product equals 600.
717. What number exceeds its reciprocal by {$.
718. Find two numbers whose sum is a and whose product equals J.
719. A needs 15 days longer to build a wall than B, and working
together they can build it in 18 days. In how many days can A build
the wall?
720. The area of a rectangle is 221 square feet and its perimeter
equals CO feet. Find the dimensions of the rectangle.
721. Find the price of an apple, if 1 more for 30/ would diminish
the price of 100 apples by $1.
722. The difference of the cubes of two consecutive numbers is
217 ; find them.
723. Find four consecutive integers whose product is 7920.
724. Find the altitude of an equilateral triangle whose side equals a.
725. A man bought a certain number of shares in a company for
$375; if he had waited a few days until each share had fallen $6.25
in value, he might have bought five more for the same money. How
many shares did he buy ?
726. What two numbers are those whose sum is 47 and product
312?
727. A man bought a certain number of pounds of tea and
10 pounds more of coffee, paying $ 12 for the tea and $9 for the coffee.
If a pound of tea cost 30 J* more than a pound of coffee, what is the
price of the coffee per pound ?
Find the numerical value of :
728. 12 -4*+ 8- l + 8 -8
729. (J)-* - (3|)* + (a + ft)' + 64- + i.
REVIEW EXERCISE 293
implify :
30. (y* + y* + y*+l)(y*.-l).
31. (a* - x*)(a* + a*x* + x*).
32. (a* -f /^ + c^ - M - aM - ft*c*)(a* + 6* + c*).
33. (rrr + w 5 n* + w s n 3 ' + ?n^n^ -f n 3 )(m* n^).
34. (i* - 2 d*m* + 4 d-)(w* + 2 rfM + 4 d*).
35. (x* - 1 + x- 2 )(x 2 4- 1 -f ^ 2 ).
36. (a-2 + &-2)( a -2_ j-2).
37. (a- 1 -6- 1 + c- 1 )(a-i + &- 1 -f c" 1 ).
38. (1 + a^ 1 + a 2 6" 2 )(l - aft* 1 + a 2 *- 2 ).
39. (4 x~* + 3 ar 2 + 2 a;- 1 + l)(>r 2 - -i + 1).
40. (64 x~ l + 27 y 2 ) -r- (4 x~^ + 3 y"*).
41. (^ - ary* + x^y - y*) * (x* - y*).
42. (a* + M -f 6) -f- (a* + a*6^ -f- &*).
43. (a^ + 6* - c* + 2 U*") -* (<i* + ^ + cb-
44. (x* - 2 a M + a 8 ) -4- (x* - 2 aM" -f a).
46. (4 a: 2 - 12 x* -f 28 x + 9 x* - 42 x* + 49)*.
48 ^i? x T ^ ^2? x sT~ x .2?
50.
n.
52. (v/x)- X
294
ELEMENTS OF ALGEBRA
753.
754.
[1 r* 1 4- O/lf r* 4- 1 "1
1 ~ x ._ i j ;. "r ^ ^ II * "*" A 1.
r^ T JU.1+J
755.
756.
757. 2^3(^-2^21 + 4^-3^:0.
758.
759.
+ V22 -
760. 4\/50 + 12 V2b8 4- SVlOOO.
761. \/G86 + v/lG-v/128.
768. vff + V^~ 4^ -2^/2
776.
2-V2
2-V3
IIEVIEW EXERCISE
295
780.
781.
782.
783.
y/a -f x + Va x
y/a + x Va x
Find the square roots of the following binomial surds:
784. 10 + 2V21. 789. 38-12VIO. 794. 3J- - VlO.
785. 16 + 2V55. 790. 14 - 4 V(j.
786. 9-2VI5. 791. 103 - 12VIT.
787. 94-42V5. 792. 75-12V21.
788. 13 - 2 V30. 793. 87 - 12 ^
799. a c + 2 Vab ac + 6t
Simplify
801.
806. 7 + 3 V5 ( 7 - 3 V'5 ( 2 + V3 | 4 - ^
7-3V5 7 + 3V5 4 + V3 5 - 2V3*
807.
Va -f x Va z Va -f x + Va a;
809.
296 ELEMENTS OF ALGEBRA
810. Find the sum and difference of
(ar + V2y-x 2 )* and (x - \/2y - a: 2 ) 4 .
811. x/aT+l - V? = 1. 813. VaT+lJ -f
812.
814.
815. \/2(r+ 1) + v/2 x + 15 = 13.
816.
817.
/3 a: - 5 + V3 a: -f 12 = 17.
/9ar + 10-3Var- 1 = 1.
818. V* + 60 = 2 Vx~-K5 + V5.
819. 2\/^"+~5 + 3Vor-7 = V25 a: - 79.
820. 3
x + 2 - V-c^lJ - V2 ar - 10.
^l - g.
829. Va: + 28 + V9 x - 28 = 4 V2 ar - 14.
830. V14 a; -f 9 -f 2 VaT+1 + V3a:+ 1 = 0.
831. \/12 a; - 3 + ViTli + V7 a: - 13 = 0.
832. V2a: + 7
833. Va: + 3 -2 Vx -f 1 = V5x + 4.
836.
834. V3 ar -f 1 V4a;-f 5 + Vx - 4 = 0.
Va:
KEVIEW EXERCISE 297
838. 5 x* -f 11 x - 12\/(ar4-4)(5z~ 9) = 36.
839. 4 a; + 4\/3^~- 7x + 3 = 3ar(a; - 1)+ 6.
840. a: 2 + Vo: 2 + 3 x -f 5 = 7 - 3 a:.
841. 4 a: 2 -f V4 x 2 - 10 x -f 1 = 10 x + L
842. * 2 - 3 a: - 3 Va: 2 - 3 x - 10 = 118.
Resolve into prime factors :
843. x 4 + 2 a; 8 + 5 a; 2 -f 18a: + 16.
844. x 6 -f x 4 + a: 8 + a: 2 -f x -f 1.
845. 5 a 4 -f 7 a 8 - 28 a - 80.
846. 4 x* - x*y + 4 xy 8 64 y*.
847. a; 8 3 x -f 2.
848. a: 8 + 3 a: 2 - 4.
849. a 8 - 4 6 2 -f 3 6 s .
850. a; 8 + 4 ar 2 + 2 x 3.
851. x* 4- 2 a; 8 ^ 2 a#* y 4 .
852. a; 8 -2a; 2 - 8 a; + 15.
853. x -f" 1 1 a^ -J- 13 a/ 4o.
854. a; 8 - 13 a; -f 12.
855. a; 8 -8a: 2 + 19a;-12.
856. x 8 4 a: 2 19 x 14.
857. 4 x 8 -f 8 a; 2 - 3 x - 9.
858. 16 a: 8 40 x 2 7 a; -f 49.
859. 8 a; 8 4- 27 y 8 . 868. a 8
860. 8 + a 8 a; 8 . 869. a* 1 * - b**.
861. 27 a; 8 -64. 870. a*" -f & 6n .
862. l-64a. 871. a + 1.
863. z*y 8 + . 872. a 8 + 216 rt
864. 275 8 -l. 873. aty - ft*.
865. 64 a - 1000 6. 874. a 10 - ab 9 .
866. 729 a: + 512 y 8 . 875. a 18 4- a.
867. 8^-27^. 876. a l0m - 1.
298 ELEMENTS OF ALGEBRA
877. Show that 99 + 1 is divisible by 100.
878. Show that 1001 79 - 1 is divisible by 1000.
879. For what value of m is 2 # 3 mx* 5 x 3 exactly divisible
by x - 3 ?
880. What must be the value of m and n to make
a; 8 + mx 2 + nx -f 42 exactly divisible by 2 2 and by x 3 ?
Solve the following systems :
881. x + 2y=\2, xy + y = 32.
882. z 3 -f- y 3 = 28, x 2 - xy + y 2 = 7.
883. a: -f y = 7, a: 8 + y 3 = 13:3.
884. a; 2 + xy = 10, y*+ xy - 15.
885. a; 2 -f ary + y 2 = 37, a: 4 + afy 2 + ?y 4 = 481.
886. --.
887. a: 2 + y 2 = 34, a: 2 - y 2 + V(j; 2 - y 2 ) = 20.
888. a: 2 + y 2 - 1 = 2 a#, xy(a:y + 1) = 6.
889. a; 2 -f ary = 8 a: + 3, ?/ 2 + ary = 8 y + 6.
890. a: 2 - xy = 2 a; + 5, a:y - y 2 = 2 y + 2.
M1 1 , 1_3. 1 1 _ 5
891 -;Vi' ? + p"ia-
892. 1 + 1 = 5; L+L=13.
x y a: 2 y 2
893. 1-1-21; l-I = 8f.
x y x z y 2
894. a: 8 - y 9 = 37 ; a: 2 + ary + ?/ 2 = 37.
895. a: 8 + y 9 = 152, a: 2 - xy + y 2 = 19.
896. x*-xy- 35, a# -f f - 18.
897. z 2 + xy = 126, y 2 4- sy = 198.
898. a; 2 + 3 y 2 = 43, a; 2 + xy = 28.
899. a: 2 + 2 f = 17, 3 x 2 - 5 xy + 4 f = 13.
900. ar(ar 2 + y 2 ) = 16 y, y(a: 2 + y 2 ) = 25 x.
901. a: 2 - a# = 2 aa: + 6 2 , xy - y 2 = 2 ay + a 2 .
REVIEW EXERCISE 299
902. x a -f y 2 x 2 y = 1, xy 2 2 x 2 y xy z + 2 a:y + 2 = 0.
903. x 4 + a; 2 y 2 -f # 4 = 243, x* + ary -f y 2 = 9.
904. a: 2 -f ar// -f y 2 = 84, or Vary + y = 6.
905. (0 x + y}(x + y) = 273, (!) * - y) (a? - y) = 33.
906. Vx -f 10 -f v^+T4 = 12, * + y = 444.
907. x + y 2 = aar, y -f- x 2 = by.
908. a; + y = 9, ^ 2 - #y + ?/ 2 = 27.
909. 23 x 2 - y 2 = 22, 7 y - 23 ^: = 200.
910. ^-f!i^2, ny - ma: = a - m*.
911. L/ay = + ft- 5? + g = ^ + g.
* o- 2 a 2 o j
912. a: 2 + 3 a:y = 2, 3 y 2 + xy = 1.
913. a: 2 + 2 ary = 39, xy + 2 y 2 = 65.
914. a: 2 5 xy = 11, y 2 + 3 ary = 2.
915. x 2 -f 2 a:y = 32, 2 y 2 -f ay/ = 16.
916. x 2 ry + y 2 = 3, # 2 + xy + y 2 = 7.
917. (* - 3) 2 -f (y - 3)2 = 34, *y - 3(* + y) = 6.
918. (3 x - y) (3 y - x) = 21, 3 :r(3 a; - 2 y) = 49 - y
919. (a; + 2 y) (2 .r + ?/) = 20, 4 a; (a; + ?/) - 16 y 2 .
920. (o; + y)(a; 3 -|-y 8 ) =1216, a: 3 - y 8 = 49(x - y).
921. a;y = a(ar + y), a; 2 y 2 = 6 2 (x 2 + y 2 ).
922. or 2 y + a:y 2 = 180, or 3 + y 8 = 189.
923. 9 a? -f 8 y -f 7 ay/ = 0, 7 a: + 4 y -f 6 ary = 0.
924. a: 2 + ary = a*, y 2 + xy = b 2 .
925. xy + x= 15, ary y = 8.
926 * 2 ~ g.V + y 2 = 03 a: 2 + xy + y a = (a? - y)^
a: + y * a: + y 12
927. 2 ar + y = 2 a 4- 6, ^ 2 + 2 a:y = a a
928.
a: y xy
929. yz = 24, zx 12, a:y = 8.
300 ELEMENTS OF ALGEBRA
930. (*+s)(* + y)=10, (* + y)(y +*)= 50, (y
931. s(y + z)=18, y( + *) = - 102, *(* + #) =24.
932. ar(a? -f y + 2) = 152, y(x + y + 2) = 133, z(* + y + 2) = 76.
933. (y + a)(* + y + z) =108, (3 + *)(ar + y + z) = 96,
934. The difference of two numbers is 3 ; the difference of their
cubes is 513. Find the numbers.
935. The difference of two numbers is 3, and the difference of
their cubes is 270. Find the numbers.
936. The sum of two numbers is 20, and the sum of their cubes is
2240. Find the numbers.
937. A certain rectangle contains 300 square feet; a second rec-
tangle is 8 feet shorter, and 10 feet broader, and also contains 300
square feet. Find the length and breadth of the first rectangle.
938. The sum of the perimeters of two squares is 23 feet, and the
sum of the areas of the squares is 16^f feet. Find the sides of the
squares.
939. The perimeter of a rectangle is 92 feet, and its diagonal is
34 feet. Find the area of the rectangle.
940. A plantation in rows consists of 10,000 trees. Tf there had
been 20 less rows, there would have been 25 more trees in a row.
How many rows are there?
941. The sum of the perimeters of two squares equals 140 feet;
the sum of their areas equals 617 square feet. Find the side of each
square.
942. The sum of the circumferences of two circles is 44 inches,
and the sum of their areas 78$- square inches. Assuming IT = -y, find
the radii of the two circles.
943. The diagonal of a rectangle equals 17 feet. If each side was
increased by 2 feet, the area of the new rectangle would equal 170
square feet. Find the sides of the rectangle.
944. A and B run a race round a two-mile course. In the first
heat B reaches the winning post 2 minutes before A. In the second
heat A increases his speed 2 miles per hour, and B diminishes his as
much ; and A then arrives at the winning post 2 minutes before B.
Find at what rate each man ran in the first heat.
REVIEW EXERCISE 301
945. The area of a certain rectangle is 2400 square feet; if its
length is decreased 10 feet and its breadth increased 10 feet, its area
will be increased 100 square feet. Find its length and breadth.
946. The area of a certain rectangle is equal to the area of a square
whose side is 3 inches longer than one of the sides of the rectangle.
If the breadth of the rectangle be decreased by 1 inch and its
length increased by 2 inches, the area is unaltered. Find the
lengths of the sides of the rectangle.
947. The diagonal of a rectangular field is 182 yards, and its perim-
eter is 476 yards. What is its area?
948. A certain number exceeds the product of its two digits by 52 and
exceeds twice the sum of its digits by 53. Find the number.
949. Find two numbers each of which is the square of the other.
950. A number consists of three digits whose sum is 14; the
square of the middle digit is equal to the product of the extreme
digits, and if 594 be added to the number, the digits are reversed.
Find the number.
951. Two men can perform a piece of work in a certain time ; one
takes 4 days longer, and the other 9 days longer to perform the work
than if both worked together. Find in what time both will do it.
952. The square described on the hypotenuse of a right triangle is
180 square inches, the difference in the lengths of the legs of the
triangle is 6. Find the legs of the triangle.
953. The sum of the contents of two cubic blocks is 407 cubic feet;
the sum of the heights of the blocks is 11 feet. Find an edge of
each block.
954. Two travelers, A and B, set out from two places, P and Q, at
the same time ; A starts from P with the design to pass through Q,
and B starts from Q and travels in the same direction as A. When
A overtook B it was found that they had together traveled 80
miles, that A had passed through Q 4 hours before, and that B, at
his rate of traveling, was 9 hours' journey distant from P. Find the
distance between P and Q.
955. A rectangular lawn whose length is 30 yards and breadth 20
yards is surrounded by a path of uniform width. Find the width of
the path if its area is 216 square yards.
302 ELEMENTS OF ALGEBRA
956. Sum to 32 terras, 4, J, \ ....
957. Sura to 24 terms, ', - , - 2.
958. Sum to 20 terms, 5, ^ 3 , V-
959. Find an A. P. such that the sum of the first five terms is one
fourth of the sum of the following 1 five terms, the first term being
unity.
Find the sums of the series :
960. 1G + 24 4- 32 4 , to 7 terms ;
961. 16 -f 21 -f 36 4- .-., to 7 terms;
962. 36 + 24 + 1G 4 ...,to infinity.
963. (iiven a - 10, d = 4, s - 88. Find n.
964. How many terms of the series 1 + 3 + 5 + amount to
123,454,321?
965. Sum to n terms, 1 3 + 5 7 +
966. Sum to n terms, 12434+ -.
967. Find the sum of j 4- 1 4- f , to infinity.
968. Sum to infinity, I f + -j$V *"
969. Sum to infinity, -^-1 + 1 - 4-14- ....
V-j - 1 2 - V2 2
970. Sum to 10 terms, (x 4- ;>/) + O 2 4 y 2 ) + O 8 + y*) .
971. Sum to n terms, x(x + y) + x-(x 2 4 y 2 } 4- x*(x 3 -f .v 8 )
972. Sum to 8 terms, (x + y) + (2x + f) + (3 x + y 8 ) 4-
973. Evaluate (a) .141414.-.; (ft) .3151515....
974. Find - -f ^ 1- , to n terms, the terms being in A. P.
n n
975. Find the difference between the sums of the series
5 + !Lni 4. !Ll^ + ... (to 2 n terms),
n n n
and ? 4- " + + - (to infinity).
n+l(n + l) T ( + !) V J '
976. The 10th and 18th terms of an A. P. are 29 and 53. Find the
first term and the common difference.
977. The 9th and llth terms of an A. P. are 1 and 5. Find the
sum of 20 terms.
REVIEW EXERCISE 303
978. Insert 22 arithmetic means between 8 and 54.
979. Insert 8 arithmetic means between 1 and 0.
980. How many terms of 18 + 17 + 10 + -, amount to 105?
981. The sum of n terms of 7 + 9 + 11+ , is 40, Find n.
982. The sum of n terms of an A. P. is "(- + lY Find the 8th
term. L V;3 '
983. The 21st term of an A. P. is 225, and the sum of the first
nine terms is equal to the square of the sum of the first two. Find the
first term, and the common difference.
984. Find four numbers in A. P. such that the product of the first
and fourth may be 55, and of the second and third 03.
985. Find the value of the infinite product 4 v'i v 7 -! v^5 ....
986. A perfect number is a number which equals the sum of
all integers by which it is divisible. If the sum of the series
2 + 2 1 + 2' 2 2 n is prime, then this sum multiplied by the last term
of the series is a perfect number. (Euclid.) Find four perfect
numbers.
987. The Arabian Araphad reports that chess was invented by
Sessa for the amusement of an Indian rajah, named Sheran, who
rewarded the inventor by promising to place 1 grain of wheat on
the 1st square of a chess-board, 2 grains on the 2d, 4 grains on the 3d,
and so on, doubling the number for each successive square on the board.
Find the number of grains which Sessa should have received.
Find the sum of the series :
11 9
988. --- - - + - - - + .-., to oo.
V2 v/2 + 1 4 + 3>/2
989. 5 + 1 + .2 + .04 + ..-, to oo .
990. 1.1 + 2.01 + 3.001 + 4.001 + ., to n terms.
992. What value must a have so that the sum of
2 a + av/2 + a + + , to infinity may be 8?
V2
304 ELEMENTS OF ALGEBRA
993. Insert 3 geometric means between 2 and 162.
994. Insert 4 geometric means between 243 and 32.
995. The fifth term of a G. P. is 4, and the fifth term is 8 times
the second ; find the series.
996. The sum and product of three numbers in G. P. are 28 and
512 ; find the numbers.
997. The sum and sum of squares of four numbers in G. P. are
45 and 765 ; find the numbers.
998. If a, ft, c, are unequal, prove that they cannot be in A. P. and
G. P. at the same time.
999. In a circle whose radius is 1 a square is inscribed, in this
square a circle, in this circle a square, and so forth to infinity. Find
(a) the sum of all circumferences, (I) the sum of the perimeters of
all squares.
1000. The side of an equilateral triangle equals 2. The sides of a
second equilateral triangle equal the altitudes of the first, the sides
of a third triangle equal the altitudes of the second, and so forth to
infinity. Find (a) the sum of all perimeters, (6) the sum of the
areas of all triangles.
1001. Each stroke of the piston of an air pump removes J of the
air contained in the receiver. What fractions of the original amount
of air is contained in the receiver, (a) after 5 strokes, (6) after n
strokes?
1002. Under the conditions of the preceding example, after how
many strokes would the density of the air be xJn ^ ^ ne original
density ?
1003. In an equilateral triangle ABC a circle is inscribed. A
second circle touches the first circle and the sides AB and AC. A
third circle touches the second circle and the same sides, and so forth
to infinity. What is the sum of the areas of all circles, if AB = n
inches.
1004. Two travelers start on the same road. One of them travels
uniformly 10 miles a day. The other travels 8 miles the first day and
increases this pace by \ mile a day each succeeding day. After how
many days will the latter overtake the former?
REVIEW EXEHCISE 305
1005. Write down the first three and the last three terms of
(a - *)".
1006. Write down the expansion of (3 - 2 a; 2 ) 5 .
1007. Expand (1-2 #) 7 .
1008. Write down the first four terms in the expansion of
(x + 2 #).
1009. Find the 9th term of (2 al - o/) 14 .
1010. Find the middle term of ( - l) w .
1011. Find the middle term of (a$ -f bfy.
1012. Find the two middle terms of ( - ft) 19 .
1013. Find the two middle terms of ( + x) 18 .
1014. Find the fifth term of (1 - a:) 9 .
1015. Find the middle term of (a + b) lQ .
1016. Find the two middle terms of (a x) 9 .
(1 \88
*2 X ---- 1
1018. Find the coefficient a: in
1019. Find the middle term of 1 5 a - |V 6 .
1020. Write down the coefficient of x 9 in (5 a 8 - 7 a: 8 ) 7 .
1021. Find the eleventh term of /4 x - -i-V 5 .
>> 2i/
INDEX
[NUMBERS REFER TO PAGES.]
Abscissa 148
Absolute term . 178
" value 4
Addition .... 15, 19, 97, 210
Aggregation, signs of . . . 9, 27
Algebraic expression .... 10
" sum 18
Alternation 123
Antecedent 120
Arithmetic mean 249
*' progression . . . 246
Arrangement of expressions . 20
Average 10
Axiom 54
.Base of a power 8
Binomial -10
" theorem 255
Brace 9
Bracket 9
Character of roots . . . . .193
Checks 20, 37, 49
Clearing equations of fractions . 108
Coefficient 8
Common difference .... 246
** multiple, lowest . . 91
* ratio 251
Completing the square . . .181
Complex fraction 105
Composition ... 8 ... 123
Conditional equations .... 53
Conjugate surds . . . . t . .210
Consequent 120
Consistent equations .... 130
Constant 155
Coordinates 148
Cross product 41
Degree of an equation . . . 232
Difference 23
Discriminant 193
Discussion of problems . . .241
Dividend 45
Division 45
Divisor 45
Elimination 130
Equations 63
' consistent . . . .130
** fractional . . . .108
" graphic representa-
tion of .... 160
11 graphic solution, 158, 160
in quadratic form . 191
' linear .... 54, 129
" literal .... 54, 112
" numerical .... 54
'* quadratic . . . .178
" simple 54
" simultaneous . 129, 232
Evolution ........ 169
807
808
INDEX
Exponent 8
Exponents, law of . . 34, 45, 195
Extraneous roots 222
Extreme 120
Factor 70
" theorem ...... 227
" II. C 81)
Factoring 70, 227
Fourth proportional .... 120
Fractional equations . . . .108
u exponent . . . .105
Fractions. 03
Geometric progression . . .251
Graphic solution of simultane-
ous equations 100
Graphic solution of simple equa-
tions 158
Graph of a function . . . .154
Grouping terms 86
Highest common factor .
Homogeneous equations .
Identities
Imaginary numbers . .
Inconsistent equations
Independent equations .
Index .
Infinite, G. P
Insertion of parentheses .
Integral expression . .
Interpretation of solutions
Inversion
Involution
Irrational numbers . .
89
235
53
109
102
130
9
253
28
70
241
123
105
205
Known numbers . . . . 1
Law of exponents . . 84, 45, 195
Laws of signs 33, 45
Like terms 17
Linear equation 65, 184
Literal equations . . . 54, 112
Lowest common multiple . . 91
Mathematical induction . . . 346
Mean proportional 120
Mean, arithmetic 338
" geometric 341
Member, first and second . . 53
Minuend 23
Monomials 10
Multiple, L.C 91
Multiplication . . .31, 102, 212
Negative exponents
11 numbers .
. 195
. 4
Order of operations . . . . 13
" of surds 205
Ordinate 148
Origin 148
Parenthesis 9, 27
Perfect square ...... 83
Polynomial 10
Polynomials, addition of . . . 19
" square of . . 42
Power 7
Prime factors 76
Problem, 1, 63, 114, 143, 180, 189, 243
Product 7
Progressions, arithmetic . . . 246
'* geometric . . . 251
Proportion 121
Proportional, directly, inversely 122
Quadratic equations .... 178
Quotient 45
Radical equations 221
Radicals 205
INDEX
309
Ratio 120
national 76, 205
Rationalizing denominators . 215
Real numbers 169
Reciprocal 104
Remainder theorem .... 228
Removal of parenthesis ... 27
Root 9
Roots of an equation . . . 54
" character of 193
" sum and product of . . 193
Rule of signs 33, 45
Series 240
Signs of aggregation ... 9, 27
Similar and dissimilar terms . 17
Similar surds 205
Simple equations 54
Simultaneous equations . 129, 232
Square of binomial .... 40
" of polynomial .... 42
Square root 171
Substitution 133
Subtraction 22
Subtrahend 23
Sum, algebraic ...... 18
Surds 205
Term 10
" absolute 178
Theorem, binomial .... 255
Third proportional 120
Transposition 54
Trinomial 10
Unknown numbers .... 1
Value, absolute 4
Variable 155
Vinculum 9
Zero exponent 197
Printed in the United States of America.
ANSWERS
TO
SCHULTZE'S ELEMENTS OF ALGEBRA
COMPILED BY THE AUTHOR
WITH THE ASSISTANCK OP
WILLIAM P. MANGUSE
STrtn gork
THE MACMILLAN COMPANY
1918
All rights reserved
COPYRIGHT, 1910,
BY THE MACMILLAN COMPANY.
Set up and electrotypcd. Published September, 1910.
Reprinted April, 1913; December, 1916; August, 1917.
NorfoooS
J. 8. Gushing Co. Berwick <fe Smith Co.
Norwood, Mass., U.S.A.
ANSWERS
Page 2. 1. 32,8. 2. $160. 3. A .$9400, B $4700. 4. South
America 46,000,000, Australia 5,000,000. 5. Seattle 12 ft., Philadel-
phia ft. 6. A $90, B $ 128, C $ 16. 7. 48 ft., 8 ft. 8. 16 in., 16 in.,
8 in. 9. 18, 18, 144. 10. 150,000,000 negroes, 15,000,000 Indians.
Page 3. 11. A $40, B $80, C $1(50. 12. A $10, B $20, C $60.
13. Bl in., 16f in. 14. A 38 mi., B 10 mi. 15. $100.
1. 7. 2. Not in arithmetic. 3. 9 is larger than 7. 4. 7,
16 - 9 = 7. 5. 3 below 0. 6. - 3. 7. 3.
Page 4. 1. Ilis expenditures.
Page 5. 3. sign. 4. sign. 5. 20 B.C., 6 yd. per sec.
westerly motion. 6. $ 1 50 loss, - 150. 7. - 1, - 1, 8. 13 S., 13.
9. 37 S., -37. 10. -10. 11. -3. 12. -2. 13. -15.
14. -14. 15. -32. 16. -7. 17. -2. 18. 6. 19. - 1.
20. + 1. 21. -1. 22. 1. 23. -3. 24. -3. 25. 1.
27. (a) 1, (/>) -2. 28. 5.
Page 6. - 29. 73, 126, 89, 13 V, 106, 59, 115.
1. 5000. 2. 9,12. 3. 12,2. 4. 85, _ 32. 5. 13 d. 6. Yes.
7. 9 m. 8. 14 b. 9. 210. 10. 9 c.
Page?. 11. 7m. 12. 14, - 14. 13. - 3 m, - 36. 14. - 2 p,
15. -30?. 16. 10g. 17. -26z. 18. 2 x. 19. - 22 x.
20. 20 jo. 21. Multiplication.
Page 8. 1. 7 2 = 49, 6* = 216, 3* = 81, 2 5 ~ 32. 2. 49. 3. 32.
4. 25. 5. 512. 6. 16. 7. 16. 8. 1,000,000. 9. 1. 10. |.
11. 0. 12. 20 \. 13. 24. 14. 25. 15. .00000001. 16. 1.21
17. 13. 18. 27. 19. 4. 20. 1. 21. T V 22. 9. 23. 9. 24. 8
25. 6. 26. 12. 27. 16. 28. 2. 29. }. 30. 3.
Page 9. 1. 16. 2. 3. 3. 8. 4. 4. 5. 12. 6. 64. 7. .
8. 16. 9. 10. 10. 576. 11. 16. 12. 256. 13. 1. 14. Ot
15. 128. 16. 192. 17. 2. 18. ^. 19 2 -
Page 10. 1. 3. 2. 3. 3. 2. 4. 2. 5. 1. 6. 6. 7. 2.
8. 3. 9. 12. 10. 20. 11. 6. 12. 0. 13. 20. 14. 24. 15. 1
16. 8. 17. 12.
i
ii ANSWERS
Page 11. 1. 13. 2. 57. 3. i:-5. 4. II. 5. (59. 6. 104.
7. 237. 8. 17. 9. 58. 10. 92. 11. 240. 12. 14:). 13. 7.
14. 50. 15. 27. 16. 49.
Page 12. 17. 7. 18. 11. 19. 35. 20. ]*. 21. 27, 22. 8.
23. 04. 24. 8. 25. 1. 26. 27. 27. 27. 28. 8. 29. 0. 30. t.
31. a 4- 4 ft. 32. <i~ - 3 /A 33. 8 x :i _- 4 ;r~ -f //-'. 34. m :; + 3(a- 6) .
35. (/* 4- ft)(X- ft-)- 36 - 2 "'* ~ 5V (a -ft)-. 38. Polynomial, Trino-
inial, Polynomial, Monomial, Binomial.
Page 13. 1. (a} 1(5 cm., (b) 135 mi., (r) 2000 m., (rf) 50,000 ft.
2. 00 04 ft., (ft) 100 ft., (V) -o^ft. 3. (a) <> sq. It., (ft) .'JO sq. in.,
(c) 24 S(i. ft.
Pagel4r.-~4. (a) 314 sq. in., (ft) 12.5f> sq. in., (r) 78.5 sq. mi.
5. (<7) $80, (ft) $40. 6. (a) 200,900,000 sq. mi., (ft) 3.14 sq. in.
(c) 314 sq.ft. 7. (a) r>23ifcu. ft., (ft) 14. 13 cu. ft., (r) 2G7,94(>,GOG,<>Gq.
8. (a) 50, (ft) 0, (r) 15.
Page 16. 1. 1. 2. -1. 3. - 0. 4. 0. 5. -2. 6. -15.
7. 8. 8. -14. 9. -22. 10. 0. 11. -15. 12. -5.
13. ,'J. 14. -2. 15. -0. 16. 0. 17. -31. 18. 0.
19. - 12. 20. - 1. 21. 2. 22. - 9. 23. 0. 24. 5.
Page 17. 25. 14. 26. 5. 27. 3. 28. -3. 29. 9. 30. - 3.
31. 12yd. 32. 12 a. 33. 6<t. 34. 14. 35. -4. 36. -I.
37. 3. 38. -4. 39. ^. 40. 51 f. 41. $3000 #1111. 42. \'\
Page 18. 1. -3a. 2. - ab. 3. 0. 4. - 9/^/rl 5. -2ftx.
6. 18.rty8. 7. -32 2 ftc. 8. 15a;. 9. 4 2 . 10. -43w// 2 .
11. 22( + ft). 12. -- 20(.r -f ^). 13. -2oVm-f?i. .
Page 19. 14. -38 ab. 15. :J!>r'. 16. -21^. 17. \(\xyz.
18. vi -f- w. 19. m n. 20. ??i- ??, 2 . 21. r/ 3 . 22. ft 1.
23. r+l. 24. ft + ft' 2 . 25. 7 - r:A 26. WIN + wiw- a .
27. -yyz+xyz*. 28. v> -f y v> 7. 29. ft. 30. a 2 15^4-4.
31. 3rf - 5e - 12.x. 32. - rt- 4- <Mft - 2ft 2 . 33. 2V^4-^/ 4- Va'+Y 2 .
34. 173. 35. 0* = 81. 36. 3* = ()501.
Page 20. 1. a - ft - 3 c. 2. - q 4- 3 * - <3 . 3. 8 x* - 9?/ 2 - 8^ 2 .
Page 21. 4. c, +/-2(/. 5. - 10 r5 2 4- 13 ft 2 - 32 c 2 . 6. 8.r 2 -5x+3.
7. 3 w" 1 + 5. 8. - jrif 4- y. 9. 21 a 3 4 > ft 3 . 10. 3 a* + at*.
11. 0. 12. 2x' 5 . 13. -3(c4-a). 14. a 4 4- a a 4-1. 15. (a 4- ft) 2 4- 1.
16. arty - x-V 2 3 x^ - y/ 4 . 17. v'ft - Vc. 18. a 3 4- a 2 4- a 4- 1.
19. $r*y 4- 3x?/ y . 20. 4 m* 4- <> run- - 2 ?t 3 .
AXSH'EJtS ii\
Page 22. 21. 3 nv> + 4 m 4 4- 8 7?i 8 - G m. 22. 4 b + c 4 d + e.
23. w a 4- 24 npy - 12 m?/' 2 - 17 y*. 24. 8 8 . ' 25. - - x - 6 a: 2 .
26. 3 a 2 + 7 . 27. - # + 12.
Page 24. 1. -2. 2. -12. 3. 2. 4. G. 5. -11. 6. -14.
7. - 17. 8. - 1. 9. 17. 10. 0. 11. - 24. 12. 31.
13. 7. 14. - 7. 15. - t). 16. - 1. 17. 1. 18. - 1.
19. -18. 20. 1. 21. 34. 22. - ri\ 23. n*. 24. 32 w 2 w.
25. 0. 26. 4wipg>' 2 . 27. ISjfat. 28. - abc. 29. 10w.r.
30. 7 a 5 . 31. a - 6. 32. a 4- h. 33. - a - b. 34 -- a 4 ft.
35. a 1 -a. 36. a 2 + a. 37. 1 4- a*. 38. ab a. 39. a 2 6 + 2 .
40.
Page 25. 41. - 10 m. 42. - 4 a - 24 b + c. 43. - 14 afy 4- 2 ?/ 3 .
44. - 2 a :5 + 4. 45. - G -\- J8 r - 25 <?. 46. 4 ?/-. 47. Z-mn + qt - 2 tf.
48. a :{ - a' 2 - a - 2. 49. G a b-d. 50. - ?>-4-tl 51. G/ 4 - 3x ;5 - x -f 1 .
52. a - c. 53. - ' ! - </- - a -f 1. 54. 3 a 2 - 5 z?/ + 3 y 2 .
55. 3 a' 1 4ab + ?/-. 56. - 1 - 2 - 2 /> - 4 d. 57. 8 + 2 a - a~.
58. -8a*b-8<tb'\ 59. - (a -f 6) + 4(1 + c) - 8(c + a).
1. a + fc + 4 r. 2. - Oa: + 10. 3. - a - 4.
Page 26. 4. .r 2 . 5. - r' 2 4- a- 1. 6. - ^ 2 4- 5 - 1. 7. 3 m.
8. a -f 2 /> f 2 c. 9. 2 y' 2 4- 2 z 2 . 10. 2 a -f 6 4- <;. 11. 2 a 3 + a 2 -f 2 a 4- 5.
12. s_r>a-5. 13. _5a-<>&4-3c. 14. 3-6. 15. 4r 2 - 4x- 2.
16. 2x 4 2y. 17. 2 2-. 18. 3x -f z. 19. x - 2// + 3z. 20. M + 10, 2m.
21. a + 6, w & 4- c.
Page 28. 1. x -f 2 ?/ - 2. 2. a 3 & + 2 r. 3. 3 a 3 - b* 4- r- 1 .
4. _2?> 2 + c 2 . 5. - 2 a 2 4- 2 ft 2 -r 2 . 6. 2 - &. 7. a - 2 &.
8. 3 a- 3 6. 9. 2m -2 ti. 10. 0. 11. -2. 12. 3 a 4- 8 b.
13. 10 x. 14. -w4-.'U4j>. 15- 2m + 2w. 16. 2 ar. 17. 2a- 2 -a;-l.
18. lOrt 364-c. 19. -37;. 20. 2a: 2 4-x. 21. 7 - a + 2 ?> + c.
22. 814.
Page 29. Exercise 16. 1. a + (ft-c4-df). 2. 2m-(4?
3. 5 2 - (7x 2 -Ox-2). 4. 4ir#-(2;r 2 + 4?/ 2 4-l).
(2n' 2 43p 2 -47 2 ). 6. x -(y-z~d}. 7. j)-(-g-
8. 5x 2 (5x4-7 a). 9. a 8 ( 2 + a4-l). Exercise 17. 1.
2. rt ft. 3. a 2 -f ft 2 . 4. m* - n*. 5. w* - w s . 6. a 4 4 ft 4 . 7. mn.
8. ?/i 3 Ji 8 . 9. 3m 2 n 2 . 10. (mn} 8 . 11. ( - ft) 2 . 12. (w4w)(w-w).
13. 9 (a + $) 2 - aft.
_ 3 4ft 8 4-c 8
Page 30. 14. a 2 4- ft 2 4 V-e. 15. x 3 - (2 x 2 - 6 x + 0) . 16. -~- J-.
iv ANSWERS
17. fa + &)(- 6) =a 2 - ft 2 . 18> ^|* = a 2 + aft 4- ft 2 . 19. ^^ =
a + ft.
Page 31. 1. -. 2. +. 3. -. 4. 15 lb., (+3)x6=+16.
5. -161b.,-15. 6. -161b., -15. 7. 16 lb., +15. 8. 20, -20, -20,
20, etc.
Page 33. 1. -30. 2. 84. 3. -12. 4. -42. 5. -18. 6. 60.
7. 15. 8. -28. 9 13. 10. 1. 11. 18. 12. 14f 13. 4. 14. -27.
15. -04. 16. -1. 17. 10,000. 18. 120. 21. 24. 22. 102. 23. -24.
24. 0. 25. -108. 26. -108. 27. 0. 28. -64. 29. -30. 30. 34.
31. 29. 32. 16.
Page 34. 33. 343. 34. 216. 35. -216. 36. 125. 1. m. 2. ci 5 .
3. 2'-'. 4. 37. 5. 6". 6. 127-"'. 7. (a6) n . 8. (x -f ?/) 8 . 9. - a 12 .
10. 7 G . 11. 12 ! ^. 12. 21 a-'&c. 13. 1400. 14. 210. 15. 3300. 16. 770.
17. 4200.
Page 35. 18. 60. 19. 30. 20. 20 aW. 21. 38wiw. 22. -ISartyW
23. 38 a*b 6 . 24. - 44 aWc 8 . 25. 30 a*b*c. 26. - 36 e*f*tj.
27. 49 p*qh*t. 28. 16 abxy. 29. 66 ? 8W 2 * 80 . 30. 30 n?b*c*.
31. -18a% : y. 32. 16 4 fc. 33. 4aWy. 34. 9 w-w. 35. - 8 a' 1 .
1. 90. 2. 360. 3. 51. 4. 42. 5. 13. 6. 1904. 7. 25,7(50.
Page 36. 8. 5 2 + 6 s + 5 8 . 9. s + O 4 -f 6 6 . 10. 7 + 7 6 + 7 1S .
11. 4^-4^-414. 12. ll9H-H 2) + ll 2 i. 13. 2 w 2 + 2 r/m + 2 mp.
14. 2 w 3 ?i 2 wiw 8 + 2 wiwp 2 . 15. 2 x*y* 2 .r% 2 2 ry.
16. 1(5 pag'V + 20 ;>(/ 2 r - 28 p'^/-. 17. 25 .r 4 4- 25 x* + 25 x 2 .
18. - 14 ;:>/ - 14 xyz + 14 a:y0. 19. - 10 3 W -f 15 aWc 6 30 a a ?> 4 c 4 .
20. - 35 a*b*c 8 -f 14 a?/e - 21 a 4 7> 3 c 2 . 21. - 18 w 4 w :j + 10 WI !} M - 14 w 2 //. 2 .
22. - 32 y s s G - 16 x2/ 4 5 4- 64 -jcy*z*>. 23. - 19p" + 19^ 10 - 57 p 6 190 p6.
24. 3 a; 2 -4 a: + 7. 25. a;(3x2_4^+7). 26. 3(*+0 + 2).
27. 3a: 2 (2a:-f iH-a; 2 ). 28. 2 a 2 (y - 2 xy -f 4 a; 2 ). 29. 5aft(a- 126- 2).
30.
Page 38. 1. Ox a -5 r?/-6j/ 2 . 2. 8 2 +26-21ft 2 . 3. 10c 2 -19rd+0c? a .
4. 14 m 2 - 6 wiw 24 n 2 . 5. 30 j9 2 + 43 jt?g -j- 15 q\ 6. 4 a 2 - 22 ac + 30 c 2 .
7. 36 ?/i 2 + 65 ww - 14 7t 2 . 8. x 2 -xy-42^. 9. 66 I' 2 - 83 In + 1 n*.
10. 39 k* - 19 A; + 2. 11. 12 x 2 ?/ 2 - 20 xyz - 8 . 12. 40 r 2 - 69 rt + 21 .
13. 132 r' 2 + r - 1. 14. a: 2 j/ 2 2 - 11 xyz - 12. 15. 2 a*6 2 c 2 + 11 a&c - 21.
16. 3 -a 2 -4-6. 17. 2*8-f x 2 -6x-4. 18. 4 m 3 + 9m 2 + m- 2.
19. 2 n 8 9 w 2 + 13 n - 12. 20. 3 a 3 - 14 a 2 29 a + 30.
21. _6g8 + 9 (? 2_i2g + 8. 22. 9z 8 -16z 2 -9z + 10. 23. 2z 8 -s 2 -3z-l.
24. 4 a 8 - 16 a 2 + 32 a - 32. 25. a 8 - 3 a 2 6 + 3 aft 2 - ft 8 .
.26. 4a-12 a 2 ft-f 5aft 2 -f 6 6. 27. iSx 8 - 6 a; 2 - 25 a: + 14.
ANSWERS V
28. -2 m 3 + 4m 2 - m - 1. 29. a 2 - ab + ft 2 . 30. lflrt 2 -8 + l.
31. 30 2 - 84 a + 49. 32. 30 >2 />-<; 2 - 00 o + 25. 33. 8 a ; W - 10 -7> a
+ 7 6 - 2. 34. ^/> 8 - 8 r. 35. m 3 ?i ;5 j) 3 - 8. 36. x* - 1.
37. 4 + 4 a & 2 + 106 4 . 38. .<* + .r* - ^ + a? + 1. 39. x*-2^-f I,
40. m 4 ?7i%' 2 + tt 4 . 41. a 3 G a-6 + 12 ab 2 8 fo*.
Page 39. 1. a' 2 + 5 + 0. 2. a- a 0. 3. ^' J - 7 -f 12.
4. # 2 -4x-21. 5. x 2 -GiC+5. 6. j3 2 -.p-132. 7. m' 2 +18?rt + 81.
8. 6 2 + 6-lf>0. 9. ?/ 2 -21 ?/-H)0. 10. Z 2 + 48Z-100. 11. 2 +10s-281.
12. a 2 - ab - 2 6' 2 . 13. a' 2 - 5 6 + ft' 2 . 14. ' 2 + ab - 2 6 2 .
15. ' 2 ~ 81. 16. m 2 - ri 2 . 17. 4 ~ 2 6 2 20 6 4 . 18. x 4 ?/ 4 - x 2 y- V2
19. ^V^ 4 - ^V"' 2 - 1: ^- 20. - 16 a 3 -f 50. 21. 10,500. 22. 132.
23. 1,000,020. 24. 10,008. 25. 1,000,994. 26. 10,606. 27. 1,009,020.
28. 10,098. 29. (x - 2) (x 1).
Page 40. 30. (x-2)(x-3). 31. (a + 4) (a + 2). 32. ( 5)
(rt-2). 33. O-5)(w + 3). 34. (m + 6)(m-3). 35. (w-4)(w + l).
36. (n 8)(?i 2). 37. (p 0)(p + 5).
1. 2 + 2 fz& + fr 2 . 2. 2 + 4 a +4. 3. a 2 - 2 x + .r 2 . 4. 2 - 4 a + 4,
5. i> 2 + p + 9. 6. x- - 10 x + 25. 7. p 2 - 14 jp + 49. 8. a' 2 - 4 a&+ 4 &*.
9. x 2 -f xy + 9 y' 2 .
Page 41. 10. 4 x 2 12 xy + 9 ?/ 2 . 11. 10 a' 2 24 ab + 9 & 2 .
12. a 4 - n 2 + 9. 13. p 4 + 10 p 2 g + 25 q*. 14. 9 /> 4 - 54 p 2 + 81.
15. 36 a 4 - 84 a' 2 ?/ 2 + 49 & 4 . 16. 4 2 ?> >2 -4 a&c + c 2 . 17. 4 .r 2 //- + l5J x// + 9 .-/ 2 .
18. 10 a 4 ?; 4 40 (i 2 V 2 c 2 + 25 r 4 . 19. 30 x 4 ?/ 4 ^ 4 -()Or 2 //-^ 2 + 25. 20. w-'-n 2 .
21. 4 m' 2 - 9. 22. a 4 - 49. 23. r*d< - 25. 24. x 4 - 121 ?/ 4 .
25. x 4 22 x 2 ?/ 2 + 121 ?/ 4 . 26. x 4 + 22 x/ 2 ?/' 2 + 121 y*. 27. 25 r 4 - 4 4 .
28. 25 y- 4 - 20 r-t 2 + 4 t*. 29. <z 2 + 10 a + 25. 30. ?/i 4 - 4 n.
31. 10,201. 32. 10,404. 33. 10,009. 34. 10,810. 35. 998,001.
36. 990,004. 37. 9801. 38. 441. 39. 484. 40. 9990. 41. 9999.
42. 999,996. 43. x + y. 44. a-b. 45. m - 1. 46. n + 2.
47. s rc - 3. 48. - 4' 6/ 49. a + 56. ' 50. ' (r + ?/) (x - I/).
51. (a + 3)(-3). 52. (w+4)(m-4). 53. (b + 5 ?i)(& - 6 )
54. (3 + 7 6)(3a~76>. 55. (46c + 5) (4 abc - 5). 56. (6 a 2 + 3)
(5 a 2 -3)., 57. (3a-66)(3a-6&).
Page 42. 1. 2 a 2 + a - 6. 2. 3wi 2 -m 2. 3. Om 2 6m -6.
4. 20a 2 -21a + 4. 5. 12 x 2 - 6 xy - 2 y*. 6. 25 a 2 6 2 - 35 ab + 12.
7. 2x 4 +7x 2 6 2 15 6 4 . 8. 2 a 4 6 4 +8 a 2 6 2 36. 9. 2xV+6x 2 y 2 ^ 2 +22; 4 .
10. 6 x 6 + 13 x 3 - 15. 11. - x 2 + 6 x 2 y 2 - 6 y 4 . 12. 30 x 4 + 19 x 3 - 6 x 2
13. 166. 14. 10,712.
VI ANSWERS
Page 43. Exercise 26. 1. m' 2 4- n 2 +p 2 4-2 mn + 2 mp 4 2 np.
2. x* 4- ?/ 2 4 z 2 2 a:// + 2 jrz 2 ?/2. 3. a 2 4- ft 2 -f 2 aft 10 a \- 25 - 10 ft.
4. a 2 4-4 ft 2 + c 2 -4 aft-2 ac + 4 ftc. 5. a 2 4-10 ft' 2 + 9 r 2 -8 ft + w-2l ftc.
6. a 2 4 ft 2 4- c 2 4 d 2 4- 2 aft 4- 2 ac - 2 wZ 4- 2 ftc - 2 ft*/ - 2 <</. 7. r 1 -
8 .r" -|- 20 S? - lit x + 4. 8. 4 a 2 4- 9 ft' 2 4- 25 c 2 - 12 aft 4 20 ac - 30 ftc.
9. r/ 2 4- 10 ?/ 2 + z 2 4- 2 - 24 .r?/ - 6 :rs 4- rw -f 8 j/.r - 8 yn - 2 .rw.
10. 4 a* 4 9 ft* + 16 r 4 + 12 a' 2 // 2 - 1*5 2 r 2 - 2-1 ft 2 /- 2 . 11. .r -f ?/ + -.
12. ??i + w ?). 13. ft c 1. Exercise 27. 1.5 a - 15.
2. _ 2 a - 8. 3. x 4- 23. 4. 7 r - 29. 5. - w. 6. 2 ft' ? 4- ft 2 - 3 ft.
7. 2 m 2 4 2 w 2 . 8. 2.
Page 44. 9. -5.r?/. 10. 4 pq. 11. - ???' 2 - 25. 12. 4 x.
13. - 34. 14. - 2 j) 2 - ;> + 77 . 15.11 sr 2 4- 2 1 r//- 8 ?/' 2 . 16. 8 r<ft - 4 ft 2 .
17. _ 5 x -i _ <) ,/ _ i o tji. 18. - // 2 4- 27 //. 19. 2 ^r 2 + 2 aft 4 tt ac - 2 ftc.
20. -4xy + 13 ?/' 2 . 21. x 2 4- 15 :r - !>. 22. <r 2 - a.
Page 46. 1. 5. 2. -5. 3. -13. 4. 8. 5. 3. 6. -9.
7. 01. 8. -49. 9. -125. 10. 12. 11. 135. 12. 50,000.
13. 12. 14. -3. 15, 3 a-. 16. 3 aft '. 17. - 4 a-c-.
Page 47. 18. - 5 mp. 19. - 12 y 2 ? 14 . 20. 1. 21. - 2 ?nc w .
22. - 4. 23. 75 a 2 . 24. 14 r 2 . 25. 20 a. 26. a 4- ft.
1. 21. 2. 29. 3. -G. 4. 15. 5. 10. 6. 2 ft - 3 c.
7. - 2 ?/ 4- 3 ry. 8. 9 a' 2 ft 2 4- 3 aft. 9. - 5 a 4 - 4 a 2 4- -
10. 5 aft - 4 ft 2 4 3. 11. 5 - 3 w 4-*7 m 2 .
Page 48. 12. 2 w 2 4- 3 mn - 5 n*. 13. - 3 x' 2 u' 2 z~ 4- 2 ar.yar - 1.
14. 7a 2 ftc 4 -4c4-2a. 15. f>r* 4- 7 arty - 3^V- x^.
16. 4 x 2 //V 2 - 3 Z2 3 - 5 z* - 2. 17. - 6 x l 4- 4 .r'^ - 3 x 2 */ 2 4- 2 ?/ 4 - y3.
Page 50. 1. x-4. 2. y + 3. 3. 8 x - 8 y. 4. 5 m - n.
5. 4 a Oft. 6. 4x4-3?/. 7. /r - 8. 8. 7 x + 5. 9. 4 c - 9 d.
10. 5^4-18(7. 11. Os-y. 12. 7a-3ft. 13. 5 a - (5 ft.
Page 51. -14. 3 aft + 4. 15. 4 x 2 y ~ 5. 16. a 2 - 3 a 4-1.
17. l4-8m4-7?n 2 . 18. ,r' 2 4- 2 x 4- 4. 19. x 2 + 2r f J.
20. 9w 2 + 0m+ 1.
1. ?--?. 2. y-fl. 3. c-3. 4. a 4- 4 ft. 5. 2 a - 3 ft.
6. ti'jry-1 z. 7. 1.3 a 2 ft 4- 9 c 3 . 8. 8 x 5 + 1.
Page 52. 9. m L 4- 1, w + 1, w - 1, ? 2 - 1. 10. <z 2 4- ft 3 , a 2 - ft 8 .
11. aftc + 1, abc - 1. 12. r 4- ?/ 8 , 6 - ?/ 8 , * 3 - y 4 , x 8 4- 1/*.
13. 6 x 2 t/ 2 4 7, 3*y2 - 7. 14. 11 a r '4-3 ft, ll'a^-S ft. 15. a 8 4- 10 xy*,
a s _ 10 x?/ 2 . 16. 1000 4- 1, 1000 - 1.
ANSWERS vil
Page 56. 1. 6. 2. !). 3. 5. 4. 8. 5. 7. 6. 6. 7. 2. 8. 2.
9. 2. 10. 1. 11. 11. 12. 4. 13. 5. 14. 4. 15. 22.
16. 00. 17. 1. 18. 7. 19. I. 20. 7. 21 % . 2. 22. 10.
23. 5. 24. 2.
Page 57. 25. 7. 26. 3. 27. 4. 28. 6. 29. - 1$. 30. -13.
31. l. 32. - 12. 33. }f. 34. 5. 35. <>. 36. 0. 37. - 2\.
38. 1&. 39. I. 40. 1. 41. ] 2 ri . 42. 0. 43. 20. 44. 20.
Page 58. 1. a - 10. 2. - x. 3. a -f- 4. 4. m + M. 5. n -
6
6. - 7. 10 -Jj- 8. >-_&. 9. 2b- ( -. lO.p+7. 11. ,100-.
12. 10, a -10. 13. b,a-b 14. d + s. 15. </ - d. 16. 3x2.
17. 7x 2 .
Page 59. 18. 2b - a. 19. + !, + 2. 20. ;r-1,rr-2.
21. x y yr. 22. y 6 yr., y -f 10 yr. 23. / + y + 12 yr.,
x + y 10 yr. 24. $ ?>i -f (I, 8 n - (>. 25. 100 d ct., 10 x ct.
26. 100 a + 10 b + c ct. 27. 100 a - b ct. 28. a-// sq. ft.
29. xy + 3x + 2y + (> sq. ft. 30. 'nj + 4 x 3 y - 12 sq. ft. 31. 2 x +
2y ft, 32. 10 act. 33. 2 ct. 34. ^ ct. 35. '^ ct.
X x .r
Page 60. 36. r>?imi. 37. rn mi. 38. "mi. 39. " lir.
4 r
40. tx mi. 41. # + 20yr. 42. ? 43. -f 44. fix.
x x y
45. iL*. 46. 10 a. 47. 48. . 49. m -+~- 50. lOx + w.
60 25 100 m
Page 61. 1. 2=10. 2. 2z + 10 = c. 3. -f 10 = 2 x.
4. p = c. b. x y + 1=a. 6. 2 4-^ = 100. 7. 4(a -ft) c = eZ 9.
3 3
8. (a -f 6)(o - />) - 90 = a ll ) -. 9. 2^ - 20 = a - 7.
7
Page 62. 10. a - 9 = 17 -a. 11. x = - 5 - x 460. 12. x = m.
13. 100= -^- x700. 14. 50= L 15. m= -- ?i 16. (a) 2x=2(3x~10),
100 100 100 ' v
(6) 2x- (3x- 10) =4, (c) 2x- 6 = *, (d) 2a + 10 = n, (c) 2a? + 3
rrax-lO, (/) (2fl5-8)-h(8ar-ia)=60, (</) 2a; + 3 = 2(3* - 7),
(A) (2 a; + 10) + 3 x -f (4 x - 10) = 100. 17. (a) 2 x - (3x - 700) = 5,
(6) 2 z-f 20 = 3^-740, (c) (2z-f 600) + (3 x- 200) -f(^ + 1700) = 12,000,
(d) 2 x + (3 sc 700) = (x -f 1200) - 200, (e) 3 x - 800 = x + 1300.
vili ANSWERS
18. (a) 5 ~=90, (6) --(6 a -30) =20, (c) J^. = _?_(2ar + 1),
V MOO _ ' v ' HXT ^ ^ ; 100 100 k ; '
(d) -^-~ ft -(5z-30) =900, (e) -i* + -A- (5z - 30) + (2s + 1)
v J 100 100 V ' v ' 100 100 v 100 '
=^8000, (/) ? 2 - = SJL+J- .
w ' 100 10
Page 64. 1. 13. 2. 15. 3. 25. 4. 18. 5. 7. 6. 9.
7. 20 yr. 8. 30 yr. 9. 90 mi. 10. 2$. 11. 1250. 12. 24J.
13. 85 ft. 14. 30 mi.
Pace 65. 15. 250. 16. 300. 17. $40. 18. 80 A. 19. 150,000.
Page 67. 1. 55,11. 2. 05,5. 3. 30,0.
Page 68. 4. 12,2. 5. 78,79. 6. 52,13. 7. 8,10. 8. 160 lb.,
480 Ib. 9. 13,7. 10. 40 yr., 10 yr. 11. 29,000 ft., 20,000 ft.
12. 4pt., 5pt. 13. 42yr., 28yr. 14. 45 in., 15 in. 15. 7 hr.
Page 70. 1. 5, 10, 25. 2. 6, 12, 14.
Page 71. 3. 12,8,24. 4. 1,3,5. 5. 3,0,16. 6. 20,21,22.
7. 8 in., 9 in., 11 in. 8. 1,000,000 Phil., 2,000,000 Berlin, 4,000,000 N. Y.
9. 30, 50, 100. 10. 21. 11. 20 yr., 10 yr., 25 yr. 12. 6, 7, 8.
13. 90,000,000 gold, 180,000,000 copper, 480,000,000 pig iron.
Page 72. 14. 5 Col., 10 Cal., 10 Mass.
Page 74. 1. 15 yd., 20 yd. 2. 10 yd. by 12 yd. 3. 200.
4. 600, 1200. 5. 200, 1200. 6. 70^,210^. 7. 5 lb., 1 lb.
8. 4. 9. 3 hr., 15 mi.
Page 75. 10. 12 mi. 11. 5$ hr. 12. 82 mi.
Page 78. 1. Oaj(o6-2cd). 2. 3x a (3r.-2). 3. 6 aty (3 + 4 6) .
4. 7a*fe(2a 2 & 2 -l). 5. 11 w(w' 2 + wi - 1). 6. z?/(4^ + 5xy - 6).
7. 17z 8 (l-3z + 2x-'). 8. 8(a6 2 +6c 2 -c 2 a 2 ). 9.
10. a a (a 8 -a+l). 11. 10aVy(2a 2 -ay4-3y 2 ). 12.
13. 6rt 2 (2a6-3?2_4 a '2 / ^) > 14. 17 7>c(2 a'^c 2 - 3aftc + 4).
15. 11 pV (2 p 8 - 5p 2 + 7 g 8 ). 16. 13 a 8 ?/ 4 * 5 (5-3 xyz + x 2 y'W).
17. ?(g 8 -? 2 -g+ 1). 18. (m + n)(a + 6). 19. 3 (a +&)(*- y").
20. (p + 7)(3a-5&). 21. 13-13. 22. 2.3.4-11. 23. 2 . 3 6 - 7.
Page 79. 1. (a -4) (a- 3). 2. (a + 4)(a + 8). 3. (ro-3)(w--2).
4. (z-5)(z-2). 5. (a + 6) (a + 3). 6. (a-5)(a-4).
PageSO. 7. (*-4)( + 2). 8. ( + 4)(*-2). 9. (y-8)(y + 2).
10. (y + 8)(y-2). 11. (y-ll)(y-4). 12. (y-7)(y + 2).
13. (y + 7)(y-3). 14. (a + 5)(a + 6). 15.
ANSWERS ix
16. (p-8)0> + l). 17. (<7 + 8)(g-3). 18. (az + 9)(ox-2).
19. (a -7 6) (a -10 6). 20. (a - 11 6) (a 4- 2 ft). 21. (a 2 + 10) (a 2 -2).
22. (ay-8)(ay-3). 23. (w + 20)(w + 5). 24. (y + 4)(y-l).
25. (a -6 6) (a 4- 4 6). 26. (n 2 + 12)(n 2 + 5). 27. (a 3 + 10)(a- 3).
28. (a 4 -10) (a 4 + 3). 29. x (z + 2)(x + 3). 30. 100(x- 3)(z-2).
31. Oa 2 (a-2)(a-l). 32. y(x- 7) (a; + 3). 33. a 2 (w-7)(w + 3).
34. 10x 2 (y-9)(y + 2). 35. 200 (x + l)(x + 1). 36. 4 (a - 11 ft)(a-6).
Page 82. 1. (2x-l)(x + f>). 2. (4a-l)(a-2). 3. (3*-2)(.r-2).
4. (5w-l)(m-5). 5. (3 n + 4) (2 - 1). 6. (3x+l)(x + 4).
7. 3(x + 2)(z-l). 8. (4y-3)(3y + 2). 9.* (2 y + 3)(y- 1).
10. (2 *+!)(* -9). 11. (5 a -2) (2 a -3). 12. (9y-4)(y + 4).
13. (2w+l)(ro + 3). 14. (5x - 7)(2z -f 1). 15. (4 a: -3)(3a; - 2).
16. (6n + l)(+2). 17. (2y-l)(y + 9). 18. (7 a + 4) (2 a - 1).
19. (3#-y)(+4y). 20. (15z-2y)(x-5y). 21. (5a-4ft)(2 a~3 ft).
22. (:5-2y)(2a!-3y). 23. (4a;-5y)(3a; + 2y). 24. 2(2s + 3)(a: + 2).
25. a(2u; + 3)(-c4-4). 26. x (5 a; + 4) (a; -f 2). 27. 10(2 a; -y) (a;- 2 y).
28. 100 (a; -y) 2 . 29. a*(5a -f l)(flr - 2). 30. 10 y 2 (\) x + l)(x~ 3).
31. 10(3 -5 6) 2 . 32. -y' 2 (2y-3)(2y-l). 33. 10 a 2 (4 - w*)(l -2 n 2 ).
34. 2(9a:-8y)(8a:-0y). 35. (2 a? 4- 3 y 2 )(2 a: 2 -f y' 2 )-
Page 83. 1. Yes, (m + w) 2 - 2 - No - 3 - Yes, (g - 6) 2 . 4. No,
(a; -8) (a; -2). 5. Yes, (a- 2 by 2 . 6. No. 7. Yes, (m-7n) 2 .
8. No. 9. No. 10. Yes, (w* + 3?i) 2 . 11. Yes, (y-8) 2 . 12. Yes,
(3a-26). 13. Yes, (5x-2y) 2 . 14. No.
Page 84. 15. Yes, (4 a- 3 by 2 . 16. Yes, (6 a; 2 + 5) a . 17. Yes,
(15a-y-2) 2 . 18. Yes, 10(a - 6) 2 . 19. Yes, x\x - y) 2 . 20. Yes,
w(?-3) 2 . 21. 9ft 2 . 22. 10. 23. 9. 24. 24 aft. 25. 216 aft.
26. 140 w 2 . 27. 9. 28. 25. 29. 9. 30. 40 x.
1. (* + y)(z-y). 2. (a + 8)(a-3). 3. (0 + 6)(6-6).
4. (2o + l)(2-l). 5. (l+7a)(l-7a). 6. (0 + 0(9-0-
7. (10a + ft)(10a-ft). 8. (ft + ll)(aft-ll). 9. (3a;+4 y)(3x-4 y).
Page 85. 10. (5xy + 9^)(oxy - 9*). 11. (7 ay + 8) (7 ay - 8).
12. (5a 2 +l)(5a 2 -l). 13. (10 aft + c 2 ) (10 aft - c 2 ).
14. (13a 2 +10)(13a 2 -10). 15. (15a + 46*)(16a-46).
16. ( 2 + ft 2 )(a + ft)(a-ft). 17. (a*& 3 + 9) (aft + 3) (aft -3).
18. (l + x 4 )(l + x 2 )(l + x)(l-x). 19. 10(a + ft)(a-ft).
20. 13x(a + ft)(a-ft). 21. x(x +y)(x -y). 22. (x + y 4 )(x - y 4 ).
23. 3a;y 8 (6x + 4)(5x-4). 24. 2 y(ll x 2 + 1)(11 x 2 - 1).
25. B 2 (12+ y 2 )(12-y 2 ). 26. 13x7. 27. 103x97.
1. (m + n +p)(m + w-p). 2. (w ~ n
3. (m + n + 4p)(w + - 4p). .4. (x + y +
X ANSWERS
5. (4x 4- y -\- -r)(4x - .y 2). 6. (f> + fo r)(5a 6 -f c).
7. (m + 2 u + (\p)(m + '2 n - Gp). 8. (m3n + a + b)(m 3n-ab).
9. (2 a ~ f> b 4- 5 <: - 9 </) (2 a - 5 b 5 c + 9 iZ) . 10. a (a + 2 6).
11. y(2x-?/). 12. x(x4-ti<0. 13. (5a+l)(9-a).
14. (5^-4- //)(5y- j).
Page 86. 1. (a + b) (r 4- y). 2. (w - ,) ( -{- &).
3. (2a-3fc)0*+ tf)- 4. (r4-20(4 o- fid). 5 - ( 2 * - y)(fi a - 36).
6. Or 4-1) (^4-2). 7. (.i-4-l)(x4-l)(x~l). 8. (w' 2 4- >*-)(:> 4- y )(.-?/).
9. (3a 2 -4// 2 )(x4->/). 10. (a - fc)( lj 4- ?>-) H. (c- 7)(^ 4- 2).
12. ( a -^)(^- ?>).
Page 87. Exercise 46. 1. O + ?/4-<?)O 4- {I q). 2. (14.^4-6)
(!__/>). 3. (a 2 /> + o) (ff 2 ?> T>). 4. (<> r4-y-3 c)((> a- ?/ 4- 3 ).
5. (3 w w 4- />; (-> m n - ah}. 6. ( 4- ^ 4- .^ 4- .'/)('< 4- & y).
7. (a - 5 & 4-# - 2 y) (a - 5 />-z + 2 //). 8. (x- 4- x 4- 1) (^ x - 1).
Exercise 47. 1. (wi 4- 4)(? - 4). 2. 8(w 2 4-2). 3. (7/1 -2) (m 4-1).
4. (6a 2 4-l)(a 2 +)- 5 - ( K + l) a (a - I)' 2 . 6. (x-f!/)' 2 . 7. a 2 (a-9).
8. 2( k 2x-2/)(x-2?/). 9. (m - 3 4- w)(m- 3- n).
Page 88. 10. x(x -f y)(jr - y). 11. 2(5 a - ft) (a - 3 ?>).
12. 1(V/ 2 4-& 2 )(tt4-/>)('e -&) 13> -(7rt-3)(7a~3).
14. 10(2 2 6 2 4-l)(a' 2 & 2 5). 15. (3 4- a + (3 7>)(3 - a fo).
16. (2x-7)(x 2 -2). 17. (14- w 4 )(l 4- w 2 )(l + ^OC 1 - 0-
18. ?i(w 4-y) ( y)- 19- r)(x- ( ,))(x - 1). 20. (< 3 l.'J)(' l + 12).
21. .r(3x' 2 -22/)((3-x). 22. 3(4-7>4-4)(^4-'> 4). 23. 10(8x' 2 4-l)
(x4-2)(x-2). 24. a(a 2 + !)(+ !)( - !) 25. (a 4- l)( a 4- 1).
26. ( rt 2_rt4-l)(a a -rt- 1). 27. (Ox - 7 ?/)(7 x- (5 y/).
28. (5 4- 1 ) (2 m 8 - 9). 29. (5 al) 3) (f> a/> 4- 8). 30. 13( + .'})(c - 4).
31. ( 4- 15 ?>)(a - 15 ?>). 32. 2(5 n - 7s) (2 n 4-3*). 33. (a 4- &4- 8)
(^ + ?>_8). 34. (16 4- - 2 y) (16- a; 4- 2 y). 35. 2 (a 2 4- 8) ( -8).
36. (x//,?-50)(xt/z- 1). 37. 17(x4-3//)(x-2y). 38. (* _ 2 )(a 4- &).
39. fi(c4-26). 40. 3p 2 (^-9)(j) 3 -4). 41. 3(.e 2 4- 3)(x 4- 2)(x - 2).
Page 89. 1. 2 a 2 ^ 2 . 5 a 2 6c 2 . 3. 13x 3 y. 4. 12. 5. 450.
6. 7.
Page 90. 7. WIM. 8. 13 x 8 ^ 4 - 9. 19. 10. 15 M. 11. 4 a 8 .
12. (w4-w) 2 - 13. 2(m4-l)' 2 . 14. 3x(x-?/) 8 .
1. 4 a s & 8 . 2. 5 x 8 . 3.5 m 2 . 4. 3 x 4 . 5. a - b. 6. x 4- y.
7. a + b. 8. x-2. 9. x4-3. 10. x - 4. 11. x + 3. 12. a -4.
13. a 4- 3. 14. y-6. 15. 2 a 4-1. 16. a 4- 3 6.
Page 92. 1. a 8 . 2. x 2 */ 3 . 3. 8x. , 4. 24x s y s . 5. 80a6
6. 42a 3 x. 7. 40 aV>*>c r >d\ 8. !)&(<* 4- &). 9. 12 m 2 (m 4- n) 2 .
ANSWERS
x
10. (a-2y 2 (a-3) 2 (a-4) 2 . 11. 6a2&(rt-6). 12. 30(3 + y) (a: -y).
13. ( + &)(-&) 14. (a-2)(a + 2)2. 15. 2 (a + &) 2 ( - 6).
+ l). 18. 2(2a-l)
- 1).
- 1).
x + 5 a 10 m + 1
22. ?-_!&. 23. 1. 24. x ^. 25. !+*?. !
a + 8 b a - 4 3 / - // b
21.
w + 2
^ 2 (!L 3 i
3^-1
rr 5 + 63
Pace 991
20
Xll ANSWERS
21 ab- 12 ft 2 -8 a 2
8. ^iie^+JoJ^^ilOa 2 . 9 92 aft - 121 1>* - 196 a 2
10 bc + qc + ab 11 238 **/* + 84 y~ z - ] 5 x2 y + :j y;2g
ftc ' . - _-
12 46 ?t ~ 30 y - 50 1/2 - 80 MP 2 30 ?/ r + t S ^
180 wv 2ft ' "' 3a 2
15. i ' ~
t+3) (wi-8)(w 2)
5 x + 13 19 rtv+Ji:'. go 3ffl 21 4m -
(x + *2)(x + 3)' ' a 2 ft 2 a + ft' 1 m 2 '
9 fi 7 ,, 26
Page 100. 26. _*^p5_^^_. 2 7.
28. 29. ' 30. -^-~
2n 9 -T 94 9
^i a 22 * x 4<i ^ 33 ^
37 -
(a b)' 2 (a + ft) a + 1 a 1 x-y
41. ~ r > 42. ^ 8 43.
'a 2 -f an -f ft' 2
2(cz-ft) ' (x
Page 101. 1. 6a-5-f^. 2. 3a-2 + ~- 3. w-1 +
a 3a w 4
4. w+4+ ? 5. rt + lH 6. ac 2H ^.
w + 3 a + 1 2x- 1
7. a _ i + _J? 8. a; 2 + 5 x + 12 + -^-.
- .
6. L 2 . 7. i^. 8. 6. 0. -A^. 10. i. 11.
4 8 c 4c
in <l ~ 10 X + L
X<6. u. j .
a - 2 b x + 3
17. 1. 18. 1.
Page 104. 1. ?.
x
6 .L+-ft.
b
Page 105. 7. ^i
^U\WF### xiii
15. 3z 1. 16.
( + I) 2
05m ' w^x
14.
15. 1.
11. 1.
12. a.
2
(a + y) 2
5. 1.
13.
(y - 1^)2
(z + 1) 2 '
2
Page 106. 1. -
6. _J_.
' x-\- y
7.
ac b
Page 107. 11.
2. .
mp n
4. .
9.
-
m lf> n
12.
14. y(x + ?/). 15. - x. 16.
2 re + 3 y
2-ft
a i
17.
5
be
10 mn
lo. -
an -f
19. ! 20. f. 21. ^_.^_.
4 + x a 2 + a + 1
ft 2 -f 1
2)
18.
Page 109. 1. 23. 2. 11. 3. flf. 4. 5. 5. 1. 6. 21.
7. 8. 8. 5. 9. 12. 10. 5. 11. J. 12. J. 13. f 14. .
15. 3. 16. 6. 17. (>. 18. 6. 19. A. 20. 7. 21. 6.
Page 110. 24. 0. 25. 4. 26. J. 27. -3. 28. 9. 29. -3.
30. 4. 31. 3. 32. 4. 33. 3. 34. 1 2 V 35. 4. 36. 7.
37. 1. 38. 11. 39. 0. 40. - 1.
Page 111. 41. -Of. 42. 4. 43.
46. 7. 47. 6.
44. -
45. -
Page 113. 1.
p-f n
6 ?-+_!?>. 7
i .
10.
11.
3
n m
8j-_m
7
o,ft c
a
3. o ft.
8. 4a-3ft.
4. 2 .
m
9. Q
12.
a + b
13.
5 14. 5
a -f 6 + c n
XIV ANSWERS
15. '/ . 16. nm. 17. - 18. a 4- ft. 19. -
w 2
20. ~ m . 21. -'"- . 22. [>> - " - & . 23< &n . 2 4. 0,
w 4- n -I- b a b
25 ~r~ . 26 w 27 ^ ~i~ ^ . 0.0 ^ . 29 . 30
' m- n ' ' a 4- ft 3 T ' 5T ^ '
31. --1^'. 32. -M_. 33. -#- 34. (a) ^', (ft) 10 -
ft 4- fl P + ^ V ~~ J r/ i PM
xx HXH
/>/
Page 114. 35. (a) ----- + ; (ft) 104, 212, -4. 36. - C -.
5 2 7T
Page 116. 1. 30. 2. 00. 3. 8,2. 4. 21,9. 5. 15,10.
6. 18,15. 7. 30 yrs. 8. 40 yrs., 10 yrs. 9. 30ft. 10. $30,000.
11. $0,000. 12. $00.
Page 117. 13. 40 mi./hr. 14. 30mi./hr. 15. 21ft min. after 4.
16. 38ft- min. after 7. 17. 5ft- min. after 7. 18. $12,000.
19. $40,000 = A's, $45,000 = IV s. 20. 9J oz. gold, 10^ oz. silver.
21. If da. 22. 1 1 da. 23. 3 da.
Page 118. 24. () 2.} da., (ft) 5 da., (r) 3^ da., (d) 4 da. 25. 13,
14, 15. 26. 18, 19, 20. 27. ^p^ r -~| 5 7, 8, 9; 10,002, 10,003,
10,004; 300,137, 300,138, 300,139.
Page 119. 28. 5,0, 29. 10,11. 30. 9ft. 31. (a) 25, 26 ;
(ft) 74, 75 ; (c) 8300, 8301 ; (d) 500,000, 500,001. 32. 11 hrs., 33, 55 mi.
33. _JL. hr., -^ m -, - d ~ n mi. (a) 12 hr., 30, 24 mi.; (ft) 5 hr.,
10, 26 mi. ; (r) 8 hr., 28, 30 mi. 34. -^- min. ; (a) 4 min. ;
w 4- ?i
(ft) 7 min. ; (c) 2 hr.
Page 121. 1. 4. 2. 3. 3. f. 4. J. 5. 3. 6. *
7. 7:9. 8. 2:1. 9. 275:108. 10. 3:2. 11. 2x:3y.
12. 4x' 2 :3?/ 2 . 13. 1:3,2. 14. x-y.x + y. 15. 1:4. 16. 1 : jj.
17. 1 :2. 18. 1: ~.
Page 124. 1. Yes. 2. Yes. 3. No. 4. Yes. 5. Yes.
6. 1:1 = 1:1, Yes. 7. 1:1 = 1:1, Yes. 8. 3 : 19 = 4 : 25, No.
9. 1:1=1:1, Yes. 10. 1:1 = 1:1, Yes.
Page 125. 11. 10. 12. 3|. 13. 9$. 14. 1|. 15. 7.
16. 20. 17. 7} ft. 18. 40wn. 19, 15. 20. 12. 81. f
ANSWERS XV
22. w - ; ?. 23. pq. 24. 1(5. 25. 40. 26. 31. 27. 6*. 28. a~. 29. 1
w rt *
30. 8. 31. J 3 - 32 - <>' 33 - 4 <^- 34 - vm-^1. 35. 5:0 = 10:12;
6 : 10 = : 12. 36. b : x y : a ; /; : y - x : a. 37. .' : y = 7 : 0.
38. :y=2:9. 39. x : y = 1 : <>. 40. jc:y = n:m. 41. ./':</ c : a -f />.
42. x : y 5 : 2. 43. ;r. : y in : n. 44. x : ?/ = 3 : 2. 45. y :y =."2:1.
46. x:y -a: b. 47. tf : ?/ - a- : 1. 48. 5:3 = 4: x. 49. 11 : 5 - 15 : x.
50. a -f 2 : 2 = 5 : x. 51. 19 : 3 - 2 : x. 52. 1 : 18 = 3 : sr. 53. 3 : 2=3 : x.
Page 126. 54. () 7 T : 7 1 ' = /> : b'. (b) C -. C' = fi .- JR'.
(,-) V : V = P> : P. (<l) A : A 1 = R~ : R>'\ (e) m : m 1 = d> : (I.
55. (a) Directly. (b) Inversely. (e) Directly. (</) Inversely.
() Directly. 56. lo mi. 57. 24 1 s<i. in. 58. 20 cu. ft.
Page 127. 59. 200 mi., 32+ mi., 174+ ini.
Page 128. 1. 8,36. 2. 13J, 31J. 3. OJ, 32j. 4. 9,15.
5. OJ, 3}. 6. 19.8 oz. copper, 2.2 oz. tin. 7. 945 cu. ft.
8. 55,160,000 sq. mi. land, 141,840,000 sq. mi. water. 9. 11 gms.
10 J -^- ^ 0?j 11 20 20 w 12 3_a 7^
' a + ?/ a + ft ' 1 + m ' 1 + m* 10 ' 10 '
, W.C 7>i// , .. 74 n , 9 , - <7^' 7)C
13. , - - . 14. 7 2 \, 7^. 15. 7 , -.
x + y x + y a + b a + b
Page 131. 1. 30,17. 2. 7,12. 3. 2,3. 4. 1,3.
Page 132. 5. 5,4. 6. 3,3. 7. -7,4. 8. 2, - 3. 9. 4, - 2.
10. 7, 9. t 11. 5, 4. 12. 5, 7. 13. 9, 4. 14. 5, 5. 15. 5, Of.
16. 7,5. 17. l,li. ig ^ 15> 19 . 28,22. 20. \\, \. 21. 19,57.
22. 27,20. 23. 5," - 5. 24. 9, OJ. 25. 3,4.
Page 133. 1. 14,1. 2. -1, -2. 3. 2, - ]. 4. 2,5. 5. 3,3.
6. 2|, 3. 7. 4, -1. 8. 2, -3. 9. 4,5. "lO. 2,1. 11. 2,2.
12. 1, - 19.
Page 134. 1. 2,3. 2. 9,7. 3. 2,3.
Page 135. 4. 4,5. 5. -3,9. 6. 7,4. 7. 2,3. 8. 5,7.
9. 7,5. 10. 16,12. 11. 41,2. 12. 36,3. 13. 6, -4. 14. 7,46.
15. 11, 7. 16. 17, 13. 17. (I, 8. 18. 4, 3.
Page 136. 19. 2,1. 20. 2,3. 21. 10,5. 22. 1,1. 23. 4,6.
24. 2, 3. 25. -7, -7. 26. 2, 3.
Page 137. 1. 1,2. 2. 2,3. 3. 2,5. 4. , $.
Page 138. 5. 2, -2. 6. 3,4. 7. 4, - 5. 8. 1, ~ 1.
9. -3, -2. 10. I, J. 11. -J.J. 12, ,*. 13. i, *. 14. J,J
XVI
Fagel39.-l. 2t2, L=. 2. SL=J r^2. 3. 0,1.
22 o ft ft
4 q - & fr-^ 5 M - w 2 6 . AzA-, _(?jrJL.
' w _ i ' w _ i m + w ' m -f u ' ' ad be ad be
7. <*-ft/ 7 af-cd m 8 . ft, a. 9. 2ft, 3 a. 10.
ae bd ae ftd
11. 2 a, 1. 12. a = J - (n - 1) rf, - Zn - -"-(^-ll
14. = -
^ a
Page 141. 1. 1, 2, 3. 2. 1, 5, 0. 3. 4, 5, (5. 4. 3, 2, 1. 5. 2, 3, 4.
6. 6, 3, 1. 7. 1, 2, 3. 8. 2, 3, 4. 9. - 4, 4, 4. 10. 2, - 3, - .
11. 9, 7, 3. 12. 20, 6, 4. 13. -3, 5, 7. 14. 11, 13, 17.
Page 142. 15. , J, |. 16. 9, 7, - 7. 17. 3, 5, 7. 18. 2, 3, 4.
19. 2,3,4. 20. 2, 3, -4. 21. 11,8,7. 22. 8,0,2. 23. 11,33,65.
24. 24, 30, 40. 25. 6, 7, 1. 26. 1, 2, 3. 27. 18, 32, 10. 28. - 9, 72,
90.
Page 143. 29. 2, 2, 2. 30. 2, 3, 4. 31. m + n -p, m - n + p,
m -f ;
Page 145. 1. 3, 7. 2.4,3. 3.0,2. 4. &. 5. -. 6.^. 7. ^.
8. 24. 9. 25. 10. 423.
Page 146. 11. A's 50 yrs., B's 40 yrs., C's 30 yrs. 12. A's 30 yrs.,
B's 15 yrs., C's 10 yrs. 13. $ 1000, $4000. 14. $500, $250. 15. .$5000,
$3000,$2000. 16. $6500at3Ji%. 17. $900 at 5%. 18. 6%, 5%.
19. 19 gms., 10^ gms.
Page 147. 20. 2 horses, 6 cows, 10 sheep. 21. 100, 00, 20.
22. 2, 4, 3. 23. 5, 4, 3. 24. 20, 40, 30. 25. 3, 4 mi./hr.
Page 149. 5. About 12f. 6. 5. 8. On a parallel to the x axis.
9. On the y axis. 10. On the x axis. 11. A parallel to the x axis
through point (0, 3). 12. The ordinate. 13. 0, 0.
Page 151. 1. (a) 12; (ft) 23 J; (c) - 1J; (<f) 5.
Page 152. 2. (a) Apr. 1, Nov. 15; (ft) May 20, Oct. 1 ; (c) Jan. 1,
Feb. 1 ; (d) Apr. 16, Nov. 6. 3. July 20, 23f . 4. Jan. 16, - 1.
5. June, July, Aug., & part of Sept. 6. Jan. & part of Feb. 7. Jan. 16
to July 20. 8. Apr. 20 & Oct. 25. 9. 18. 10. 4. 11. Apr. & May.
12. Nov. 13. Jan. 14. Jan. 15. July. 16.10. 17. Apr. 20 to Oct. 26.
Page 153. 18. Nov. 16.
AN 'S WE US xvii
Page 157. 21. (a) 12.25; (ft) 2.25; (c) 7.84; (ci) 3^; (<?) 2.5;
(/) 3.6; (gr) 2.24; (ft) 3.25. 22. (a) 4.25, - 1.75, - 1.75 ; (ft) 2,3.73
ami .27, 3.87 and .13; (c) -2; (d) 2; (e) 3.41 and .59; (/) 3.41 and
.59 ; (0) 3 and 1 ; (ft) and 4. 23. (a) 2.75, - 3.25, 1.5 ; (ft) 3.24 ;
-1.24; (c) 2.73, -.7; (d) 2.73, -.73; (e) 2.4, -4.
Page 158. 24. (ft) -18C., -13C., - 10 C., 0C. ; (c) 14 F. ;
32F.,34F.
Page 159. 1. 1.75. 2. -2.5. 3. G. 4. 2.67. 5. 2. 6. 3.
7. -1. 8. 1|. 9. 3, -2. 10. 2.79, -1.79. 11. 3.83, -1.83.
12. () 3, 3 ; \ft) 5. 83, .17 ; (c) 1, 5. 13. (a) 5. 1, - 1.2 ; (ft) 5, - 1.
14. (rt) 1.64, -3.64; (ft) -4, 2.
Page 163. 9. 2, 1. 10. 2, 3. 11. 3, - 1. 12. H, 1. 13. - 1, -2.
14. 3, 1^. 15. 3, - 1. 16. 5, 5. 17. , 0. 18. Inconsistent. 19. 4, 3.
20. Inconsistent. 21. |, f. 22. 4}, 2|." 23. 3, 2.
Page 164. 24. |, 1. 25. Indeterminate. 26. 3, 2. 27. 2, f.
28. Indeterminate. 29. 2, 5 and 2, 3. 30. 3, 9 and 2, 4.
Page 166. 1. a- 15 . 2. x g . 3. -a 10 . 4. a ll V&. 5.
6. -8mW. 7. -125 a 8 . 8. 04 x 12 */ 1 '^ 1 ' 2 - 9- -x 1 ^ 27 . 10.
11. SlstyW 12. _ a 6o&i85 c i5o t 13 . I21a 4 ftc 2 . 14. -27
15. ImW. 16. xW. 17. H. 18. -". 19.
*
. . . .
9 81 27 81. i/*
25 a 8 jgiooyiio
343x30 -1 26 :=_!. 27 64_
125a 12 ft ' 125 xW ' a 121 a 12 ft 27 '
28. - 29. a 4TO . 30. a &m - 31.
a 4) ft 44 81
Page 167. 1. a 3 + 3 2 ft 4- 3 aft 2 + ft 8 . 2. x 3 -3x 2 y + 3x?/ 2 -2
3. a 3 +3a 2 +3a + l. 4. m 8 6w 2 -f-12 wi 8. 5.
9. 27a 3 -27 2 +9a-l. 10.
11. 343 a 6 147 a 4 -f 21 a 2 _ i. 12. 1 + I5a 3 + 75a 6 + 126a 9 .
13. 27 aH64 a 2 ft + 36 aft 2 +8 ft 8 . 14. 125 m 8 + 150 m% -f 60 win 2 4- 8 n 8 .
15. 27a-135a 2 ft4-225aft 2 -125ft 8 . 16. 27 a 6 ft - 27 a 4 ft 4 -f 9a 2 ft 2 -l.
19. a + ft. 20. x-y. 21. a-1. 22. 1 -f m. 23. 2ft- 1.
Page 168. 1. |) 4 -f-4p 8 7+6p 2 g 2 -f-4pg 8 -f g*. 2. m 4 4 m%+6 w 2 n 2 .
4wn 8 + n 4 . 3. x*-f 4x 8 + 6x 2 -f-4 x-f 1. 4. 1 4-4 1/ + 6t/ 2 -f 4*/ + t/*.
5. m 4 -4m 8 H-6m 2 -4m4-l. 6. 1 - 4 aft -h a 2 ft 2 - 4 a^ft 3 + a 4 ft*
XV111 AN S WE no
7. i + 4 x 2 + Ox 4 +4^ + x 8 . 8. 5 -5
9. c 5 +5 c*d+ 10 c 3 tf 2 + 10 c 2 d+6 c<7 4 + d 5 . 10. m* m*>n + 16 w 4 2 -
20 in s + 15 w 2 w 4 -G mw 6 + w 6 . 11. a 7 + 7 b fe + 21 *?>-' + 36 4 & 8 + 35a 3 & 4
-f 21 rt'-6 6 + 7 ?> 6 -f 6 7 . 12. rt fi - 5 a 4 + 10 a 9 - 10 a~ + 5 a - 1.
13. 32+ 80 a +80 a* +40 a 3 + 10 a 4 -fa 5 . 14. ro 12 + 4 m+ w + 4 w + l.
15. 1 3 w 2 H 2 + 3 ? 4 n 4 - i c 6 . 16. ?7i 16 6 w 1 - -f 1 m 9 1 w + 5 m' 3 - 1.
17. w 8 + 8 in 7 n + 28 ? G w- + 50 m*w* + 70 w 4 4 + f>6 ?n : *w 6 +28 >-/* + 8 mn 1 + w 8 .
18. w 5 w 5 + 5 W 4 w 4 c + 10 ??i 3 w 8 c 2 + 10 w 2 ?i 2 c 3 + 5 mwc 4 + r 5 . 19. j/^/t^/'
r> wi 4 n 4 p*+ 10 w 8 w : y 10 wi 2 w 2 7> 2 +6 w/ip- 1. 20. 32 r^ 10 + 80 w 8 + 80 wt c +
40 m 4 + 10 m' 2 + 1 . 21. 81 + 540 + 1360 a 4 + 1500 a 2 + 025.
22. 10 x 8 100 * 6 + GOO x 4 1000 se 2 + G25. 23. 1 G a 4 - 1 GO a :J c +
GOO 2 c 2 - 1000 ac 3 + (J25 c 4 . 24. 1 + 8 z + 24 a: 2 + 32 r l + 10 x 4 .
25. 1 + 5 a?b* + 10 a 4 b* + 10 a& + 5 a/> 8 + a 10 /; 10 .
Page 170. 1. 2. 2. 9. 3. -5. 4. 4. 5. +35. 6. 30.
7. 00. 8. -20. 9. 180 a. 10. 90.
Page 171. 11. +X 2 . 12. m 4 . 13. 25 . 14. 49 a. 15. 10*.
16. - 10. 17. 3 '-. 18. 2 ^-. 19. 2 6 :l . 20. 2 ?>i?< >2 .
21. Zll. a . 22. -^i. 23. 0. 24. . 25. - x. 26. (x + y\
b 3 6 c 10
27. 2(> + 7>). 28. (a + Z). 29. 420. 30. 90. 31. 72. 32. 90.
33. 70. 34. 300. 35. 40. 36. 23.
1. ( + l). 2. (l-y). 3. (x-2y). 4. (3a; + y).
5. (x' 2 -l). 6. (1 + 8x2).
Page 172. 7. (2 a - 1 1 ?>). 8. (2 a + ft). 9. + (win - 7 /)).
10. (rt'- I -/> 4 ). 11. (7 4 -3M. 12. (4a 2 -9& 2 . 13.
14. (a + y+l). 15. (a-b + c). 16. ( a + fr
Page 174. 1. +(^ 2 -3^ + 2). 2. ( 2 -+l). 3.
4. (4rt 2 +3 + i)). 5. (5m 2 Cm + 3). 6. (2-3 alt + 7 -ft 2 ).
7. (5^ 2 + 4x?/ + 3?/ 2 ). 8. (48 + 6.r 2 + 6jt). 9. (l + x + .^).
10. (i + 2a;+3^ + 4.x 8 ). 11. (Gn 3 + 5 a 2 + 4 a). 12. (Gx 3 +
3.r 2 i/ + S:r2/ 2 ). 13. (2 wi 8 + 3m 2 - 5). 14. (7 a 2 - 3 ab + 2 /> 2 ).
15. ( x 3 + 2 x 2 -2z + 4). 16. (2a-3a: 2 + a;-2). 17. (:' 2 +
xy - y 2 ). 18. (27 + 3 a 2 - a 8 ). 19. (6 a + 5 a 4 + 4 a 2 ).
20. (6a 8 + 4a + 3a + 2). 21. ^ + | + ?V 22.
23 . (l+? + l + lV 24. (- 6 +3 + 5
V x & x s J \x
Page 176. 1. 76. 2. 64. 3. 57. 4. 71. 5. 84. 6. 98. 7. 99.
8. 119. 9. 101. 10. 237. 11. 309. 12. 247. 13. 763. 14. 978.
15. 2.83. 16. 6.5. 17. 8.6. 18. .94. 19. .037. 20. 1247. 21. 2038.
ANS WERS xix
22. 7563. 23. 5083. 24. 6561. 25. 15,367. 26. 6J. 27. If
28. ^.
Page 177. 29. 2.236. 30. 3.60. 31. .469. 32. 1.237. 33. 1.005.
34. .935. 35. .645. 36. .243. 37. 7.522 ft. 38. 9.798 yds. 39. 3.925 ft.
40. 4.690.
Page 179. 1. 13. 2. .5. 3. 17. 4. 5. 5. 2. 6. 1.
7. 3. 8. 4. 9. V2. 10. v / l~8. 11. 5. 12. f . 13. ^-.
14. 5. 15. 5. 16. V- 17. J f l. is. vYb. 19. (< + ?>).
_ ^
20. iVaft. ^_
Page 180. 21. J. 22. (a-fl). 23. vV-'-TA 24. A f^.
25- *. 26. " 27. . 28. W**.
>TT >i 4 TT M 4 > w
29. JJI.
*
1. 1. 2. 15. 3. 10, 15. 4. 2, 4, 6. 5. 9 ft., 15 ft.
6. 21yds., 6yds.
Page 181. 7. 21 in., 28 in. 8. 39 in., 36 in., 270 sq. in. 9. V2.
10. 6, 8. 11. 2 sec.., 2] see. 12. () 7 in.; (6) Vl4 or 3.742 ft. 13. 21 ft.,
28 ft. 14. V35 or 5.916 yds.
Page 183. 1. 10, - 2. 2. -7,5. 3. 3, -9. 4. 7, - 1.
5. 12, -5. 6. 7,4. 7. 1 \/3. 8. 6V21. 9. 7, 10.
10. 6|, i. 11. 8, -2.}. 12. 5, -4J. 13. 1, - *. 14. ZLlAi K
15. |, -V. 16. J,-?,. 17. 7}, -2. 18. 11,1. 19. 7, - 3.
20. 6V'2J. 21. -^,-6. 22. 7, - 12. 23. 11, -3. 24. 2,
-|f. 25. 14,6. 26. 7, - f 27. 3, |. 28. 3, -4.
29. 4, - /.
Page 184. 30. 7, - 4. 31. 3, -4. 32. 12, - f 33. 14, -10.
34. 3, - 1. 35. 9, 1. 36. 4,13. 37. 1 Vl3. 38. 9, 1,V
39. 10,18. 40. 4,i. 41. 3,6. 42. 5f, 5. 43. 5, - 8-j.
44. -6, 2. 45. 18,6. 46. 2, 1&. 47. 1 - V17. 48. - a, 4 a.
49. 7 m, w. 50. a, 5.
Page 185. 1. 2, f 2. 2, f. 3. 3, 3. 4. - 5, 3J. 5. - 5, - f.
6- i-i _7. 4, -16. 8. 10,1. 9. 12,6. 10. 6, {. 11. }, f.
12. :J -^-^- 13. !, -i 14. ^,-4. 15 f -f 16. -V.S-
17. 7m, -m. 18. 4 n, -16n. 19. ~ 1 V^3. 20. -n.---
w
21. 1, a + 6-1.
XX ANSWERS
Page 186. 22. .02, - 1.62. 23. 1.37, - 1.70. 24. 9.48, - 1.48
25. 1.2;], -3.23. 26. 3.41, .59. 27. 2.74, - 3. H. 28. .23.', - .0*8.
Page 187. 1. - 6, - 1. 2. 3,7. 3. 6,2. 4. -2,12.
5. -0, -4. 6. 2, -12. 7. 6,4. 8. -1, -7. 9. ,-6.
10. 7, U. 11. 0, - 4, - 5. 12. 0, - 4, - 1. 13. 1_^L
s ' ' 2 (5
14. 0, 3, f. 15. 7, 2. 16. 0, - 2, If.
Page 188. - 17. 0, - 3, f . 18. 1 , - 2. 19. 0, 3, - 2. 20. a, ft - a.
21. - 1, 3. 22. 6, - 2. 23. 0, - 1, 2. 24. 0, V7. 25. 0, 2, 3.
26. 0, a + 64-c. 27. 1, 2, - 3. 28. 3, 7. 29. 6V-64.
30. %, - 1. 31. 28, -3. 32. 6, 3. 33. -4, -21. 34. -3, 2.
35. 2, 3|. 36. -1|, 2. 37. 4,7. 38. 4,7. 39. 1), 3if.
40. f. 41. |,3. 42. -i,l. 43. 0, . 44. a + 6,-
45. 3 a. 46. r* -f 6 v^^fcT"^. 47. - a, ft. 48. a - 2 ft,
5-2 a. 49. - 5, 2, - 2. 50. 3, 3.
Page 189. 51. jr 2 4jr + 3 = 0. 52. x 2 + B - 12 = 0.
53. x* + 7 x + 10 = 0. 64. x 2 - 9 x = 0. 55. a 8 - 2 x 2 - 5 x + 6 = 0.
56. x*-x 2 -6x = <). 57. a; 8 - Oa;* + 11 x- 6 = 0. 58. x*-4x=0.
1. or . 2. 25, 35. 3. 2, 6. 4. 12, 24. 5. 6, 7. 6. 14, 15.
7. orf. 8.4,10. 9. 10 or 19. 10. 10 in., 19 in. 11. 70 ft., 120 ft.
Page 190. 12. AB = 204 ft., ^l/>> = 85 ft. 13. 8\/2 in., 6^2 in.
14. ,$40 or $60. 15. $30 or $70. 16. $80. 17. 10 mi./hr.,
10 mi./hr. 18. 20 nii./hr. 19. 8 or 12 mi./hr. 20.5^. 21. $ 120.
Page 191. 22. AB = 26, #<7=3. 23. 5 ft. 24. 15 ft. 25. 4 da.
26. 2 V3 in. 27. 20 eggs.
Page 192. 1. 3, 1. 2. 3, 2. 3. 2, 1. 4. V^l,
5. 5. 2, V^~2. 6. 2, - 1, -
VV- 8. *'-' i, 3. 10. 2, -3, 3, -4.
11. - 1, 2, - 2, 3. 12. 1, - 2, - 4. 13. 0, 1, 1 V2.
14. i, i t x V ~ 16 . is. V^l, V2. 16. 1,2,4,5. 17.1,2,2,3.
4
18. 0, 1, - 1, _ 2. 19. - 1, 2, - 5. 20. - 2, - 1, 0, 1.
Page 194. 1. Real, unequal, rational. 2. Real, unequal, rational.
3. Real, unequal, irrational. 4. Imaginary, unequal. 5. Real,
equal, rational. 6. Imaginary, unequal. 7. Real, equal, rational.
8. Imaginary, unequal. 9. Real, unequal, irrational. 10. Real,
unequal, rational, 11. Imaginary, unequal. 12. Real, equal,
ANtiWEUS xxi
rational. 13. 5,2. 14. 9, -3. 15. 2, - f . 16. 3. - f .
17. m, p. 18. , J. 19. 4, 15. 20. -_! V3. 21. - 5, 3.
22. -2,6. 23. -1V-1. 24. J JV37.
Page 196. 1. 3. 2. 2. 3. 4. 4. 2. 5. 3. 6. 4.
7. 2. 8. 4. 9. 8. 10. 9. 11. 5. 12. 4. 13. 1. 14. -2.
15. 8. 16. 0. 17. . 18. J. 19. |. 20. (m - ).
21. z + y, 22. v'frc 7 . 23. Vr. 24. vV. 25. \a\ 26. \/^.
27. \/3. 28. v'frW. 29. \/. 30. V^ : . 31. vm.
Page 197. 32. x$. 33. aW\ 34. r*. 35. m'. 36. a' 2 .
37. *V. 38. wA 39. 2. 40. \. 41. 9. 42. 8. 43. 3.
44. \. 45. 10. 46. 49. 47. 20. 48. 10. 49. 29. 50. &.
Page 199. 1. ? 1 , ) - 2 - i 3 -jV- 4 l - 5 - J - 6 - A-
7. 1. 8. ,1 5 . 9. >J y . 10. 4." 11. 8. 12. j. 13. Af.
14. i. 15. J. 16. Jb \. 17. 32. 18. 49. 19. '-J.
20. -44. 21. - 22. - 23. - 24.
3 a* tf tfb 1
26. _$L. 27. 30 ^:- 28. i*^ 4 . 29. Oafe-'.
25. ^.
30. f> a-^c- 2 .
25 a 2 u^c^xy
a b t> 7
31. a 3 ^ 8 ^" 3 ^/" 3 . 32. afoe*/' 2
33. m-' 2 ^n*. 34.
7- 1 m~ 1 w- 1 p- 1 .
35. Vm. 36. -^r- 37. 3
\/x. 38. A. 39. -
40. -JL_.
Vra
Vx vw
S/^2 wi
41. - 42. Vrn^. 43.
J_. 44. y^. 45.
3 / A
^/m.
Page 200. 46. 5. 47. 2. 48. -1. 49. -2. 50. 0.
51. 125. 52. 10. 53. -3. 54. -1. 55. -3. 56. -3.
57. 33. 58. - 1. 59. 25. 60. - 17. 61. 0.
Page 201. 1. 8. 2. 3. 3. 1. 4. J. 5. a. 6. 12* 2 .
7. 30 a. 8. 5. 9. 7V7. 10. \/r\ 11. 5\/5. 12. 243.
13. -ifa. 14. 3. 15. a 18 . 16. 2V a. 17. fx'-^z'l 18. .
19. 11. 20. v. 21. x/25. 22. v/3. 23. ^49. 24. V.
25. ar 1 . 26. v^T 4 . 27. x. 28. m. 29. 1. 30. ( 2 L - 31. r;//^.
n\/*
P a ge202.-32. I. 33.^7. 84. ^S- f|- 3
$7. 1. 38. \ 39. -.. 40. 1.
XXii ANSWERS
Page 203. 1. a 2 - a Vft - 2 b. 2. x- Vxy - y. 3.
34
4. 03r* 121
101 Vr 35. 5. a 4 +-* + !.
r
6. x + 2 Vzy + y 1. 7. x - 1. 8. a-** + or 2 " + 1. 9. 4 z 2 -
13 a;3 + 40 3 - 12 *^ + 9. 10. x^ - x^y* + y%. 11. 4 or 2 + 9 <r l + 3.
12. 7/> + 8V/) + 1. 13. a + 2 a^b* + ft. 14. 3 a~ 3 5 a~ 2 ft~ 1 +
2 or 1 ?;- 2 . 15. Va Vft. 16. (Va Vft+Vc). 17. (x 3 + 2 ar 1 ).
18. (x' 1 1+x). 19. (5x- J - 2 + 3:r,). 20. (;r-^ 3^ + 2).
21. (l + Vic + ). 22. 1+2 v/i + 3\/!^ + 4 x. 23. V2 + 2.
24. -2. 25. 19-5V3. 26. -13-5V6. 27. 17 - 4 V22.
Page 204. 28. x + 5 x 3 + 0. 29. x 2 x r * 15. 30. x y,
31. 2. 32. 6-2V(J. 33. x - 25. 34. a^ + 2^+1.
35. x% - 3 ^ + 3 x^y - y : l 36. 7 - 2 VlO. 37. 13 + 2 V22.
39. (o* + 2). 40. m* -n*.
16.
Page 207. 1. 3V3. 2. 3V5. 3. 4\/2. 4. 2\/7. 5. 2v / (T
6. 1) V3. 7. 11V3. 8. .rV:r. 9. 3 V. 10. abVab. 11. 2aVf.
12. 8a6V5. 13. 2 a?>V2 a. 14. 20&V6. 15. l
5 ( .'\
17. - 2 '"V5. 18. 2>X2. 19. 3^2. 20.
21. 5
25. (a
y
30. yV35.
22.
26.
27.
24. (a + ft) V2.
k/2. 28. JIV6. 29.
31. ^: V2c. 32. V.r. 33. ?tV?w. 34. a
3,f.
Page 208. 37. ^7 \AOx-. 38. 3 \
41. 3V^T. 42. 4aV^J
c
45. 3 ftV 2. 46. V 5 . 47. .577.
50. .648. 51. .692.
1. V80. 2. V63. 3. ^88". 4. v'TM.
7. -v/^r 5 - 8. A/^~. 9
10.
16,
43.
39. Vz 2 -?/ 2 . 40.
r -T. 44.
48. .707. 49. .632.
5. v^. 6. Va 2 "ft.
ANSWERS
xx
_ __ rw
Page 209. 1. V/.T*. 2. x/w^ 3- \~s~'
*
6. "v/wi ??, .
11. \^r^b-
16. Vabc..
21.
7. \/04a. 8. \/w/t 4 . 9. ] \/abc*.
12. v / 25^ 4 7 14 . 13. \/a6c. 14.
17. V2"a. 18. fl^Vac. 19.
22. 3\/wi. 23.
24. V8.
Page 210. 25. x/27, x/4. 26. \/8, x/8L 27. v/27, \^6.
28. W), ^32. 29. \/128, v/9. 30. x/8, vT), v"5. 31. x/8l, x/125, ^9.
32. : % ;r , v/^ v^fr*. 33. ^27, v/i), v/8. 34. v/lO, v^O, "^8000.
35. v^a, V2. 36. v^4, \^6. 37. V5, v^f. 38. v/l2, V5, v^lf.
39. 5V2, 4\/5. 40. \/2, v^30, v"3.
Page 211. 1. 4VO. 2. 8V2. 3. -13\/3. 4. 0. 5. 2\/7.
6. 6V2. 7. v 7 ^. 8. 3 V2. 9. ^\/3. 10. ^ v 7 15. 11. 8v2T
12. 8V7- \V3. 13. a\/5c. 14. a^\/a7>. 15. V3"m. 16. -V-Jla.
17. 3\/IO. 18. V LV/ ^-
Page 212. -21 i?i=
a: ;J
23. 6aV2-\^. 24. 0.
Page 213. 1. 6. 2. 10. 3. 3\/2. 4. 5V2. 5. 7\/(l
6. r)\/(l 7. 7VTO. 8. 10V(). 9. 2. 10. 3. 11. 3v^2. 12. 5v/2.
13. Vrt-r. 14. rtv/5. 15. 2yV2?/. 16. 2 ate-. 17. 4 a*.
18. 14c 4 V5. 19. Vdbc. 20.
21. '-. 22. V3.
23. -^ 24. 2
3 b
27. 5 \XO-fl-6Vi5. 28. 1-V5.
31. 6x-2?/. 32. 32m-27n.
36. 46. 37. - 3. 38. m
40. 6+2V5. 41. 7-4\/Jl
25. Vtf + 3 + 2 V'3.
26. 3 Vl5 + 20.
29. a 2 - b. 30. w-?i.
33. 1. 34. b. 35. .
?i-2Vm/t. 39. 6 2\/0.
Page 214. 42. 120 - 30 VT5. 43.
D
ab
44. 8 - \/15.
45. 3 V15 - 4.
46.
- 30 Vl4. 47. 9 VlO + 5 \/2. 48. 2 \/2.
49. 3\/15 - 6. 50. 2 + VlO - v y (5. 51. 4 >/3. 52. Vat. 53. aVa.
Page 216. 1. V2. 2. x/3. 3. V3. 4. 2. 5. Vn. 6. 3.
XXIV ANSWERS
7. 2V3. 8. f. 9. ^\/2. 10. V3. 11. Va. 12. p 13. nVTl,
14. j V5. 15. V3. 16. V^TTfc. 17. fV2. 18. 19. .7071.
* 6
20. 1.732. 21, .3535. 22. 4.4722. 23. 1.0606. 24. 1.1547. 25. .2828.
Page217. 1. 2-V3. 2. 3V2-3. 3. 15 -f 3 V2L 4. |(V2 + 1).
5. Vf6-f|Vtf. 6. V3-V2. 7. (Vf + V2). 8. 4V3 + 6.
9. 8 + 5V2. 10. 3(7+3V5). 11. (4 + 3 V2). 12. 6 V3 - 13.
Page 218. - 13. i^Lzi. 14. 5 + 2 vU 15.
4
16. 6 V;W + 12 v/7 - 3 \/15 - 6 V3. 17. V35. 18.
19. i^ v ^-. 20. ^r. 2 _^JflJ?. 21 Vob 22 ~ Vac
1 ~ x x-y ' ' i _ c
23. 2.4142. 24. .732. 25. 0.7083. 26. .601. 27. 0.464.
28. 5.5530. 29. 1.1805. 30. -26.389. 31. 7 -f 5 V2.
Page 219. 1. 9 mn. 2. 2x^2^. 3. 2ajV2*. 4. 125. 5. 512.
6. 4. 7. 8. 8. 25. 9. 15,625 n*. 10. V6c. 11. Va. 12. \/57t.
Page220. 1. (V5-f \/3). 2. (V2-1). 3. (V5-1).
4. (3+ v/2). 5. (V8 + V2;. 6. (\/5-V2). 7. K>/0 + \/2).
8. (\/3-f 1). 9. (2- V3). 10. ^(VlO-\/2). 11. (2-V2).
12. (2-f V"5). 13. (2-Vll). 14. j- ^ (\/22 4- \/TO).
15. (V6 + 2V2). 16. i(V-f Vft). 17. (VaT^-v 7 a).
Page221. 18. (Vll-V2). 19. -. 20. V6- 1.
5
21. 2V2. 22. 4. 23. 2- V3.
Page 223. - 1. 4. 2. ^. 3. (a + 6) 2 . 4. 27. 5. 7. 6. 5. 7. 4.
8. 3. 9. 10. 10. 0. 11. 10. 12. 4. 13. 2. 14. - f. 15. 16. 16. 5.
17. 24. 18. 5. 19. 6. 20. 1.
Page 224. 21. 7. 22. 5. 23. 4. 24. . 25. J, \. 26. 25. 27. 25.
28. ^. 29. 9. 30. 64. 81. 100. 32. !^ 33. 4. 34. {. 35. -2!5_.
5 6 m -f- w
36. f. 37. J.
Page 225. 1. 4. 2. 9. 3. 16. 4. 25,81. 5. 216, - 64. 6. 16.
7. 4. 8. 4. 9. 5. 10. 3.
Page 226. 11. -1,9. 12. 8, - 3. 13. -3,2. 14. -4,5.
15. 2, - . 16. - 2, - 3. 17. 8, - 1|. 18. 3, - 6. 19. 9, - 2.
20. 7, 4. 81. V, A-
ANSWERS XXV
Page 228. - 1. -7. 2. 30. 3. 50. 4. 1. 5. 0. 6. 2 &.
7. 0. 8. 100.
Page 229. 4. (w - l)(m - 2)(m- 3). 5. (a + 2) (a 2 - 2a + 2).
6 . (p-l)(p-2)(p-2). 7. (p-l)(p-3)(p-6). 8. (+!)( -2)
(B4-3). 9. (w-2)(m-3)(2m + 5). 10. (a - l)(a-3)(a - 4).
11. (6-3)(6' 2 -56-l). 12. (w-p)(w-2p)(wi-3p)(w*4-2p).
18. (ro-w)(w-4w)(w a + 6mw -f w 2 ). 14. (a 4- 2)(* - 2 <? +
4a 2 - 8a -t- 16). 15. - 1, - 2, - 3. 16. 3, 1 \/0. 17. 1, 4, f>.
18. 3, 3, - 1. 19. 1, o, 4. 20. 1,2, 3. 21. 3, 3, - 3.
22. 2, 2, 3. 23. 2, - 4, - 5. 24. - 1, 1, 1, 3. 25. - 2, - J, 3.
Page 231. 1. (a- l)(a 2 + a -f 1). 2. (s + l)(x 2 -:r + 1).
3. (l-a&)(l46 + 2 & 2 ). 4. (a-2)(:iB 2 -f 2a44). 5. (r.-f 2)(sc 2 -2 r + 4).
6. (2a + l)(4a*-2a + l). 7. (2 a- 1)(4 a 2 + 2 a + 1).
8. (a -4 &)(* + 4 & + !&*). 9. 27(2 a 4-fc)( 4 2 -2 & + 6 2 ).
10. (10 -#0(100 + 10^ + a: 4 ). 11. a(.r - l)(z 2 + z + 1).
12. a(l+a)(l_a-fa 2 ). 13. (a+&)( 2 -
14. (rt- + ^)( 4 -a 2 6 2 -h6 4 ). 15. (
(&y-2a#H-4). 16. (a- l)(a 4 + a :} + a 2 -f a -f 1).
17. (a + l)(a*-a 8 + a--a + l). 18. (m 4 + l)(ro- w 4 + 1).
19. (1 +a 2 6 2 )(l -a 2 6 2 +a 4 6 4 ). 20. (8- a) (04 + 8 a + a 2 ).
21. (xy + 5) (x*y* - 5 xy + 25) . 22. (4 mn - t/ 2 ) ( 10 w 2 n 2 -f 4 winy 2
23. (3m 3 7)(9w 6 +21m*+49). 24. (-?>
o& + 6 4 )(a*-a' J 6 + a 2 6 2 -a& 8 H-6*). 25. 2, - 1 x/^3. 26. -2,
87 . 3 , =A|^Z3. 28. l,nl^EI.
Page 233. 1. 4, 2 ; 2, 4. 2. 5, 3 ; - 3, - 5. 3. 6, - 3 ; - 3, 5.
4. 10, 30 ; 30, 10. 5. 22, 3 ; - 3, - 22. 6. 25, 4 ; 4, 25.
7. 73,12; -12, -73. 8. 3,4; 4,3. 9. 7,2; 2,7. 10. 11,10; -11, -10.
11. 13,3; -P,, -13. 12. 20,0; -0, -20.
Page 234. 13. 4, 4 ; 4, 4. 14. 0, 1 ; 1, 0. 15. 4, 1 ; - 1, - 4.
16. 6, 2 ; 2, 5. 17. 3, 4 ; 4, 3. 18. a - 6, b ; 6, a - //.
19. 0, 30 ; 30, 0.
1. 4, 1 ; - 4, - 1. 2. 3, 1 ; - 3, - 1. 3. 2, 1 ; - J, - f .
4. 8, 2 ; -V^ ~ f- 5. 2, 3 ; y, J. 6. 1, 2 ; 1, 2.
Page 235. 7. 2, 1 ; - Y, - V- 8 - 6 > & J 0, 0. 9. '- 24, 12 ;
f , |. 10. 1, 3 ; 5, - 1. 11. 3, 4 ; 4, 3. 12. 5, 2 ; - ^a, - 2 y 4
13. 5, 2 ; - 6, - 2. 14. - 2, - 3 ; J, 2.
Page 236. 1. 1, 2; 2, 3. 2. 2, 3; 3, 4
3. 1, 2; 3, 5. 4. 1, 2; 2, qpl. 5. 1, 5; 7, 2
. 6, 2. 7. 2, 4; 3, 5. 8. 2, 3; 3, 4.
xxvi ANSWERS
Page 237. 9. 2, 5; 2>/3, V3~. 10. 3, 2; \/6,
3V5. 11. f>, 2 ; V7, 2V7. 12. jj, $; f, J.
13. i }, i i ; i j, 1. 14. 7, 5 ; 5, 1. 15. 2, 1.
16. 3, 2; -$VO, i'ljVU. 17. 4, 3; 1, 12.
Page 238. 1. 6, 4 ; - 4, - 6. 2. 3,1. 3. 5, 4 ; 4, 5. 4. 2,3.
5. 6, 2 ; 2, 5. 6. 4, 1 ; 1, 4. 7. 3, 2. 8. 4, 3 ; - 3, - 4.
9. 2, 1 ; 1, 2.
Page 239. 1. 1, 2 ; 1, - 2 ; - 1, 2 ; - 1, -2. 2. 1, 8; 2, 1.
3. 4, 1 ; 1, 4. 4. (>, 3 ; 3, 0. 5. 2, 3 ; 3. 2. 6. $, | ; |, J. 7. 512, 8 ;
8, 512. 8. 1, 3; 4, - 3.
Page 240. 9. 7, 11. 10. 5, 2 ; - '>, - -y. 11. 4, 3; 3, 4.
12. 3, 2 ; Y> V 7 . 13. 1, 2 ; 3, 5. 14. 1, 2 ; 2, 1.
15. ft, 12 ; 12, 9. 16. 4, 3; - j, - f*. 17. 6, 3; - 3, - 0.
18. 2, 1. 19. 1, 125 ; 125, 1. ' 20. |, 1 ; ^ J.
21. 1, 2. 22. 1,2; 3, 4. 23. 15, 5 ; - 15, _ 5. 24. m + n, m- n ;
m - n, tn + n. 25. 1, 4; 3, 2. 26. 5, 4 ; 4, 5. 27. }, } ; J, \.
28. 2, 1. 29. 5, 3 ; 3, 5.
Page 241. 30. 2, 4 ; 4, 2. 31. |, j. 32. 3, 1 ; 1, 3.
33. 4, 1 ; - 1, - 4. 34. 8, 0. 35^ 5, 3 ; 5, - 3 ; - 5, 3 ; -5, -3.
36. 1,3; 3,1. 37. 2, 3. 38. 2, 8 ; 8, 2. 39. ( ^~- 40. 1,3;
-1J, 7f 41. 3,4; 4,3.
Page 243, Exercise 113. 1. Indeterminate. 2. x = QO , i.e. no
solution. 3. oo . 4. Indeterminate. 5. GO , oo ; 4, 4. 6. co , co .
7. oo.
Exercise 114. 1. 37,30. 2. 20,13.
Page 244. 3. 18, 1. 4. 17, 15. 5. 8ft., 12ft. 6. 55, 48.
7. 12 ft., 35 ft. 8. 40 ft., 25 ft. 9. 40 ft., 9 ft, 10. 28yd., 45yd.
11. 2,18. 12. 40 in., 30 in. 13. 1} in., 3 in. 14. 5 cm., 3 cm.
Page 245. 15. 35 in., 12 in. 16. 20 in., 21 in. 17. 30.
Page 247. 1. a, c, and d. 2. (a) 5, 8, 11, 14, 17, 20 ; (&) 2, - 1,
_ 4, _ 7, _ 10, _ 13 ; (0 - 1, - 3, - 5, - 7, - 0, - 11. 3. 14.
4. 35. 5. 4|. 6. 30. 7. -37. 8. 201. 9. 2n.
Page 248. 10. 31,136. 11. -14, -50. 12. 84,020. 13. -3,0.
14. 900. 15. 288. 16. -400. 17. 50. 18. -10. 19. 69.6.
21. 5050. 22. "_. 23. ri*. 24. 78.
2 2
26. (a) $3400; (/>) $46,200.
ANSWERS xxvii
Page 250. 1. a. 2. x + Vy. 3. **-+-. 4. ~ v 5. 5,0,7
6. 9|, 9, 8J, 8, 7|, 7, 6i. 7. 16. 8. 20. 9. 5, 50. 10. 5.
11. 4. 12. 4. 13. 3,470. 14. 35. 15. 6. 16. 7f. 17. 2i * ~ an .
} 3. $ 50. 19. 25, 35, 45, 55, 05, 75.
Page 252. 1. , c, and d. 2. 3, 12, 48, 192, 708. 3. 16, 8, 4, 2,
1, I. 4. vy. 5. 1. 6. 1JH. 7. 327,680. .8. 16,384.
9. -,v 10. Y-
Page 253. 11. 005. 12. 304. 13. 45,920. 14. 43. 15. ^\
16. |- 17. 910. 18. y 5 ^ 5 - 1 ). 19. 10,210. 20. 81,120.
21. 10, 100. 22. 5, 405. '23. 45.
Page 254. 1. 2. 2. 1. 3. 04. 4. 2|. 5. 6|. 6. 410|. 7. 27.
8. 70. ' 9. i 10, <|. 11. Jj? r 12. r j. .13. &' 14. JSg. 15. 0.
16. A. 17. (). 8 sq. in. ; (?>) 8(2 4- V2).
Page 257. 1. a 5 4 5 4 Z> 4- 10 a 3 ?/ 2 4- 10 aW 4- 5 M 4- fc 5 . 2. r 6 -
x r '?/ 4- 15 x 4 //- - 20 rV J 4- 15 xV x>/ 5 4- <J ' 3. 1 4 8 x -f 28 x~ 4-
60 .r* 4- 70 .K 4 4 50 x 5 4- 28 x 4- 8 .r 7 4- ^ 8 . 4. :r 7 - 14 x 4- 84 .r^ - 280 x 4
+ 500 x 3 - 072 a? 4- 448 x - 128. 5. .r f 4- 5 J 4 4- ^ r 5 4- ^ x 2 4- ^ x 4- ^ 2 .
6. 10 4 4- 3 .^ d*b 4- 1 a' 2 /') 3 4- 5*7 ^ a + -g 1 ! ^ 4 7- 12 w 4 | w- i^S ? 8
4- 2 \ w 4 - ^ ?>i 6 4- 7 ?/i 6 . 8. x llj - 5 y 1 4 10 ;<-2 10 x' 2 4- 5 x- 6 - x' 10 .
9. 24-12x4- 2 x' 2 . 10. 10 % 4 20 ab* 42 //^. 11. 120 aW,
12. 45 a 8 /)-. 13. 220 w 9 8 . 14. 330 x 4 . 15. 280 />*. 16. -53,130 x 30 .
17. 70. 18. 189 a 4 . 19. 105. 20. 1820. 21. -15,504. 22. - 101,700.
23. 495. 24. J 2 // 2 - 25. - 20 flW. 26. 70. 27. 12,870 m*n*.
28. 4950 M 2 b y *. 29. 1000 aW.
REVIEW EXERCISE
Page 258. 1. 1, 0, 125, 04, 27, 8, 343, 125, 343. 2. -8, - 53, 8,
- 53, - 192, 27, - 170, - 419. 3. 1, 1, 1, 4, 1, 9, 4, 4. 4. 0, 8, 0, 3,
0, 32, - 1, 0. 5. 0, 7, 7, 10, 8, 12, 12, 18. 6. 6, 3, 8, 2, 2, 4, G, 0.
7. 0, - 4, 3, - 6, 4, 5, - 5, 2.
Page 259. 8 4, 5, 2, 4, 6, 2, 3, 1. 9. 1, 4, 3, 3, 6, 4. 10. 2 x 4-
Ja. 11. 6x' 2 . 12. x 4 . 13. B -f y 8 + z* - 3 a-ys. 14. 0. 15. 7 x 1 4 x 8
- x^ 4- 15 x 4- 5. 16. 7 x 4 . 17. 3 a 4 4- 15 a 8 4- 11 2 4- 14 a - 1
18. 16 x 8 4- 12 x*y - 2 y* - 4 &z 4- x>&.
xxviii ANSWERS
Page 260. 19. VF-Tx + vTfy + 1. 20. - 3 x 2 . 21. - 5 a 2 x-8 x 3 .
22. 14 x - 7. 23. r 5 4 xy. 24. 1 + as - x 2 . 25. 2 a. 26. x } 4- x 2 4- x.
27. 0. 28. 12 x. 29. 0. 30. (a) 2 x 4- 2 ?/ ; (ft) 2 y - 2 g ; (c) 3 x + y
+ z; (d) x - y 4- 3 2 ; () 2 x + 2 ?/ -f 2 2 ; (/) 2 - 2 y + 2 s. 31. 11 a:
-16t/. 32. 3a~2c. 33. -8x 3 -8x. 34. 13 - 12 a. 35. -9x.
36. - 5 b + c - 2.
Page 261. 37. 3a' 2 -f5+7. 38. 5 + 8. 39. -5x + 2y~z.
40. 3 a - 3 b - c. 41. x 2 - 9 x - 4. 42. 7 a - 9 b- c. 43. 6 c - e +/.
44. 3 a _ 5 5 _ 7 c . 45. a - 6 b. 46. - 3 6 + 9 c - 9 df. 47. x 8 + 3 x 2
+ 3 x-f 2. 48. x 3 - 3 x 2 + 3 x - 1. 49. x 3 4- x 2 -- 7 x - - 15. 50. 2 - 15 x
+ 28 x 2 - 15 x 5 . 51. 1 - x -!- x' J - x 4 . 52. x 4 4- 2 x 2 4 0. 53. 6 x 4 -
13x 3 4- Ilx 2 -5x4- 1. 54. ft-2ft 4 4-l. 55. a 8 -a -- 13 a 4 + ll a 2 -2.
56. a 8 4- ft 3 -I- c ;{ - 3 aftc. 57. a* - ft 3 - c 3 - 3 aftc. 58. x 4 + 2 x 2 ?/ 2 -
8 y 4 - 18 ?/ 2 2 2 -9 ^ 4 . 59. ^ 6 8 -j-27 4-0 ab. 60. 8 x* + 27 y -f- 2 3 - 18 x?/0.
61. x 3 4- 15 x 2 4-71x4- 105. 62. x 3 - 15 x 2 4- 71 x - 105. 63. x 3 - 15 x 2
4- 62 x - 72. 64. x 8 + x 4 4- * 65. x 8 - a 8 . 66. x 8 + x 4 y* 4- ?/ 8 .
67. 1-x 8 . 68. xt-y 6 .
Page 262. 69. a 24 - x 24 . 70. a o_a 4 -a 2 +l. 71. a*-2a 2 6 2 +& 4 ,
72. 4 x 3 - 3 x?/ 2 4- 1/ 3 - 73. 4 - 4 a !! /> + 16 a/> 8 - 10 ft 4 . 74. x 4 - 36 xfy 2
4- 108 xy 3 81 ?/ 4 . 75. - a 4 - a- 4-1. 76. x' - 2/ 1w . 77. a 8 ?* 3 4- ?> 3 .
78. a 3m 4- 5 3n 4- c 3 4- 3 a 2 '6 w 4- 3 a'ft-. 79. m 3 " + n 36 + P 3c - 3 mn b p c .
80. 24 a 2 6 - 12 a/; 2 . 81. /> 3 4 &p t q. 82. - 4 a 3 + 3 a?; 2 - 6 3 .
83. 3 x 3 4- 4 x 2 y - 4 x?/ 2 - 3 ?/ 3 . 84. 6 p' 2 q - 54 ? 3 . 85. 2 x 2 ?/ 2 4- 2 ?/ 4 .
86. 0. 87. 4 fee 4- 4 ac. 88. x?/ 2 + y' 2 z 4- 2 x^. 89. - x 4 - y 4 z*
4- 2 2/V 2 4- 2 x 2 z 2 4- 2 x'V 2 . 90. a~b - a6 2 4- fc' 2 c - fee 2 4- ^a - a' 2 c. 91. 0.
92. 6a6c. 93. 10 a -12 b. 94. 6 a 2 - 15 ab 4- Oac 4- 6 be.
Page 263. 96. 2(a.+ ft) 2 -(a 4- ft). 97. (a + ft)" - 2. 98. 2 2 *-
a z 4- 5. 99. 2 2a - 2 4- 2 30 -- 2. 100. 3~ n 4- 3 - 3 2n - 1. 101. 5 4 4-,5 3
_5 2 4-5 = 73(). 102. 2 x 2 4- 5x4- 4. 103. 5x 2 -2x4-3. 104. 2 y 2 -
6 y - 4. 105. x 2 - 2 xy + 4 y 2 . 106. a 2 4- 2 aft 4- ft-. 107. 2 x 2 - 5 ax -
2 a 2 . 108. x 2 -3x2/-?/ 2 . 109. x 2 -5r*x 5 a' 2 . 110. 2 ?/ - ?/ - 4.
111. 3 a -5 a -5. 112. 2 a 2 -4 aft -5 ft 2 . 113. - 3 r 2 - 2 :ry 4- 3 y 2 .
114. 2z 2 ~3xy-?/. 115. y-3. 116. x 2 +^ + 2/' 2 . W. 2 - 4 a 4-
16. 118. a 4 4- a 2 ft 2 4- ft 4 . 119. a 2 4- 4 ft 2 4- c 2 4-2 fi - ac 4- 2 aft.
120. x 2 + 9 y 2 4- 1 4- 3 jry 4- x - 3 y. 121. x* - ;rty 4- xy 3 - y*.
122. x 7 4- x 4 4- x. 123. a"-*- 1 4- a n - a"- 1 . 124. a* 4- a' -'ft 4- a*--ft 2 .
Page 264. 125. 1 + a 4- a 2 - 2 ax 4- 2 x 4- 4 x 2 . 126. x 2 4- 2/' 2 + 2' 2
xy-xz-yz. 127. x 2a - x -f- x* - . 128. 1 4- 9 2w 4- 4 ft*" 4- 3 ~ + 2 ft n
- 6 a m b\ 129. 2 aft 3 4- 3 ft 4 . 130. x 6 - 3 x 5 4- 9 x 4 - 27 x 3 4- 81 x 2 -
243x4-729. 131. a 2 -2 aft -2 ft 2 . 132. x 4- 1. 133. 1|. 134. 0.
ANSWEKti xxix
135. 2. 136. m. 137. 1. 138. 12. 139. 24. 140. 6. HI. 7.
142. -- 3. 143. 2^. 144. - 2. 145. 2. 146. 5. 147. - ?.
148. 1. 149. 2. 150. Iff
Page 265. 151. -1. 152. -1. 153. 1. 154. 3. 155. 15.
156. 1. 157. 2. 158. 3. 159. 4. 160. 6. 161. - 1. 162. 50.
163. 20. 164. y-3, y - 2, y- 1, y. 165. 15 - a. -166. 23 -k.
167. 30 + xyr. 168. 18, 19, 20, 21, 22. 169. 15, 18, 30.
170. 37 1, 75, G7|. 171. 10 in. 172. (a} 59; (6) -40; (c) 160 C.
Page 266. 173. 1. 174. 4. 175. 147 mi. 176. 36ft.
177. 30 yr. 178. 12, 10, 6 panes. 179. 40 yr., 12 yr. 180. 32 h. p.,
48 h. p. 181. 12 ft., 10 ft. 182. 12 yr., yr. 183. 42 yr.
184. Or + 2)0e-l). 185. (y-7)(y + 6), 186. (x - 12) (j; + 3).
187. (a + 11) (a -10). 188. (y - 17)(y + (>). 189. (ab - 11)
(a& + 22). 190. (4 a +!)( + 3). 191. (5x - 2)(2x + 3).
192. (at + 8) ( -7).
Page 267. 193. (y - 7f))(y - 2). 194. (2a + l)(a - 10).
195. ( + 7)(rt-4). 196. (3x + 2y)(2x-3y). 197. C3 y _l)(,/_4).
198. (a; 2 + 3) (x 2 -f f>) . 199. (2 as + 6 y) (x 2 y) . 200. ( jc - 1(5)
(.r + 4). 201. (y_24)(y-5). 202. (x - y + 3)(r - // - 3).
203. a- 8 (r-7y)(ai- I2y). 204. (5 x - 3 y)(a - y). 205. (7x-2//)
(2/-3y). 206. (x + 2)(x - 2) (3 x - 1). 207. (4 x 2 + 9)(2 x 4- 3)
(23-3). 208. 2a(4-2ft)(-2fo). 209. (y _ ft)(y 4- 6 - 1),
210. z(x-10)(x-l). 211. r>x 2 (4x- 2 -4x-l). 212. 3(x - 9)(x-f 2).
213. a 2 (15- 6)(4 + 6). 214. 2 xy(3 x - 6) (2 x + 1). 215. x(x + 6)
(a-1). 216. 2(x-8)(x-3). 217. 2 ?/(x - ll)fx -3).
218. x(x f 3)(x+ 10). 219. a(a- l)(a 2 4- a + 2x 2 ). 220. ry(x + 2)
(y + 2). 221. (3-2)(2a +3). 222. v?y(x + y + 9a;y 2 ).
223. (3x-2?/)(2.r + 3y). 224. (7 a- y)(jc + 7 y). 225. (a - c)
(a + & + c)(a 2 + aft- 2ac + ftc + c 2 ). 226. (JT +)(x 2 -x^+2; 2 )(x+y)
or OB + y) Oe* 1 + s). 227. (7 x - y)(z - 3 y). 228. yfor-y)
(2 * 4 ary + y 2 )- 229. (x + y + 2)(x + y- 2). 230. (5x + 2) (3 x +4).
231. a(3 + 26)(3a-2ft). 232. fc'2a+6-2c). 233. (y + 1)
(?/+l)(y-l). 234. (3a + 4?> + o(5c-rt). 235. (x^ + x-1)
(r^-x + 1). 236. (8x + 3)(3x-4). 237. (,r - l)(x 8 + x 2 - 1).
238. (x + l)(x-l)(y + l)(y-l). 239. (ox + ? /)(5x-y)(x+3 y)(x-3 y).
240. (a >2 ft 2 + 2a6.--c 2 )(-a 2 ?) 2 +2a/) + c 2 ). 241. 2(-d)(rt + ft + c+c2).
242. (x-yX^+y" 1 )- 243. (4 x 2 - y) (\r-3y). 244. (2x-3y)
(x + 6)(x-6). 245. 8(a-r}(/)-27). 246. 3 xy(x- l)(y + 1).
247. (a 2 +l)(a*+ 1). 218. (/ - m)(x- y + a).
AN 8l\' Eli S
Page 268. 249. (7 x-3y 4 1)( 4 2 *>). 250. Ca&c 1)(-M 3).
251. 2x(x-)(x 4 &). 252. (a 4 A)(2 x 4 II 4 -)- 253. o(x4l).
254. 3x44. 255. 5x4-2. 256. 3xy-f 5. 257. rt - 1. 258. ^/>-J.
259. 2 x - 3. 260. 7 x - 5. 261. 3 x// - 7. 262. x 4 5. 263. x - 7.
264. a; -10. 265. x-12. 266. (x 4 \')(s 4 2)(x 4 3). 267. (x 4 1)
(x 4 3) (x - 4). 268. (2 x - 5) (x 1) (3 x - 4) . 269. (x 4 5)
275.
X47
276. ^- + J/.
1 x~y
Page 269.-
,-,
278. '-^--;- M *.
o x," 4 x
m- 4 ?i 2
280. m - 7 L-?'
>. 281.
_^. 282. ^^.
m , 4 /
) 3
x c
284 2 X o
r 1
28ft .
Qflfi 1 2 X
2 x 4 3
X41
1 4- 3 x
288 3a4 5fc
289 *** ~^ ^^
^ . 290 ^ ~^ y ~ z
x-y 4
z x y 4 3
292. -L^ir:^.
293. ^^" ' 7 .
294. (&4- c -)(c-
x- 1
ax />
9.ftfi <^ 4- TiV/'.
a. ^ 2ft7. 1
42x^
279. *. + f.
x 4 3
283. -'^rJ'.
x ?/ z
a~ b*). 295. 1.
Page 270. -298. 4. ' ^ 299.^^-^-^^^^^.
0.2* > (x44)(x-3)(?/47)
300. i^^ 2 !^- 4 ^ ^-^. 301. - -? 302.
3;(x 4) (x4 1)
303. - - -- -. 304. 2. 305. 0. 306. 7^T
307. ?^ZLiZ 308. ^-^A^ + w^ 2 . 309. o.
(^- 4) (x 4 11) (x- 13) n(w4 n) a
310. J^l^^J..
x' 2 -5x46
x s - 1 a 8 ft 3
014 1 . QIK 1 . aia 2
(X-
317
3) (x _ 4) (x - 5) I42x (x 2 l)(x >2 -4)
1 Q 1 /v.8,, 1
1 . 318 ^ . 319 *
(X-
320. 1.
l)(x 42)(x 4 5) x 4 -y 4 """ x(l X 4 )(l4^)
001 rt c , 322 323
****' r -L 4- h 4- ^
XXXI
Page 272. 324. - ; A^L-5L-. 325. 5
326. ^-^K^+M^ 2 ). 327. 2(q-.".)Cr-4)__ j
328. AC^Ln?).. 329. A^izA??r+J!j/?_ . 330. ^_- f>
331. 2 ?/+:>. 332. 1. 333. *-- ^ . 334. ^"" 4 . 335. :r '--
6 ?/ + 3 x + 3 // s + 4 x- -f- 7/ 2
Page 273. 336. (a + b + c\ 337. a' 2 + 4 & 4- \ b.
339. .vin a -T+^. 340. 27 ^" .
r J // J li'oy 3
341. m - H ' 2wi ^-' / ' ;J . **" ~ /r " y 4 -I- 2 >- 3 + 8 x* -f 2 y -4- 1 ^
350. . 351. If.
343
00
Page 274. 355. - --- 356. - i 2 ^. 357. 4 r-w.
x 54 1 1 + x
359. a 2 . 360. 1. 361. 0. 362. ?--=.!. 363. 1.
2 ?/
364. -1. 365. L . 366. *
?>*. 370.1.
j-J{j ^> ^-'2 j^
al(<i-~l>]
)' 2
Page 275. 367. &* + <.*.
368. -/-'. r ...
369. 4 + &<
071 ,,.. ^p " ^
373. 1 + x.
374 ^ + ? '
Oil. //~. O < <*.
rr* ^
2 ab - a-
Page 276. 375. _-*L'L+. 1 ')_ 376. 12. 377. 4 A. 378. -4.
x' 2 + 4x + l " 4
379. - -7. 380. - f,. 381. 2. 382. S. 383. 2 r 3 6 . 384. 0.
385. -. 386. 9^. 387. 1. 388. l'j. 389. 0. 390. 3.
Page 277. 391. 7. 392. 4. 393. - 1. 394. -2.3. 395. 3.
396. T \. 397. 0. 398. 20. 399. 3. 400. 11. 401. 402. i-
13 c a
403. - ^ . 404. a-b. 405. 6-a. 406. A^_. 407. 2(a + 6).
7/i (5 _ a) a + b
408. - <L -- 409. a +6.
XXX11 ANSWERS
Page 278. -410. ab. 411. 1 /?$-+&?. 412.
. - __
a + b a + b c
413.
2 ' w -f w '
417. c. 418. 0. 419. a + ft + c. 420. A 5 mi , B 4 mi. 421. 42.
422. ISjmi., ISJini. 423. 14 miles.
Page 279. 424. 24 days. 425. 5-1,55. 426. z 8 +?/ 3 . 427. |.
428. a* - />a. 429. 21. 430. .rz - -- 431. 7 : 5, m - 1 : wi,
vz
(d - 6) : (a - c). 432. --- m - in., -^- in. 433. 40, 50, 90.
m + M r?i + ft
434. f. 435. m. 436. ^. 438. () not true, (&) true, (c) not
true, (d) true.
Page 280. 439. (a) 1, (&) 2 fc, (c) 3a- 4 5. 440. 40| oz.
441. 3, 10. 442. 2, 3. 443. - L \, I. 444. 10.}, 10$. 445. |, 1.
446. 1$, 2$. 447. 33, 2. 448. \ I. 449. , 3 ,, ^V. 450. i, i
b a b
451. 0, - 7. 452. 7, - 5. 453. - 7, 8. 454. 0, - 2. 455. - 2,
- 7. 456. - 1, 1. 457. 6, - 1. 458. 10, 7. 459. 10, - 2.
Page 281. 460. 8,0. 461. -10,7. 462. 0,12. 463. 5, - ',}.
464. 0,7. 465. 11, - 1. 466. |, 2. 467. 0, - f>. 468. 1,0.
472. <L+ 6 ,
(.te + />c
c(f- be,} c^ac-j-d} ^
fcfZ ' a/- ?>rf
+ ( ;
86
Page 282. 476. fj. 477. |. 478. |. 479. 9,7,5. 480. 12,4.
481. 20 yr., 32 yr. 482. 53 yr., 28 yr. 483. A $ 3500, B $ 2500.
484. 17,4. 485. $2000 at 0%. 486. $260 at 0%. 487. 84.
Page283.-488. 63. 489. *+-, !L=4. 490. ' * , L- - 6 .
& 22 2 a 2 a
491. 10, 7|, 5J. 492. 18, 32, 10. 493. 2, 2, 2. 494. 5, 6, 8.
495. 1, 4, 6. 496. 5, 6, 7. 497. 20, 10, 5. 498. 5, 6, - 7. 499. 17,
22,46. 500. - - - , - ? - , - ~ . 501. 10,2,3.
a 6 -f- c a + & c 6-fc a
502. 2, 1, 4. 503. , 2, 2f 504. 2, - 3, 0. 505. 4, - 1, - 2.
506. 7, 2, 0. 507. - 3, 2, 6. 508. 8, - 4, 6.
2ft 2r 5n
2
2
ft c ' a b
519. 1, 1, 1. 520. -i
R9.1 ' ~~ - - .
-f ft - (
;' fc _ rt-c'
ANSWERS xxxin
Page 284. 509. 10,6,0. 510. 5,3,8. 511. 3,0,5. 512. 8, - 1,
- 2. 513. - 7, 6, 14. 514. 1J, 4, 1. 515. f, f, f 516. -?-- ,
2 __ 518. ^ 4.
a -f ft + c
+ 26 + C), - J(a -f ft + 2c).
, , . S82 ;_ 523 M
2 ftc 2 c 2 ft
3465, 0000, 6435. 524. a -f ft + c, a + ft - c, a - ft -f c, - a + ft + c.
Page 285. 525. (a) 74 Ib. tin, 40 Ib. lead. (ft) Ill Ib. tin, 115 Ib.
lead. 526. 4* da., 3 da., 24 da. 527. 3f , 4f, 24. 528. J7] min.
529. g(rc+ 6-c), a+ 6. 530. da., 7^ da. 531. 232. 532. 3,4,9.
533. 8 mi. per hour, 4 mi. per hr.
Page 286. 536. 2ft da. 537. 1.56 sec. 550. (a) 2.75, - 3.25, 1.5.
(6) 3.24, - 1.24. (c) 3. (d) 1. (e) 2.4, - .4.
Page 287. 551. (6) 31.25m. (c) 2.24 sec. 552. 3.83, -1.83.
553. 3, 3. 554. .7, - 5.7. 555. 5.54, - .54. 556. 4.37, - 1.37.
557. - 2.10, 4.16. 558. - 1.03, 2.03. 559. - 1.02, 3.02. 560. - 5.62,
.62. 561. -1.31,3.31. 562. - 1.53, - .35, 1.88. 563. 1.78, 2 imag.
564. - 1.04, .55, 1.30. 565. .04, 3.02. 566. - 2.5, 1.73, 2 imag.
567. 2.00, - 1.15, .21, 1.0, 3.05. 568. 1.38. 569. (a) - 4.51,
- 1.75, 1.20. (ft) -4.12, -2.4, 1.52. (c) -4.78, -1.14, .02. (d) - 5.10,
2 imag. (e) 3. (/) - 10 to 8.5 -f. (gr) -10 or 8.5+. (h) 8.5. (i) -3.33.
570. 1$, 2$. 571. |, If 572. 3.6, - 1.6 ; - 1.6, 3.6. 573. 3, 2 ; 2, 3.
574. 4, 3 . _ 3, _ 4. 575. 4, 2 ; - 2, - 4.
Page 288. 576. 4.3,1.4; -1.8, -3.4. 577. 2.3,1.15; -2.3,
-1.15. 578. 4.8, 1.3; 1.3, 4.8. 579. H, 1 ; o> , T
580. Roots imaginary. 581. - 7, 1 ; 1, 7. 582. 2, 4 ; 0, 0.
f\4 ,% rr\*
583. 3,0; 1, -2. 584. -21*_. 585. 586. a 7
27 y* y% Z *
+ 86 <z 4 & 8 + 35 ft 4 + 21 2&6 + 7 rt?> 6 + ^ 687i a 7 _ 7 rt e & + 2 1 a 5 ft 2 -
35 4 ft 3 + 86 3 ft 4 - 21 a 2 fts + 7 a ^ _ 57. 588i tt e _ 3 rt 4^2 + 3 ^254 _ ^
589. 1 -3x + 3x 2 -rA 590. 8 + 12 x + 6 tf -f ir 3 . 591. 27-54x
-f36a-2-8x 8 . 592. l+4x+0x 2 +4x 8 -f x 4 . 593. a*-8a + 24tf i -82a;-f 1.
594. 2 a 8 x 8 + 6 ax&fy 2 . 595. 2 4 x 4 + 12 a 2 xt 2 b*y' 2 + 2 6 4 ?/ 4 .
596. 2(6 + 10a: 8 + a-6) > 597. i -_ 4 sc 2 + 6 ^ - 4 x + .r 8 .
598. 1 4-2x + 8x 2 + 2x 8 4-x 4 . 599. 1 - 2 x -f 3 x' 2 - 2 x 8 -f x 4 .
600. 2(4 + 26x 2 + 10 x 4 ). 601. 1 + 3 x -f 6 x 2 + 7 x 3 4- 6 x 4 + 3 x G -f-x.
602. 8 a 6 - 8 a 4 & + 6 a 2 & 2 - 4 6. 603. 2 - 2 xt/ 2 4- 8 x 2 ?/ 4 - 8 x 3 ^.
604. x 8 - xj/ -f xV - xV -f y 4 . 605. a 4 - 2 a*b + 3 a6 3 - ft*.
XX XIV ANSWERS
606. 3 - x -(- 2 -S 2 . 607. 2 a-b + a - &-. 608. x 3 4- 3 x - 4- '
x X*
609. 2 x- 4-4 ^/- 4-3^4- >A 610. (*_ a: + ft). 611. (a-3&4-5rj.
612.
Page 289. 613. a 4- b. 614. 2 a: 4- 3. 615. 3 x 2 - y. 616. 2 a - 6*.
617. a-b. 618. y - 2. 619. 971. 620. 5002. 621. 78.04. 622. 2092.
623. 7003. 624. 210.0. 625. 1010. 626. 898. 627. 7.002. 628. 009.3.
629. 25.203. 630. 14.702. 631. 703,001. 632. If 633. If. 634. 1$
635. 2.049. 636. 3.001. 637. 0.303. 638. 8 f 639. 4330 da. 640. 5, - 14
641. 25, - 4. 642. 12, - 13. 643. 50, 3. 644. 1, - 4. 645. 4, - 25.
646. 2, -8. 647. 2, -11. 648. 11, -0. 649. 9, - 9*. 650. 2, - 2f
651. - 2, 10.
Page 290. 652. 2, - ft. 653. 2,3. 654. -3, |. 655. If - 2f
656. 7, - If 657. - 1, If 658. 3, - 4$. 659. 3, I}'/. 660. 2, 4 r V
661. -a, - b. 662. a + b,a- b. 663. {Z , - 664. 4, 0. 665. If 0.
b b
666. 13, f 667. 0, - 3f 668. 5, - l/' 3 , 669. 5, If 670. 5, -if
671. 2f, 0. 672. a l ~- 673. (ab)-. 674.
. . . .. ,
a 2 / 4- />
a 2 4- 7/ 2 + w ( 4- 7>) <T! .- ,
676. i, - - ~ 677 ' ^~ 3 V3 ^' 678< i x a ' 2 + /
679. V^+lO^M-"^-. 680. 1J. 681. 3. 682. 2 + , - 1.
683. ( 2 + 36 )K3 + 2&). 684. ^, ^. 685. 0, ~^' 2 + fe ' 2 ).
6 a a 4- b
686. |o, 3 a. 687. a + 6, - 2
(i -f- A
Page 291. 688. c, n-^l + -^. 689. - a, -6. 690. \, \.
ab + be ac
691. 2 a - 6, 8 6 - . 692. ^-^ ^ fe + ^ ~ < a -+^
2 a
694. fo ~ a , 1m*. 695. ^A, J^-^. 696. 1, __
1 a/> ft-a l-a6 \+ab a-f6-fc
__ ( a 4. /> 4- c ) _
697. 1, ^ z - 698. \/3 V2. 699. V5 1.
700. - 1, - 2f - 4, 2f. 701. 1, - 2, - 4. 702. 1, 1|, - |, - f
703. 3, 2, (5 V^~3). 704. 2, - 3, K- 1 3 V^3).
705. - 2 V2, 1 V2. 706. ^ - - . 7Q7> j(_ 5
b
(_ 5 x/^15). 708. 1 V7, 1 V"^TJ.
ANSWMHti
a db Va^T
Page 292. 709. 3, i 2. 710. i^, -;- 711.
712. VV> -L4V-34. 713. v'll, iv/Jj. 714. ^
715. 1, 1, ?->.^. 716. 24, 25. 717. -5. 718.
3 <
719. 45da. 720. 17ft., 13ft. 721. 00. 722. 8,9. 723. 8,9,10,11.
724. "V3- 725. 15 shares. 726. 39,8. 727. 300. 728. I 1 ,.
729. 13,
Page 293. 730. y - 1. 731. a 2 - x 2 . 732. a + 6 + c - 3 a*&M.
733. m-n. 734. w" + 4 d" + Hid.
736. cr*-lr*. 737. a"- 2 + 2 tf-'c- 1 -/>-- + <-
739. 4 x' 5 - x 4 4- 3 x~* -(- 2 ar- -f
741. x-\-y. 742. ^ Ti a*tj~ 6 4- ^
j i
-i-3x 4 rt + 2u: 8 'o' a +a 2 .
747. rt3 + 2 1 ^-i-^-- > -,
750. 2 V2 , 24 4 , .r-J w L
735. x* -[- 1 + J- 4 .
738. 1 + (t' 2 lr'~ + a 4 6- 4 .
f- 1. 740. Hi a;~ - 12 af -V^ 4 '> f^'
743. ^^ -f ^3 _ r } t 744. x i _L 2 .,-V/^
3 746. 2x-3^ 4-7.
748. a- 5 , a:-* , 1. 749. a;- 1 , r, a 2 6^.
751.
752.
755.
1
756. 34.
758. 5.
Page 294. 753. .r. 754. -^
;r+y-> 1-^ _
759. 29\/3. 760. 30\/10 4- 104 v/2.
763. 33^2. 764. 59,257.
767. fya-w&cu. 768. -3^. 769. T %.
773. ^7x-y.
776. 24-V2. 777. 34-2V3. 778. 3-2>/2.
782 *+V( 2 -"r 2 )'
x ^
785. VT14-V5. 786. v/7 - V2. 787. 7 - 3V5. 788. VIO-\/3.
789. 2\/5-3V2. 790- 2v / 3-v / 2. 791. 3 VlT - 2. 792. 3V7-2V3.
793. 3V7-2VO. 794. J(v'lO-2V 795. 3V5-2V3. 796. 3-V7.
XXXVI ANSWERS
797. 6+V7. 798. Va + 6 + Vtt-"fc. 799. Va - 6 4- V& - e
800. X/^-^+A^ + V 801. \/5. 802. ^VG. 803.0. 804. \/2.
" o 2 *a 2
l +j!>.. 806. 48. 807. **. 808. ^E*!. 8 09. x.
2 *x a 2 x
Page 296. 810. 8(?/ 2 + 2x 2 ?/ - x 4 ), 16xyV2*/^-~x 2 . 811. 1C.
812. 1. 813. 19. 814. ~ tt - . 815. 17. 816. 23. 817. 10.
4 b' 2
818. 4. 819. 11. 820. 13. 821. 6. 822. 3. 823. 7. 824. 7.
825. 7. 826. 0. 827. - Va. 828. -1. 829. 17. 830. Hoots
o
are extraneous. 831. Hoots are extraneous. 832. 1). 833. |.
834. o. 835. 0,3. 836. ((' + d)*. 837. 25.
Page 297. 838. 4, 6, - 7, H. 839. 2, 3, - |, $. 840. 4, 1.
841. 0, 2|. 842. - 11, 14. 843. (x + 2)^ 3 + 5 .r -f- 8). 844. (:r 4 1)
(x 2 4-*4- 1)(V 2 -X4 1). 845. (a 4- 50(i -2)(5 a* 4- 7 04- 20).
846. (x4-2?/)(x-2y)(4x 2 -.r?/ + l(l^). 847. (a - l)(x - l)(x + 2).
848. (x - 1) (x 4- 2) (x 4 2). 849. ( - ?>) (a + ( ^> _ 3 ^ , 860 . (, r + 3)
(x 2 + * - 1). 851. (x + y) (x -f y) (x + y) (x - y) . 852. (x 2 -f r - 5)
(x-3). 853. (x + 0X0-24. 2 a- -5). 854. (x - l)(x - 3)(x + 4).
855. (x-l)(^~3)(.r-4). 856. (x 4- l)(x 4 -2)(x - 7). 857. (x - 1)
(2x + 3)(2x4-3). 858. (r. 4- 1)(4 x - 7)(4 x- 7). 859. (2 x + 3 y}
(4 x 2 - 6 xy + 9 2/ 2 )- 860. (2 4- rae) (4 - 2 ax 4- a 2 jc 2 ). 861. (3 jr - 4)
(9x 2 4l2x + 10). 862. (1 - 4)(1 -f 4 a 4 10a 2 ). 863. a(ry + ::}
(x*y*-ryz + z*). 864. (3 b - 1)(0 ?> -f 3 ?> 4 1). 865. (2 a - 5 b)
(4 c 4- 10 ab 4- 25 ?>2), 866> (9 %7 . + 8 ?/ ) ( 81 ^3 _ 72 xy 4 04 ^).
867. (2x-3?/^)(4x 2 4-Ox^ + 9//%2). 868. (a + 2 ?>c)(a 2 2fl?>44 ?/V 2 ).
869. (a m - />") (a'- >wt + m ft m -f & 2m ). 870. ( 2wt -f ft 2 ") (a* m - a 2m W" 4 ^ 4 '0
871. ( 4 4- 1) (a 8 - a 4 + 1). 872. (a 4 rc 3 ) (a' 2 - an 4 3(> n 6 ).
873. 6(a-6)(o a + + &). 874. a(-ft)(
875. a^*4l)(a-a 4 + l). 876. (a m -f- l)(a m -
4- m + 1) (a 4m - a 3 " 1 4- '* 2w - o m 4- 1)
Page 298. 879. 4. 880. m = 2, n = - 29. 881. 4, 4 ; - 4, 8.
882. 3, 1 ; 1, 3. 883. 5, 2 ; 2, 5. 884. 2, 3. 885. 4, 3,
3, 4. 886. 2, | ; 1, V 887. 5, 4- 3. 888. 2, 1 ; 1, 2 j
- 1, - 2 ; - 2, - 1. ' 889. 3, 6 ; - , - f. 890. 5, 2 ; - f , - f
891. 2, 4 ; 4, 2. 892. $,;, $. 893. ^, 2. 894. 4, 3 ; - 3, - 4.
895. 5, 3; 3, 6. 896. 7, 2. 897. db 7, 11. 898. 4, 3; jV3
899. 8, 2 ; J V|, ^ V}. 900. 0, ;
ANSWERS
XXXVll
oJ--- 6
V 41
901.
1 i ' , r ' '
a 6 a a + ft <
Page 299. 902. 2, 1 ; - 1, 1; $(l V^3), 1 V.I. 903. 3,
T3. 904. 8, 2; 2, 8. 905. 4, 3 ; 1, 12. 906. 115,
329; 333, 111. S07. 0, ; v/(ai- !)(&- I) 2 , v/Ca^T)^ - 1).
908. 3, 6; 6, 3. 909. 19,01; - fj, ^y. 910. w, w ;
911. Z
tt2 , ^. 912. |, i-1-
6 a "
2 ?/i 2 w
913. 3, 5. 914. 1,
V5 V5
T 2 ; Y, =F J. 915. 4, 2. 916. 2, 1 ; 1, 2 ; - 2, - 1 ; - 1, - 2.
917. 8, 6 ; 6, 8. 918. 3, 2. 919. 1, 2. 920. 5, 3 ; 3, 6.
921. 0,0;
924.
923. 0, ; 4, - 1. ,
Va- -f ft-
926. 8, - 4 ; 28, 66 ; - 4, 8 ; 56, 28.
_2 -i-fcVira^
;' 2
927. a, 6
922. 6, 4 ; 4, 5.
925. 3, 4 ; 5, 2.
a + 2 ft 4 &
928. 1, 1 ;
3 ' 3
2, 4, 6.
Page 300. 930. 3, 7, T 2. 931. 0, T 6, 8. 932. i 8,
7, dL 4. 933. i 3, 4, 6. 934. i>, 6. 935. 7, 4. 936. 12, 8.
937. 20 ft., 15 ft. 938. 3 ft., 2| ft. 939. 480 sq. ft. 940. 100 rows.
941. 19 ft., 16 ft. 942. '3 in., 4 in. 943. 15 ft., 8 ft. 944. 10, 12 mi./hr.
Page 301. 945. 60 ft., 40 ft. 946. 16 in., 9 in. 947. 11,760 sq.
yd. 948. 73. 949. i(- 1 + V- 3), J(_ 1 - V-~~3). 950. 248.
951. 6 da. 952. 12 in., 6 in. 953. 7 ft., 4 ft. 954. 6 mi. 955. 2 yd.
Page 302. 956. 4. 957. -333. 958. - 26-j. 959. 1, 2, -5 ....
960. 280. 961. 5l4f. 962. 108. 963. 4. 964. 11,111.
977. 80.
Page 303. 978. 10, 12, 14 .... 979. |, J, f, -. J. 980. 7 or 30.
981. 4. 982. 3. 983. 5, 11. 984. 5, 7, 9, 11. 985. 8.
986. 6, 28, 496, 8128. 987. %* - 1 = 18,446,744,073,709,651,615.
xxxviii ANSWERS
988. ^ f (2-f-3V2). 989. 0. 990. - W 1 W -JI + 1 / 1 _ -_L\ . 991. .
992. 2(2 -v/2). 993. (5,18,51.
Page 304. 994. 162, 108, 72, 48. 995. a ~ \, r = 2. 996. 4, 8, 1(5.
997. X, (J, 12, 24. 999. (a) 2^ + \/2), (6) 8(1 + v 2).
1000. () 12(2+V3), (Z>) 4V3. 1001. (a) -- 1 , (6) -^ 1002. 0.
1003. ^Trsq. in. 1004. 9 da. 1005. a 13 - 13 ax + 78 a3 2 ... -
32
78 a' 2 x n + 13 (tx 1 - - a- 13 .
Page 305. 1006. 243 - 810 x 2 + 1080 x* - 720 * 4- 240 r 8 - :J2 r 10 .
1007. t - 14 y + 84 y* - 280 ?/ + 5(>0 y 4 - 72 ?/ 5 + 448 y*> - 128 ?/ 7 .
-" C - I } - ( '^^ 5
1008. K 4- 2 MJ--iy - -
1009. 192,192rt?)r 8 r? 8 . 1010. 1 2,870 a 8 6 8 . 1011. 70
1012. - 92,378 <W and 92,378 9 /> l . 1013. 1710 rtV and 1710
1014. 120 *. 1015. 252 6 /> 6 . " 1016. 120 6 a; 4 and 120
1017. i-^l^. 1018. 35. 1019. 12,870 z 8 . 1020. - 5 & 7
1021. 3003.
ELEMENTARY ALGEBRA
By ARTHUR SCHULTZE. i2mo. Half leather, xi 4- 373 pages. $1.10
The treatment of elementary algebra here is simple and practical, without
the sacrifice of scientific accuracy and thoroughness. Particular care has been
bestowed upon those chapters which in the customary courses offer the great-
est difficulties to the beginner, especially Problems and Factoring. The intro-
duction into Problem Work is very much simpler and more natural than the
methods given heretofore. In Factoring, comparatively few methods are
given, but these few are treated so thoroughly and are illustrated by so many
varied examples that the student will be much better prepared for further
work, than by the superficial study of a great many cases. The Exercises are
very numerous and well graded ; there is a sufficient number of easy examples
of each kind to enable the weakest students to do some work. A great many
examples are taken from geometry, physics, and commercial life, but none of
the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry. To meet the requirements of
the College Entrance Examination Board, proportions and graphical methods
are introduced into the first year's course, but the work in the latter subject
has been so arranged that teachers who wish a shorter course may omit it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE, Ph.D. lamo. Half leather. xiv+563 pages.
$1.25
The Advanced Algebra is an amplification of the Elementary. All subjects
not now required for admission by the College Entrance Examination Board
have been omitted from the present volume, save Inequalities, which has been
retained to serve as a basis for higher work. The more important subjects
which have been omitted from the body of the work Indeterminate Equa-
tions, Logarithms, etc. have been relegated to the Appendix, so that the
book is a thoroughly practical and comprehensive text-book. The author
has emphasized Graphical Methods more than is usual in text-books of this
grade, and the Summation of Series is here presented in a novel form.
THE MACMILLAN COMPANY
PUBLISHERS, 64-66 FIFTH AVBNTC, HEW TOSS
ELEMENTARY ALGEBRA
By ARTHUR Sen ULTZE. 12010. Half leather, xi -f- 373 pages. $1.10
The treatment of elementary algebra here is simple and practical, without
the sacrifice of scientific accuracy and thoroughness. Particular care has been
bestowed upon those chapters which in the customary courses offer the great-
est difficulties to the beginner, especially Problems and Factoring. The intro-
duction into Problem Work is very much simpler and more natural than the
methods given heretofore. In Factoring, comparatively few methods are
given, but these few are treated so thoroughly and are illustrated by so many
varied examples that the student will be much better prepared for further
work, than by the superficial study of a great many cases. The Exercises are
very numerous and well graded; there is a sufficient number of easy examples
of each kind to enable the weakest students to do some work. A great many
examples are taken from geometry, physics, and commercial life, but none of
the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry. To meet the requirements of
the College Entrance Examination Board, proportions and graphical methods
are introduced into the first year's course, but the work in the latter subject
has been so arranged that teachers who wish a shorter course may omit it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE, Ph.D. i2mo. HatF leather. xiv+56a pages,
$1.25
The Advanced Algebra is an amplification of the Elementary. All subjects
not now required for admission by the College Entrance Examination Board
have been omitted from the present volume, save Inequalities, which has been
retained to serve as a basis for higher work. The more important subjects
which have been omitted from the body of the work Indeterminate Equa-
tions, Logarithms, etc. have been relegated to the Appendix, so that the
book is a thoroughly practical and comprehensive text-book. The author
has emphasized Graphical Methods more than is usual in text-books of this
grade, and the Summation of Series is here presented in a novel form.
THE MACMILLAN COMPANY
PUBLISHBSS. 64-66 7HTH AVENUE, HEW YOKE
. PLANE AND SOLID GEOMETRY
By ARTHUR SCHULTZE and F. L. SEVENOAK. i2mo. Half leather, xtt-t
370 pages, $1.10
PLANE GEOMETRY
Separate, izmo. Cloth, xii + 233 pages. 80 cents
This Geometry introduces the student systematically to the solution of geo-
metrical exercises. It provides a course which stimulates him to do original
wor.r and, at the same time, guides him in putting forth his efforts to the best
advantage.
The Schultze and Sevenoak Geometry is in use in a large number of the
leading schools of the country. Attention is invited to the following impor-
tant features : I. Preliminary Propositions are presented in a simple manner ;
2. The numerous and well-graded Exercises more than 1200 in number in
the complete book. These are introduced from the beginning ; 3. State-
ments from which General Principles may be obtained are inserted in the
Exercises, under the heading " Remarks"; 4. Proofs that are special cases
of general principles obtained from the Exercises are not given in detail.
Hints as to the manner of completing the work are inserted ; 5. The Order
of Propositions has a distinct pedagogical value. Propositions easily under-
stood are given first and more difficult ones follow ; 6. The Analysis of
Problems and of Theorems is more concrete and practical than in any other
text-book in Geometry ; 7. Many proofs are presented in a simpler and
more direct manner than in most text-books in Geometry 8. Difficult Prop-
ositions are made somewhat? easier by applying simple Notation ; 9. 7 he
Algebraic Solution of Geometrical Exercises is treated in the Appendix to the
Plane Geometry ; 10. Pains have been taken to give Excellent Figures
throughout the book.
KEY TO THE EXERCISES
in Schultze and Sevenoak's Plane and Solid Geometry. By ARTHUR
SCHULTZE, Ph.D. iamo. Cloth, aoo pages. $1.10
This key will be helpful to teachers who cannot give sufficient time to the
solution of the exercises in the text-book. Most solutions are merely out-
lines, and no attempt has been made to present these solutions in such form
that they can be used as models for class-room work.
THE MACMILLAN COMPANY
PUBLISHERS. 64-66 FIFTH AVENUE, NEW YORK
The Teaching of Mathematics in
Secondary Schools
ARTHUR SCHULTZE
Formerly Head of the Department of Mathematics in the High School of
Commerce, New York City, and Assistant Professor of
Mathematics in New York University
Cloth, 12mo, 370 pages, $1.25
The author's long and successful experience as a teacher
of mathematics in secondary schools and his careful study of
the subject from the pedagogical point of view, enable him to
speak with unusual authority. " The chief object of the
book/ he says in the preface, " is to contribute towards
making mathematical teaching less informational and more
disciplinary. Most teachers admit that mathematical instruc-
tion derives its importance from the mental training that it
affords, and not from the information that it imparts. But in
spite of these theoretical views, a great deal of mathematical
teaching is still informational. Students still learn demon-
strations instead of learning how to demonstrate."
The treatment is concrete and practical. Typical topics
treated are : the value and the aims of mathematical teach-
ing ; causes of the inefficiency of mathematical teaching;
methods of teaching mathematics ; the first propositions in
geometry ; the original exercise ; parallel lines ; methods of
attacking problems ; the circle ; impossible constructions ;
applied problems ; typical parts of algebra.
THE MACMILLAN COMPANY
64-66 Fifth Avenue, New York
CHICAGO BOSTON SAN FRANCISCO DALLAS ATLANTA
AMERICAN HISTORY
For Use fa Secondary Schools
By ROSCOE LEWIS ASHLEY
Illustrated, Cloth. i2mo. $1.40
This book is distinguished from a large number of American
history text-books in that its main theme is the development of
the nation. The author's aim is to keep constantly before the
pupil's mind the general movements in American history and
their relative value in the development of our nation. All
smaller movements and single events are clearly grouped under
these general movements.
An exhaustive system of marginal references, which have
been selected with great care and can be found in the average
high school library, supply the student with plenty of historical
narrative on which to base the general statements and other
classifications made in the text.
Topics, Studies and Questions at the end of each chapter take
the place of the individual teacher's lesson plans.
This book is up-to-date not only in its matter and method,
but in being fully illustrated with many excellent maps, diagrams,
photographs, etc.
" This volume is an excellent example of the newer type of
school histories, which put the main stress upon national develop-
ment rather than upon military campaigns. Maps, diagrams,
and a full index are provided. The book deserves the attention
of history teachers/' Journal of Pedagogy.
THE MACMILLAN COMPANY
64-66 Fifth Avenue, New York
BOSTON CHICAGO ATLANTA SAN FRANCISCO
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11