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K
BRA
EACHER'S HAND-BOOK
LIBRARY
OF THE
University of California.
Gl FT OF
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THE .TEACHER'S
Hand-Book of Algebra ;
CONTAINING
/
METHODS, SOLUTIONS AND EXERCISES
ILLUSTRATING
THE liATEST AND BEST TREATMENT OF THE ELEMENTS
OF ALGEBRA.
BY
J. A. McLELLAN, M.A., LL.l).,
M
HIGH SCHOOL INSPECTOE FOR ONTABIO.
The object of pirre Mafliemati'r<>.ivhicli is mwther vanipfor Alfiehra,istheunfoldivo
of tlie laws of the human intelligence.'' — SyiiViiHTEB.
FIFTH EDITION-REV ^SEO AND ENLARGED.
W. -T. GAGE & COMPANY,
TOKONTO AND WINNIPEG.
Entered according to Act of Pt^rliament of Canada in the year
1880 by W. J. Gage & Coitpany. in the office or tne iviinisier
of Agriculture.
PREFACE.
This boot — embodyincr the substance of Lectures at Teachers'
A.ssociations — has been prepared at the almost unanimous request
o^ the teachers of Ontario, who have long felt the need of a work
to supplement the elementary text-books in common use. The
following are some of its special features :
It gives a large number of solutions in illustration of the best
methods of algebraic resolution and reduction, some of which are
not found in any text-book.
It gives, classified under proper heads and preceded by type-
solutions, a exeat number of exercises, many of them illustrating
methods and principles which ai'e unaccountably ignored in
elementary Algebras.
It presents these solutions and exercises in such a way that
the student not only sees how Algebraic transformations are
effected, but also perceives how to form for himself as many
additional examples as he may desire.
It shows the student how simple principles with which he is
quite familiar, may be applied to the solution of questions which
he has thought beyond their reach.
It gives complete explanations and illustrations of important
topics which are strangely omitted or barely touched upon in the
ordinary books, such as the Principle of Symmetry, Theory of
Divisors, Factoring, Applications of Horner's Division, &c.
A few of the exercises are chiefly supplementary to those pro-
posed in the text-books, but the intelligent student will find that
even these examples have not been selected in the usual appar-
ently aimless fashion; he. will recognise that they are really
expressions of certain laws ; they are in fact proposed with a view
l(w220
IV PREFACE.
to lead liim fr, investigate these laws for himself as soon as he
has sufficiently advanced in his course. Nos. 8, 9, 10 and 11
afford instances of such exercises.
Others of the questions proposed are preparatory or interpreta-
tion exercises. These might well have been omitted, were it not
that they are generally omitted from the text-books and too often
neglected by teachers. Practice in the interpretation of a new
notation and in expression bv means of it, should always precede
Nits use as a symbolism itself subject to operations. Nos. 23 to
36 of Ex. iii., and nearly the whole of Ex. xv. may serve for
instances.
By far the greater number of the exercises, are intended for
practice in the methods exhibited in the solved examples. As
many, as possible of these have been selected for their intrinsic
value. They have been gathered from the works of the great
masters of analysis, and the student who proceeds to the higher
branches of mathematics will meet again with these examples
and exercises, and he will find his progress aided by his familiar-
ity with them, and will not have to inten-upt his advanced
studies to learn processes properly belonging to elementary
Algebra. In making this selection, it has been found that the
most widely useful transformations are, at the same time, those
that best exhibit the methods of reduction here explained, so that
they have thus a double advantage. A great part of the exercises
have, of necessity, been prepared specially for this work.
Articles and exercises have been prepared on the theory of
substitutions, on Elimination, &c., but it has finally been decided
to hold these over for Pt. ii,, which will probably appear if the
prSsent work be favorably received.
CONTENTS.
Chapter I. — Sobstitution, Horner's Division. &c.
PAGE.
Sect. 1. — Nnmeiical and Literal Substitution 1
Sect 2.— Fundamental Formulas and their Applications 10
Sect. 3. — Horner's Methods of Multiplication and Division, and their
Applications 21
Chapter II. — Principle of Symmetry, &c.
Sect. 1.— The Principle of Symmetry and its Applications 33
Sect. 2. — The Theory of Divisors and its AppUeations..- 39
Chapter in. — Factoring.
Sect. 1. — Direct Application of the Fundamental Formulas 62
Sect. 2. — Extended Application of the Formulas 71
Sect. 3.— Factoring by Parts 79
Sect. 4. — Application of the Theory of Divisors 83
Sbct. 0. — Factoring a Polynome by Trial Divisors 90
Chapter IV. — Measures and Multiples, &c.
Sect. 1. — Division, Measures and Multiples 101
Sect. 2. — Fractions IO9
Sect. d. — Ratios 122
Sect. 4. — Complete Squares, &o 130
Chapter V.
Simple Equations op One Unknown Quantity 138
Preliminary Equations. Resolution by Factors. Fractional Equa-
tions. Application of Ratios. Equations involving Suids,
Higher Equations, Sec.
Chapter VI.
Simultaneous Equations I70
Equations of Two Unknown Quantities. Systems of Equations.
Application of Symmetry. Equations of Three Unknowns.
Systems of Equations.
Chapter VQ.
Examination Papers 207
:ir.R^R>t^
CHAPTER I.
Bection I. — Substitution.
Exercise i.
1. If a = 1, 6 = 2, c = 3, rf = 4, a; = 9, ?/ = 8, find the
value of the following expressions : —
l_a_(l-l-a;)}.
a-{x-y)-{b-c){d-a)-{y-b){x + c).
x - !j^y - {y — a)\d + c(h-- c)v']■
{x+d){y+b+c)+{x-d){a—b-(^)+(y+d](a-x -dy
{d-x)^ + {c + v^*
la-b){c^ -b-'x) -{c-d){b^ -a'-'x) + {d-b-c){d^ -f'*'^
d — ad + c (\d + b
d + a d—G d — b
2. If a = 3, 6 = - 4, c = - 9, and 2s' = a + b + c, find the
value of the following expressions : —
s{s — a){s — b){s-c).
««H-(5-a)2+(s-5)^+(s-c)2.
«« - (s-a){s — h) - {s — b){s -c) — (s — c)(s— a).
2{s — a)[s — b){s — c)-j-a{s-b){s-c) -\-b{s-c){s~a)+c{s- a){s-b).
8. If ffi = 2, 6 = - 3, c = 1, a; = 4^, find the value of the
following expressions : —
a2_62 „2+/,2 (a-^,)2 (a-h)^ *
c^^Tb^' ~a^b^' (a + by' (a + by'
a^^ab + b^ a^-b^ a; (2x-3 3«-l ).x-l
a'^-ab^b^ a^-b^ 2
{a + b)\{a + by-c^\^
Ab^c^-{a^-b^-c^)^'
a'{b - g) Vb''{c-a)^",^{a-b)^
- Ja-b)(b-G){c-a)
2 SUBSTITUTION.
\
4. If a = G. ^ = 5, e == - 4, rf = — 3, find tlie value of tin
following expressions : —
y{b-' +ac)+ y(c2— 2ac), yib^+ac+ ^(c^— 2«c)}.
5. If a; = 3, i/ = 4, 2; = 0, find the value of: —
{3a;-v/(a;3+?/2)}2{2a;+v/fa:2+2/2+2)}.
G. Calculate the values of {x+y+z)^-^{x^+y^ + z^, ^^^^
xyz
(a) 9^=1, y = 2, z = H.
(b) x = %y = 3,z = 4:.
(c) a; = 3, 2/ = 4, 2 = 5.
(J) a: =10, .v = ll, 2=12.
7. Givcu x=S, y = 4:, z= —5, calculate the values of
{x+y+x}^ -S{x+y+z) (xy+yz+zx).
x^{y+z)-{-y^{z+x)+z^{x+y) + 2xyz.
x'^[y-z)-{:ij^z-x}+z^x-y).
(5«-4z)2+9(4a;-2)2-(18a;-5«)2.
{Sx + Ay + 5z)^ + {Ax + 8y + 12z)Z-(5x+5y + lSz)*,
8. If s = a+ 5 +c, find the value of
(2s_a)2 + (2s-ft)2-(2s+c)2, given
(1) a = S, 6 = 4, c = o, (2) a = 21, 6 = 20, c = 29,
(3) a=119, 6 = 120, c = 169, (4) a = S, b= -4, c = 5,
(5) ^=5, 6 = 12, c=-13.
9. If a = l, 6 = 3, c = 5, f? = 7, f = 9,,/"=ll, prove that
<i + 6 + c + (i + e+/=
2
1+1+1+1+1=1(1-1
ab be cd cle ef 2 \ a /
A+ V + l_+±=l/l-l
abc bed cd^ def 4 i fl6 et
J_+J^+_l.=l(l- M.
o6';i 6cde cdef 6 \a6p t/(//
SUBSTITUTION.
a2^h2.\.c2_ab — bc — ca = h^+c^+fl^ — bc — ed — db =
t.2 4. ,f2 _|_ ^2 _ cd - de -ec = d^+e^ -\-P -de- ef-fd.
10. lia=l,h = % 6 = 3, d = A, e=o,f=&,(j = l, prove that
a+b + c = U-d, a-{-b + c-\-d = ^de,
a^b+c+cl+e= W, a + b+c+d-\-e+f= ^fg,
aa^fc3+c.3= ^4l+^), a2_{.b3 4.^2+^3=:
de{d±^)^ «3+^,3 +,3+^3 +,3 = ^0
«3 +i2 +,2 +,/3 ^.,2 + /^ = M+'/J.
a3_|_i3 4.^3 4-^3 = (« + J4.(.+ rf)8,
a^-\.b^4.rS + d^+e^-{-p = {a+b+c + d-\-t+f)»
A , lA . A. cd(c+d)(c^d—l)
6c(o+c)
a4 4-64 4.^4 4_J4 =
6ie((i+e)(crfe— 1)
^44.,44.,4+J4 + ,4 = €Ai-^i^),
,4 + ,4+,44.d4+,44./4= ^(Z+^) (^^^tI).
c2+t^2=e3^ C3+J3 4.,3=y3
11. Assume any numerical values for x, y, and z, and calculate
the values of the following expressions : —
(a;5 - 10:c3 4. 5a;j 2 + (5a;4 - 10a;2 + 1)3 - (a;3 + 1) 5 .
(a,4.i)3_2(x+5)3-(a;+0)34-2(a;+ll)3 + (x+12)3-(a;+16)3.
(^2 _^3;3 4.(2x^)2 _ ^^2 4.^2)2
(a;3_3x^3)34.(^);c32/-^3)3_(a;3 4.^2)3.
(3x3 +4a;i/+2/2)3 + (4a;3 +2a;(/)2 - (5a;S+4a;.'/+i/2)2
(a;_2/)34-(v-5)34-(5-a;)3-3ia;-2/)(2/-5;(«-aj).
Art. I. If X = any number, aS, for example, 3, then x-
(which = x.a;) - 3*, x^ (which = x.x'^) = ?..(;2, a;* (which = x.x^) =
Bx-3, &c. Oi- 3 = x, 3j; = a;2, 3a;3 =0;^, '6x'^ = x^, &c. Hence prob-
4 SUBSTITUTION.
lems like the follomng may be solved like ordinary arithmetical
problems in " Keduction Descending."
Examples.
1. Find the value oi x^ — '2.x — 9 when x^H.
x^-2x-9
5
5x
-2x
3»
5
15 Explanation,
-9 x^^6x,
.-. a?2-2a; = 8a;=15, and
6 .-. a;2-2a;-9=:15-9 = 6.
2. Find the value of a;* — *^ — ^x"^ —dx—b when x~'^
x'^—x^ — 4.x'^—Sx — 5
3
p^ 3x3
Tj 2a;3
3
Pa ^*^
-4a;3
r„ 2ar2
3 .-. x4-a;3-4j;«-3a:-6 = 4
— if ic = 3.
7>s ^-^
-3a;
Tj 3a!
3
P4 9
-5
% *•
SUBSTITUTION,
Explanation.
:. x^—x'^ = 2x^ = 6x^ ,
,•. x^-x^ — ix'^ —2x^ — 6x,
:. x'^-x'^-ix^-Sx = 3x^-9,
:. cc^ — x^ — 4a;2 — 3a; — 5 = 4.
8. Find the value oi 2x^ + V2x^ 4-6a;3 -12x+ 10,
Using coefficients only, we have
2+12+6-12+10
-5
Pi ••
. -10
+ 12
r,
+ 2
- 5
Va ...
-10
^2 ■••
+ 6
Vm. ...
- 4
'9 ••'
- 5
1t>, ...
20
r3
-12
fg ...
8
-6
Pa ...
-40
r4
+10
r. ...
-30
the quantity = —30 if ar= -5.
Art. II. If the coefficients, and aiso the values of x are small
numbers, much of the above may be done mentally, and the work
will then be very compact. Thus, performing mentally the mul-
tiplications and additions (or subtractions) of the coefficients,
and merely recording the partial reductions r^, r^, r^, and the,
result r^, the last example would appear as follows : —
6
SUBSTITUTION.
-5)2 4-12 +6 -12 +10
2
-4
8
Art. III. In the above examples, the coefficients are "brought
iown" and written below the wodncts p^, p^, p^, p^, and are
added or subtracted, as the case may require, to get the partial
reductions r^, r^, r^, and the result r^. Instead of thus " bring-
ing down " the coefficients, we may " carry up " the products j9^,
P2' Ps' Pv writing tliem beneath their corresponding coefficients,
and thus get r^, r^, r^, r^ in a third (horizontal) line. Arranged
in this way Ex. 2 will appear
1 -1 _4 _3 _5
+ 3 +6 +6 +9
1 +2 +2 +3; 4;
and Ex. 3 will appear
2 +12 +6 -12 +10
-10 -10 +20 -40
-5
2 +2 -4 +8; -30
Comparing these arrangements with those first given (Ex. 2
and 3), it will be seen that they are figure for figure the same,
except that the multiplier is not repeated.
Art. IV. When there are several figures in the value of a;,
they may be arranged in a column, and each figure used sepa-
rately, as in common multiplication. Where only approximate
values are required, *• contracted multipHcation " may be used.
4. Find the value of 3a;5-lG0a;4 + 344a;3_^700a;3-1910a;+
1200, given a;= 51.
3 -160 +844 +700 -1910 +1200
1 3-7-18 37 -23
50 150 -350 -650 1850 -1150
•7 -13 +37
result is 27.
-23; +27
SUBSTITUTION.
5. G'ven »;= 1-1S3, find the value of CAx*-lUx+4:5 correct to
three decimal places.
64 0 0 -144 +45
64 75-712 89-5673 -38-0419
6-4 7-5712 8-9567 -3-8042
5-12 6-0570 7-1654 -3.0434
•192 -2271 -2687 --1141
1
1
8
8
64, 75-712, 89-5673, -88-0419,
.'. result is —-004.
•0036.
1.
2.
3.
4.
5.
Exercise ii.
Find the value of
xi-Ux^-Ux^-lSx+n, fora;=12.
xi + r)0x^-lGx^-16x-61, for ic= -17.'
2x4 + 249a;3-125a;2-|.l00, fora:= -125.
2.^3 _478a;3_ 234a;- 711, for a; = 200.
x'^—iix'^-8, for a; = 4.
6. a;6-515a;S-3127a;4+525a;3-2090a;2+315Ga;- 15792, for a:
= 521.
7. 2a;5+401a;4-199a;3 + 390a;3_602a:+211, for «= -201.
8. 1000x4 - 81a;, for a; = •!.
9. 99a;4 + 117a;3-257a;2-325a;-50,- fora;=lf.
10. 5a;^+497a;4 + 200a;3 + 19Sa;3- 218a;- 2000, fora;=-99.
11. 5a;5-620a;4-1030a;3 + 1045a;2-4120a:+9000, fora; = 205.
Calculate, correct to three places of decimals, —
12. a;3 4-3a;2-18a;-38 for a; = 3-58443, for a; =- 3-77931, and
for a; = -2-80512.
13. 7/4- 142/2 +1/+ 38 for t/ = 3-13131, for y= -1-84813, and
for ?/= -3-28319.
Exercise iii.
What do the following expressions become (1) when x = a, (2)
when x= -a?
1. a;4 -4ra3 + 6fl2a;3_4fl3a;-l-rt,4,
2. y\x^-ax+a^). 3. y(x^ + 2ax+a^).
4. (a;2-|-aa; + rt3)3-(a;2-ax- + r/2)3.
If a; = 1/ = 3 = «,^nd the value of the following expressions r
8 SUBSTITUTION.
5. {x-y) (y-z) (z-x).
6. (x-i-y)^ {y + z-a) (x+z-a).
7. x{y + z) (y2 +z2 -x^) +^ ^^, + x] {z^ +x^ -y^-)+z{x ^y) (x^ -^
8. -^ + JL + ^_.
y+z x+z x+y
Find the value of
n ■^ , X 1 abc
9. — + — .wtienjc=
a b a-{-b
10. + — + — -, when x= — (a-b+c),
a[b — x) b[c — x) a{x — c) a
11. ^+ -^-, wheua.= ^(Ll«)_.
a b — a b[b+a)
12. (a + x) {b+x)-n{b + c)+x^, when a; = —.
b
13. bx-\-cy-\-az, when x = b-\-c — a, y = c + a — b, z = 'i-^-b -c.
14. <l±^l±b^ - __1^ _, when a:= -a.
a(l+6) —bx a-2bx
15. — — - —! —, when a; =*(/;- ^0-
\x+bj x — a — 2,b
16. (p — q) {x+2r) + {r — x) (p+q), when a; = ^ ^ ^^ ~ ^■'■.
17. ffl3(6-c)-f62(c-a) + c2(a-6), when a- 6 = 0.
18. (a-\-b + c) (6c+m+rt6) - (rt+6) (6-f-c) (e+a), when rt= — 6.
19. (a+6 + c)3-(a3+63_^c3), whena+6 = 0.
20. {x+y +z)'^ - (x+y)'^ - {y+z^ - {z+x}'^ -\-x'^ +y* +z*, when
x-i-y-r-z = 0.
21. a3(c-62) + 53(a_c2)_}.c3(j_a2)+a6c(a&c-l), when6-a»
= 0.
22. aW«_!+!iV + 6^/^^l±lT. when a^ +6^=0.
23. Express in words the fact that
(a-&)2=a2_2rt6+62.
24. Express algebraically the fact " that the snra of two quan-
tities multiplied by their difference is equal to tjie difference of
the squares of the numbers."
SUBSTITUTIOSI. 9
• 25. The firea of the walls of a room is equal to the height mul-
tiplied by twice the sum of the length and breadth : what are the
areas of the walls in the following cases :
(1) leuglh I, height h, breadth h.
(2) height x, length b feet more than the height, and breadth
6 feet less than the height.
26. Express in words the statement that
{x-^-a) {x+h)=x^-^{a+b)z+ab.
27. Express in symbols the statement that " the square of the
sum of two quantities exceeds the sum of their squares by twice
their product."
28. Express in words the algebraic statement,
(x+y)^ =x^ +y^ + Sxij{x-\-y).
29. Express algebraically the fact that "the cube of the differ-
ence of two quantities is equal to the difference of the cubes of
the quantities diminished by three times the product of the
quantities multiplied by their difference,"
30. If the sum of the cubes of two quantities be divided by
the sum of the quantities, the quotient is equal to tl: ^ -square of
their difference increased by their product ; express this algebrai-
cally.
81. Express in words the following algebraic statement:
""Lzyl^ix+yy-xy.
x-y
32. The square on the diagonal of a cube is equal to three
times the square on the edge ; express this in symbols, using
I for length of the edge, and d for length of the diagonal.
83. Express in symbols that " the length of the edge of the
greatest cube that can be cut from a sphere is equal to the square
root of one-third the square of the diameter."
34. Express in symbols that any "rectangle is half the rectan-
gle eoutained by the diagonals of the squares upon two adjacent
sides." [The square on the diagonal of a square is double the
square on a side.]
85. The area of a ckcle is equal to x multiplied into the square
10 SUBSTITUTION.
of the radius ; express this in symbols. Also express in symbols
the area of the ring between two concentric circles.
36. The volume of a cylinder is equal to product of its height
into the area of the base, that of a cone is one-third of this, and
that of a sphere is two-thiids of the volume of the cii-cumscribing
cylinder ; express these facts in symbols, using h for the height
of the cylinder, and r for the radius of its base.
Exercise iv.
Perform the additions in the following cases :
1. {b-a)x+{c-b)y, and {a+b)z+{h+c)y.
2. ax-lnj, {a — b)x-{a+b)y, and {a + b)x-{b—a)y,
3. (y~z)a^+iz-x)ab + {x-y)b^, and {x-y)a^-{z -y)ab-{x
4. ax+by + rz, bx+cy+nz, an."c.>t4-.''.'y + &z.
5. (a+b)x^+{b+c)y^+{a+c)z^, {b + c^x^ +{a+c)y^ + {a + b)z9,
{a+c)x^ + {a + b)y^ + {b+c)^^, and-(a+/;+c) {x^+y^+z^).
6. x(a-b)2 +y{b-c)2-\.z{c-a)*, y(^a-b)^ +z{b-c)^+z{c-
^)^, and z{a - b)^+x{b - c)3 +y(c-a)^,
7. {a-b)x^+{b-c)y^+{c-a)z^,{b-c)x^ + {c-a)y^+{a-b)z^,
and [c-a)x^ + in-b)y^+(b-c)z^.
8. {a-^b)x + {b+c)y-{c + a)z, {b + r,)z + {G + a)x-{n + b)y, and
(a+c)y+(a + b)z-{b + c)x.
9. rt3-3rt6-^^/;2, 263-363+C3. ab-^,b^+b^, and 2a&-^?;».
10. ax^-nbx'', -Qaaf+lbaf, and - 8bx" + I0ax'' .
11. What will {ax-by + cz)-\-{bx + cy-((z) -{cx + mj+bz) be-
come when X - y - z = l ?
Section II. — Funovmental Formulas and theib Appcication.
4. By Multiph cation we get
{x + r) (x + s)=x^ + {r + s) x + rs A.
(x-hr){x-ts){x + t) = x'' + {r + s + t)x'' + {rs-tst + tr)x + rst B,
From A we immediately get
(a; -}-j/) 2=^2 +2x^+2/2 [1]
FUNDAMENTAL FORMULAS. 11
{x+7j + z)^=x^+2xy + 2xz + y^ + 2yz t z^ [2]
(2«)^ = 2rt2 + 2 nab [3]
{x+y){x — y)^x'-^—y^ [4]
From B we derive
{x±y)^ ^x^±8x^y + Sxy^±ir [6]
= x^±/j^±3xij {x±y) [6]
{x + y4 »)^=a;2 +y'-^+z^ +'Sx-{y+z) + 8y^{z i- x) i-Sz^{x + y)
+ 6xyz • [7J
= x^+y^ + z^ + 3 {x + y) (y^-z) (z + x) [8]
= x^ +y^+z^ +8 {x+y + z) {xy -\- yz + zx) — Sxyz... [9]
(2a)3^2a3 + 8^a^b -\- Q^ahc [10]
[The symbol £ means the sum of all such terms as]
Formula [1] . — Examples.
1. We have at once {x -\- y)'^ + (^ — y)'^ = 2(^2 _j_ ^2^^ a^^j
{x + yY —{x — yY=4.xy.
2. (a + 6 + c + d) '^ +{a — h — c 4- d) '^ may be written
{{n + d) + (6 + c)}-^ -h {{a + d) — (6 + c)}2, which (Ex. 1; ==
. 2{('< + J)2i-(i + f)^} ; similarly
l^a — h -f c— (/)2+ [a + b—c — d)'^ = {{a — d) — {b-c)}-^-^
.-. (a+ 6 + C+ tf)2 + [a — h — G -ir-dY -f {o. — h + c — d^Jr
(a + 6-c-(/)3 = 2{(a + J)-^ + (6+c)s+(a-c/)2 + (/, _c)a} ^
(again by Ex. 1) 4(«3+63+,.2_i.,i2).
3. Simplify (« + ?j-fc)s - 2{a+b+c)c + c^ ;
This is the square of a binomial of which the first term is
^aJrb-^c) and the second -c; the given quantity .•. =
{(«+6+c)-c[- = (a
12 FUNDAMENTAL FORMULAS.
4. Simplify {a+b)* -^a'-i + b^) [a-i-b)^ + 2{a* + h^).
By Ex. 1. 2[aA + b't)=:{ct2^b^)^ + {a» -b^)^ ; :. given quan-
tity = (« + 6)4- 2[U^ + //2) (« + 6)2 4- („2 +02)2 + („2 _ ^>2^a =
{{a + 6)2 _^a3+62j}2 + ^,i2_63)2=rt4+2«2y3^ /;i = (a2 +62j2.
Exercise v.
1. (a;+3//2)2 + (a; -32/3)3, |i,,,3 + 3J2)2 _ ^,,2 _ 37,2)2.
2. Shew that {mx-{-n;j)2 + {nx-my)^ = {m^-^n^) [x^ + i/^).
B. " " {mx — ny)'^—{Hx — iny)^ = \^m''^—n^){x^-i/^).
4. Simplify ;./ + 3ij2_^2(«H-36) ((i-^jH-(a-&)3) {a-b\^.
6. " (a;+ 3)3^ (x -1-4)2 -(a;+ 6)3, and (4^3-2^2^2 -
(i(/2+2a;3)2.
6. Simplify (a + 6+c)2 + (6 + (;)»-2(6+c) {a + b+c)
7. Shew that ['ix + by)^ + {cx-\-dyY-^{ay - bxY -\- [cy - dx)^ =
(rt2 + i2 + c3+(Z3j (^3.34.^3^.
8." Simplify (a;-3y2)3^.(3^2 _^)3 _ 2(3a;3-^) (a:-3v/2).
9. " (a;3-i-a;^-^2)3_(^3_a.^„^3^3^and(l + 2a;+4a;3)2
-|-(l-2a;+4x2)«.
10. If « + />= - Jc, shew that (2(t-6)2 + (26 -c)3 + (2c-a)2 +
2(2«-6) (26-c) + 2(2Z>-c) (2c-«)-|-2(2c-a) (2«-?>) = /^jc2.
11. Simplify- 2 (ff- 6) 2 -(a -26) 2; (a^+^ah-^b^)^ -{a^-\-b^y.
12. " ((/ + M2-(6-H6-)3 + (c + t/)2-(^+«)2.
13. " (^a;-2/)3+ai/-«)2+a^-a;)2+2(ia:-2/) (Az-cr^
+ 2(A2/-^)(i-^-^) + 2Q^-2/)(i2/-=)-
14. Prove that [x- yf + {y -zf +{z-xY = '2{x-y) {z- y) +
^{y-x) {z-x) + ''l{z-y) {z-x).
15. Simplify (l + o;)* -2(1 +a;2) (1+^)3 + 2(1 +x4).
16. " (a;-l-.(/+,^)2-(.c+?/-z)3-(^+2-a;)2-(z+a;-y)2.
17. " (a;-2y+3z)3 + (3z-27/)3+2(a;-2?/ + 3z)i2?/-32).
18. " (r/2 4.62-,,. 3)3 _f.(c2_ 62)2 +2(/;2_^.2)(rt2_(.62_c2),
19. " {x+yy + {x-y)^-\x-y}-[x + y)».
FUNDAMENTAL FORMTJIiAS. 18
20. " {5a+3b)^ + 16{da+by^-{lSa + 5b)9.
21. Shew that (3a-ft)2+(3fe-6)34-(3c-a)3-2(i-3fl)(36-c)
+ 2{'db-c){Sc-a)-2{a-Si.){3u-b)-i{a + b + c)2=0.
22. If z2= 2x7/, prove that (2x3 -2/2)2 + (22 -2^ 3)2 +(a;3-2z2)3
-2(2x3-2/2)(22-2^2)^_2(x2-2z2)(23_2j/3)_
2(x3 - 222) (2x2 -1/2) = (x+2/)*.
23. Simplify (l+x+x3+x3)2 + (l-x-x2+x3)2 +
(1-X + X3-X3)2 + (1+X-X2-X3)2.
24. Simplify {ax-{-by)^-2{a^x^ + b^y'^) (ax+by)^ +
2{a^x^ + bhj^).
Formulas [2} and [3] . — Examples.
1. (l-2x + 3x3;2 = l_4x+6x2
+ 4x3-12x3
+9x*
= l-4x+l(ix-- 12x3 +9x4.
2. (ah + bc + ra)^=a^h^-\-2ab^e + 2a^bc-{-b^c^-{-^abc^ + c^a^ =
a2b^+b^c^+c^a^+2abc{a + b + c).
3. \{x + y)^+x^ + y^}^^{x+ij)^ + 2(xi-y)^{x2+l/^)-\-x^ + 2x^
y2^^4 = (a. + j^)4 + (^|.y)3|(a;-H^)2 ^ (x-y)^} t X-^ + 2x^7/3 + z/4
= 2,X + ^)4+(x2 - 7/'3, 3 +^4+2x2^/3 +yi = ^{ix+y)^ + X* + 7/4}.
4. (x3+X?/ + /y3j3=x*+i^.X-'^ + 2x2^/3 +x3^2 -f 2x7/3 + ^/^ =
(x+7/)2x2+a;'//3+7/2(x + 7/)3.
5. In Ex. 3, substitute 5 -c for x, f -a forj/, and consequently
b — a for x+7/, then since (b — a)^ = {a — b)^, Ex. 3 gives
|(rt_/,)2 + (6_c)2+(c-a)2}2 = 2{(a-Z>}4 + (5-c)4 + (c-fl)4}.
6. Making the same substitutions in Ex. 4, we have
(a2+63^c2 -ah-bc -caY = {a —b)^{b - c)2 + (6-c)2(c- a)2 +
(c — a)2(a — fi)2, or, multiplying both sides by 4,
{(a_6)3^(5_c)3 + (c-a)3}2=4(rt_6)2(J_c)2 + 4(fe_c)2 X
(c-a)2+4(c-rt)2(a-6)2. and .-. from Ex. 5, (a-i)4 + (^6-o)4 +
(c-a)4 = 2(«-6)2(i>-c)2+2(fe-c)8(c-a)2 + 2(c-a)3>(a-6)3.
14 FUNDAMENTAL PORMULAS.
Exercise vi.
1. (l-2.'c+3a;3 -4x3)3, {l-x-tx^-x^)».
2. {l~2x+2x^-3x^-x^)^, (l + 3a;+3a;3+a;3)2.
3. (2a-6-c2_l)2, (l_;^ + y + 2)3, (la;- 1?/ + 03) 2.
4. (:c3-x2^+a;^3_^3)3, („a;+te2+cx3 + (/a;4)3.
o. Shew that (a^ ^b'^ +c^) (.c^ i- y2 ^z2 ^ _ (^^x + bij + •z)^ =
{a/j - 6a;) 3 + (ex - flz) 3 + (65 - cy ) 3 .
6. Prove that {a + b)x + (6 + c)y + (c + ^/)3 multiplied by {a — b)x
-^[b — c)y + (c — a)z, is equal to the difference of the squares of
two trinomials. .
7. Shew that (a-b) (a-c) + {b-c) (b-a) + {c-a) (c-b) -
i-{(a-6)2 + (6-c)34-(c-a)3}=0.
8. Simplify {a-(6-c)}3 +{6-(c- a)}» + {c-(a-6)}».
9. Shew that {a^+b^ -x^- )^ +{ai + bl-x^)^ + 2{aa^ +bb,y
^(,,2 ^ ,t2_a;3)2+(t2_l_i,2_^2)2+2(a6 + ai6i)3.
10. ^rovethsii{{a-b){b-c) + {b-c){c-a)-]-{c-a){a-b)}^ =
(a-bY(b-GY + {b-cY (c-a)3-h(c-rt)3 (a-6)2.
1 1 . Square 2(« — \bx —^cx-\-2dx.
12. If a; + ?/ + 2 = 0, shew that x^ + //* + 2* = (^2 -7/2)3 +
(^3_22)3+(22_^2)2.
13. Provethata3(6 + c)2-}.//2(c + rt)3-fc3(rt + 6)3 + 2a6c(a + 6 + c)
= 2(rt6 + 6c+crt)2.
Art. V. To apply formula [4] to obtain the product of two
factors which differ only in the signs of some of their terms : —
group together all the terms whose signs are the same in one fac-
tor as they are in the other, and then form into a second group
all the other terms.
Examples.
1. Multiply a + 6 — c-f t^ by a-6 — c — d; here the first group is
a — c, the second b-\-Ll\ :. we have
{(a -c) + (6 + <0} {{a-c)-{b+d)}={a-cy-{b-\-d)^.
PUNDAMENTAIi FORMULAS. 15
2. (1 J- Sx-^Sx^ + x^) (1 - nx + Bx^ - x^.) = {(1.+ ?>x"} +
{Sx-^x-)} {(1+3x3) - ('3a;+«3)}={l+3a;3)2 - (3a;+a-)3 = 1-
3x^ +8x4 -a;«.
3. Find the continued product of a +/'+c. h+c — a, c+n —b and
n + h — c.
The first pair of factors gives {{b + c)-\-a} {{h-\-c) — <-i\ ={6+c)^
-o8 = 62+26c+c2-«2.
The second pair gives {a — {h — c)] {a + {h — <:)] =a^ — h^-\-1hc
— c2 ; the only term whose sign is the same in both these results
is 26c ; hence, grouping the other terms, we have
{26c + (/y2+22_a2)} |2k-(63+c3 - a^)] =
(263)2 -(63+c2-a2)3 = 2a262 + 2^2,•3+2c3»2 _„4_ft4_c4.
4. Prove (^i.^-^ah^h)^ -aU^ = {a'^+ahY + {ah + b^)^ .
The expression ^{>t^-\-h^) {n^ + 2ah+h^) = {a'^ +b^) (^+6)2 =
a2(a+6)2+63(« + 6)2 = (a2_f.a5)2_}-(a6+63)2.
Exercise vii.
1. (a2+2a6 + 62) („2_2a6+62).
2. {:L;c^-xy+y^){hx^-\-y^+xy).
8. (r<2_n6+262) (o3^.«?,+262); (.'c4+4.T2/) (a:4_4a;y).
4. {(x + y)x-y{x-y)] {{x-y) x-y{y-x)).
5. Simplify: (.c+3) (x-8) + (a; + 4) (a;- 4)-(a; + 5) (a- 5).
6. " (l+a-)'i+(i-a;)4-2(l-a;2)2.
7. (a;2+j/^)2-(2^?/)3-(a;3_y2)3.
8. (2a2-362+4c2) (2«3+3J2_4c2).
9. (2a+6-3c) (6 + 3c-2a); (2a— 6-3c)(/^-3o— 2a).
10. (x4+2/4) (a;2+i/2) (x+?/) (x-7/).
11. (a;2+a;2/+?/2) (aj3 _a;?/+?/3) (x* -a:3?/2+y4).
12. (a+A-aZ;-l) (a+6+a6+l).
13. Prove (a2 +62 +6-2)(6'-i+(-'- _ as^^^s+as _fe2) ^^2 _j./,2_c2j
= 46*0* when a4 = 64 _j_c4.
14. (a!2+t/2-|a;i/) (a;2 + //2+|xy).
15. (a-*— 2x3^3x2— 2a; +1) (a;* +2x3+3x2 +2a;+l).
16 FUNDAMENTAL FORMULAS.
16. Multiply (2:k— 2/)a2 - {x-\-2j)ax +x^ by (2x-y)a'' +
{x+y)ax-x^.
Prove the following :
17. {a^A-h^+c^+ab + bc+m)^-(ah + bc + ca)- = {a + h +c)3
x(a3+62-|-r.2).
18. (a3-|-/>2 +c3 +a6-j.5c+ca)3 - {a^ +ah + ca-hcf =
{{a+h)[b^c)]^ + {(b+c) (c + «)}2.
19. 4.{ab+cdY-{a^+h^-c^-d^)^ =
<a->rb-\-c-d) {a + b-c + d) {c + d->ra~h) {c-\.d — n-{-h).
20. Find tlje product of a;2+//2 +2'- 2x// + 2xz-2?/0 and a;= +
.,3+22 ._ 2xy - 2x3+2^/2.
21. (a;2 +;/2 +.r//|/2) [x^ -xy^2+y^) (a;* -//").
22. (l-6a+9ri2)(i + 2a+3a2).
23. {(m+») + (/?+r^)} (m-9+p-nj.
24. Obtain the product of 1+x+x^, x^ +x — l, x" — x-f-l, and
H-a;-a:3.
25. («-ft2)2 (a+i8)3 (rtS4-64)3 (a4^.ft8)3.
26 Shew that {xP- + a;y + ^/^^s (^,2 _ xij -f ;y2)2 _ {x^y^y =
Formula A. — Examples.
1. Multiply x' —x+5 by x^ —x — 1 : here tKe common term is
a;2 _3.^ ^}ie other terms +5, and — 7, hence the product = (x^ —xY
4_(_7+5) (a;?-a;)+(-7x5) = (ic^-a;)2-2(a;2 - x)- %o=x^-
2x^-x^+2x-S5.
2. {x—a) (x—3a) (a;+4a) {x+6a): taking the first and third
factors together, and the .second and fourth, we have the product
= {x''+Sax - 4a2)(a;2 + 3aa;-18a2)=n(a;2+3aa;;2 _ (ia^ + lSa^)
x(a;2+3aa;)-72a* = &c.
Exercise viii.
1. (a;2+2a;+3) (a;2 + 2a;-4); (x-y + Zz) (x-y-^^z).
2. (x+1) (a;+5) (x-l-2)(a;+4); {x^ Jra-b) {x^+^b-aX
3. (a2-3)(a2_l)(a2 + 5) (a2 + 7); {x^ + x' +l){x*+x'' -2).
4. {(a;+2/)^-4a;i/)} {(x+y)^ +5a;//}.
FUNDAMENTAL FORMULAS, 17
6. (?«a;+?/+3) (nic+y+7).
7. (x+a-y) (x+a+dy).
8. (x-" +a;" -a) (x^" +a;'' -*).
9. (ia;4-2/-' + 2)(|aj4-2/2-4).
10. (-+--;5 -+- + 2i) .
\x ' y 2j \x ' y ' 'I
11. Multiply together a- 2 -f 1/2, a;- 2 + 1/3, a;-2--i/2, and
K-2- v/8.
12. (x+a + b) (x+h-c) {x~a + b) [x + b + c).
13. (a + b+c) {a + b-{-l)-\-{a+c-\-d) {b-\-c+d) - {a+b+c+cl)K
14. Prove that
(2aH-26-c)(2//+2c-a)-H(2c + 2a-6)(2a4-2i-o)+(2// + 2c-a)
(2c + 2a -b) = 9{ab+bc+cu,).
Formulas [5] and [6] . — Examples.
1. We get at once
(a;+^)3-(a;-2/)3 = 22/(3a;^+2/2).
2. Simplify (rt+6+tf)3-3(« + i+6')2c4.3(fl+fc + c)c2-r3.
This plainly comes under formula [5] , the first term being a+5
+c, the second —c; hence the expression is {(a + i+c) — c}' =
{a+b)K
3. Shew thsit {x-" +xii+y-)^ + {xij -x- -y^)^ -
6xy{x* +a;2y3 +?/*) = 8x^y^.
This comes under formula [G] , the first term being
{x^+X7/+y'^), and the second- (a;- —xy+y^) ; we have therefore
{{x^+xy+y-)- {x^-xy+y^)}^ = {2xyy = 8x-y».
Exercise ix.
Simplify
1. (l-a;')3 4.(i+a;2)3^ (x^ +xy^)^ -(x^ -xy^)^.
2. (a + 2fc)»-(rt-6)3, (3a_6)3_(3«,_2i)3.
18 FUNDAMENTAL FORMULAS.
4. (a-fc)3+(a + &)3+6«(a2_62).
5. (x-i/)^+{x+!/)^ + d{x-yy- {x+y)^3{7j-x){x+y)\
6. (l+a;+x^)s-(l-a;+a;2)3_6a;(l+a;-+a;4).
8. (3a;-4?/-r02)3-(52 - 4?/)3 + 3(52 - 4^)2(3a:-4^ + 5z}~
■S(3x-4.y + 5zy{5z-iy).
9. (l+^+x-)3 + 3(l-x3)(2+a;-) + (l-:c)3.
10. Shewtbat «(a-26)3-^6-2«)3 = (rt-^/y)(a+c/)3.
11. Shewthata3(rt3_263)3+i,3(2a3-i3)3 = (rt3_i3)(«3^i3)3
12. (a;3+u:?/+i/'-*)3 + 6(a;-+i/-) {x^+xy+y^)-\-{x^ -xy+y^)^.
13. Shew that aS^^s + 2^3)3 + b3{2a- + 63)3 4- (8a-62)3 =r
14. Simplify {aa!;-\-l>y)^+0'^y^ +b^x^ ~Sabxy{ax-i-by).
15. What will a^+b^+c^ —3abc become when a+b+c = 0 ?
16. Find the value of x^ -y'^+z'^ + 'dx-y^z'^ when x- -y- -fz^
= 0.
Formulas [7] , [8] and [9] . — Examples,
1. Simplify (2x-3?/)3+(4^_ 5a;)^4-(3-c-y)3-
.■i(2x-3^)(4^-5a;)(3a;-^).
By [8] this is seen to be {(2a;- 3jr)+(4^ — 5a;) + (3a; - ^) } 3 =
(0)3 = 0.
2. Prove that («-/;)3 + (6-t)3 + (c-a)3 = 3(a-6) ib-c){c-a).
In [8] substitute a — 6 for x, b — c for y, and c — aforz; for
these values x+y+z=-0, and the identity appears at once.
3. Prove (a + 6 + c)3 -(6 + 6--a)3_(a + c-i)3-(a-h6-c)3 =
24Labc.
In [7] let a; = 6+c — a, y = c+a — b, z = a-\-b—c, and therefore at
+ 7/ = 2c, y-{-z = 2a, z-\-x = ''Ib, and this identity at once appears.
- FUNDAMENTAL FORMULAS. 19
Exercise x.
1. Cube the following: 1 -x-\-x^ . a-b-c 1 - 2:r+3a;2_4a;3.
2. SimpUfy (a?2 ^ 9a:_l)3 + (2a;-l)(a:^ + 2a;- 2) —
(a;3 + 3a;2-l)3.
3. *Prove that. {x+y){y+z)(z+x) +xf/z = {z + y-\-z){.r]/ -\-i/z+zx).
4. Prove that {ax — b?/)^ + a^i/'^ — b^x^ •^'dahxy{ax — bij) =
5. Simplify (a;-22/)3 + (?/-2z)3 + (z-2a:)3 + 3(2'-?/-22) x
(y-z -2a;) (2-a:-2?/)+(a;+?/+z)3.
6. Simplify (2a;-' - Sy^ +422)3^(2^2 _ g^s + 43^2)34-
(222-3x2+4|/-)3.
7. Simplify (2fl;B-fe?/)34-(2?>7/-c5)3 + (2rz— ffa;)3 +
8f2fla;+6i/ - rz) {^by+cz — o^a;) (2r2+rta; - by).
8. Prove (x^ + Sx'^y-y^y + \'^xy{x +jj)]^ = \(x- y)^+^x^y]
x{a;'^+a^+|/%'-3.
9. ' Prove 9{x^ +y^+z^) - (x-l-y + z)^ = (4a; -f 4y-!- z) {x - tj)^ +
{4y + -iz+x) (y-z)-+{iz+4:x+y) (z-x)^.
10. U x+y-\-z = 0, shew that a;3 4-7/3 _|_23 :=3a^2_
11. If a;= 27/+3z shew that a;^ -8y3 _27z3 - 18a;^2 = 0.
12. Shew that (x^+xy + y^)^ + {x^ -xy+y'')^ + 8^^-
^z'' (a;4 +a;22/2 +?/4) = 0, if x^ +?/2 +2^ = 0.
13. Prove that 8{a + b + r)3 - (a + 6)3 - (6 + c)3 - (c+ a)3 =
3(2a+6 + c) (a + 2i4-c) (a + 6+2c).
Prove the following :
14 (ax — by)^ +0^;/^ =a^i;^ -\-?>abxy{by — ax).
. . . ^
*Note that the right-hand member is formed from the left-band one by changing
additions into multiplications, and miMipHcations into additions ; hence in (x+y+
s).(x.y-i-y.z+z.xi the signs + and . maybe interchanged throughout without alter-
ing the value of the expression.
20 FUNDAMENTAL FORMULAS.
15. a^+b^+c^-3nbc=i{{a-b)* + {b-c)^ + (c-a)^} X
{a+b + c).
16. {a + b + c) {{a + b-r) {b +c-a} + {b + c-a) {c+a-b) +
{c+-a-b) {a + b — c)} = {a + b — c) {b + c — a) (c+a — b)-+8abc.
17. a^ + b^+c^+2iabc = {a + b + c)^-S{a{b-cy+b{c-a)^-\-
c{a-b)^}.
18. (fl. + />+7c)(a-i)3 + (5 + c+7a)(6-c)« + (c+a+76)(c-fl)a
= 2(a + fe + f)3 -54rt/>c.
19. (a + b + c) {{2a-b) {2b-c) + {2b-c) {2c-a) + {2c-a)x
{2a-b)} = i2a-b) {2b-c) {2c-a) + {2a + b -c) {'Ib + c-a)x
{2c + a-b).
20. li x^{y-'rz) = a^, y^{z+x) = b^, z^(x + y) = c^, and xyz = abc,
shew that a^ -{-b^ + c^ +-2abc = {x+y) {y-\-z) (z+x)
Expansion of Binomials. •
We have from formula [5]
{a+b)^ —n^ +Sa^b + 3ab^ +b^ ; multiplying by a + b we get
la+b)'^=a^ + 4rt.36 + Sa^b'^^ +iab^+-b^ ; multiplying this by
o+i we get
[a-\-b)^=a^ + 5a^b+10a3b^ + 10a^b^ + 5ab^ + b*.
From tiiese examples we derive the following law for the form-
ation of the terms in the expansion of a+6 to any requu-ed
power : —
(1). The index of a, in the ^rsf term, is that of the given power,
and decreases by unity in each succeeding term ; the index of b
begins with unity in the second term and increases by unity in
each succeeding term.
(2). The coefficient of the first term is unity, and the coefficient
of any other term is found by multiplying the coefficient of the
immediately preceding term by the index of a in that term, and
dividing the product by the number of that preceding term. It
will be observed that the coefficients equally distant from the
extremes of the expansion, are equal.
MULTIPLICATION AND DIVISION, 21
Exercise xi.
1. Expand (ic+i/)«, (a; + 2/)^ («+?/)% {x+ijY*.
2. What will be the law of signs if — y be -written for y in (\) ^
3. Expand (a-Z>)^ (a-26)4, (26-a)4.
4. Expand (l + m)6, (w^-fl)^ (2m+l)e.
5. What is the coefficient of the 4th term in {a—by° ?
6. Expand (a;2- 2/) 4, (a-2''.^)^ (a3-2/>3)G.
7. In the expansion of [a — bY^, the thii-d term is %Qa'^*^lt^, iind
the 5th and 6th terms,
8. Shew that {x-{-yY -x^ -y^ —^xyix+y) {x^ +x>i-{-y'^).
9. From (8) shew that 2 {(a - by -^r {h - cY + [^c - ay] =-.
5(a-6) (6-c) {c-a) {{a-hy + {b-cY + {c- aY).
Section III. — Hoknek's Methods of Multiplication and
Division.
Examples.
1. Find the product of kx^-\-ix'^ -'rvix-\-n and ax'^-\-hx-^r.
Write the multiplier in a column to the left ot the muitipiicaud,
placing each term in the same horizontal line with the partial
product it gives •
kx^ . +ix'^ +mx -fw Q
ax^
•^bx
+ c
akx^ +'.'?j;* +a)iix^ -\-a}ix^ Pj
-\-hhx'^ -\-'ilx^ -\-bmx~-\-bnx ■■■'P^
-\-rkx-^ ~\-(:tx^ ■\-cinx-'rcn p^
akx^ -f (rt/ + 6/.-)a;* + ( '.nn + bl + ck)x^ -f- {an + bm + cl)x' +
(b7i+cm)x-{-cn P.
Art, VI. The above example has been given in full, the pow-
ers of X being inserted ; in the following example detached coeffi-
cients are»used. It is evident that if the coefficient of the first
term of the multiplier be unity, the coefficients of the multiplicand
will be the same as .those of the first partial product, and may be
used for them, thus saving the repetition of a line.
22
MULTIPLICATION AND DIVISION.
2. Multiply 3^4 -2.7-3- 2a: + 3 bya;3+3a;-2.
1
+3
-2
3 -2 +0 -2 +3
+9 -G +0 -G +9
-6 +4 -0 +4 -6
3x^ + 7x^-12x^+2x^-Sx^ + rdx-Q.
3. Find the product, of (^-3) >- + 4) (a: -2) (a; -5).
+^
1
-3
+4 -12
+ 21
-2
1
+ 1 -12
-2-2
-6
1
-1 -14
-5 +5
+ 24
+ 70
-120
a:4
-6x3_9a;3
+ 94a;
-liO.
4. Multiply a;3 - 4a;3 + 2x - 3 by 2a;3 - 3
1
-4 +2
-3
2
0
0
-3
2
-8 +4
0 0
0
-6
0 0
0 0
-8 +12
0
-6 + 9
2.'c«
-8a;-^ + 4.r4.
-.9a;3+12a;2
-63; +9
[x^ X Z^ =X*]
In this example the missing terms of the multiplier are suppliecl
by zeros ; but instead of writing the zeros as in the example, we
may, as in ordinary arithmetical multiplication, " skip a line "
for every missing term.
5. Multiply a;4 - 2a;3 + 1 by a;4 _ a;2 + 3.
1
-1
3
1 +0 -2 +0 +1 [x*xa'*'-xn
_1 _o +2 -0 -1 L« xa- -X J
+ 3 +0 -6 +0 +3
x°
-3aj«
+ 6a?*
•7a,-2 +3
MULTIPLICATION AND MYISION,
ss
G. Find the value of {x+ 2) (a;+3)(a;+4) {x + 5)- 9(x + 2)(a;-i- 3}
X^^^-4) + 3(x•+2)(.^■^-3) + 77(a; + 2)-85.
1 +5
-0
+ 4
+ 3
+ 2
1 -4
+4
-IG
+ 3
-39
+77
1 +0
+3
-13
+ 0
1 +3
+ 2
-13
+ 6
+38
-26
+ 79
-85
+ 5.^3- 7.f3 + 12a; - 9
7. Find the coeflficieut of a;* in the product of x —ax^+bx^~
cx + d and x^+px-\-q.
1 —a +6 —c
— ap
+ ?
-{-d
+ {b-ap-{-q)
Exercise xii.
Find the product of
1. (l+x+x3+a;3+a;'ij(l ~x+x^ -x^ +x^ -x'^^+x^^).
2. {l'+x^){l -x^+x'^){l+x+x-^+X'i+x^).
3. (x-5) (x+Q) {x-7){x+8); (2^--a;3 + i) (^4_^+2).
4. (aj3 + 5a;3 - 16a; - 1) (a;3- 5^-3- l(3.c+l).
6. {6x^-x'^-j--2x^-2x^ + 2x^ + Wx+6) (3a;2+4a;+l).
Obtain the coefficients of x'^ and lower powers in
6. {l + ix-ix^-\-^\x^-^§^xi) (1 - .^x-ix^-^^^x-^-^^^x*).
7. Multiply 2x'' -x^+Sx -4:hj 3z^ -2x^ -x-l.
24 MULTIPLICATION AND ERISION.
Simplify the following :
8. (x+l) {x-\-2) (a;+3) + 3(a;4-l) (a;+2) - 10(a;+l)+9.
9. x{x+l) (£c + 2) {x-h'd)--dx{x+l) [x+2)-2x{x-hl) + 2x.
10. x{x-l){x-2){x-3} + dx{x-l){x-2)-2x{x-l)-2z,
11. {x-1) (x+l) {x+3) {x + 5)-U{x-l) [x-tl)-rl.
VI. Given that the sum of the four following factors is — 1, find
(1) the product of the first pair ; (2) the product of the second
pair ; and (3) the product of the sum of the first pair- by tiie sum
of the second pair.
(1)
X
+^^
+xi3
+ X^*
(2)
x^
+ X8
+ ^9
4. .-IS
(3)
x^
+^-»
+a;i3
+x^^
(4)
x^
+ a;'
H-r'o
+a;ii
13. Given that the sum of the three following factors is equal
to — 1, find their product.
(1) X 4a;' 4a;8 4a;»8
(2) ic2 -ha;3 +xio 4a;* '
(3) x^ +x*^ +x^ +x^.
Art. VII. Were it required to divide the product P in the
first of the above examples by ax'^ +bx+c, it is evident that could
we find and subtract from P the partial products ^g, ^3, (or what
would give the same result, could we add them with the sign of
each term changed), there would remam the partial product ;>, ,
which, divided by the monomial ax^, would give the quotient Q,
This is what Horner's method does, the change of sign being
secured by changing the signs of b and c, which are factors m
eacli term of _pg, p^, respectively.
BITJLTIP LIGATION AND DIVISION.
25
+
H « 3
+ '
+
^^ ^
•^ 5 "1^
+
+
+ I
+
"5
*
- O
a
.2
*-i3
r-l
03
Co
a-i
CO
o
O
<D
CD
a
* 1— H
o
f-H
■ ^
• ^H
p— <
O
r-4
>
CD
"^
-<-i»
S
a
>
P^
>
c3
o
r---l
-4-3
a
fcD
J?
'Hi
•8
a
• <— 1
r-i
o
O
n3
•"3
IN
1— ~i
•1-H
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O
CO
-1-3
a
a
• »-4
CQ
P*
CO
o
r^
"«
}-\
.2
a"
<s
Pi
• *
Sh
'Ph
S»
'^
Q)
'o
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aj
Ph
_a
• 1— (
c3
a
o
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a
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«
&.
fl
r^
g
CD
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■
2
OQ
■ 1— 1
1 — I
* 1-^
CO
■4^
o
-1-3
1
c
a
CD
^-4
-4J3
^
3
O)
o
"*^
O
TS
1— H
-4-3
C^"*
-f
- f
a
o
CD
a
1— ■
o
3
o
i
o
-k3
o
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P^
a
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00
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to
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cr"
CD
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r— *
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■4-4
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a
• r-(
1 1
o
^
4
' 1
- 4
"73
■73
a
c3
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00
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cu
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a-t
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00
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■M-l
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i e;
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a
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a
CD
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-1-3
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r-»
-a
-t.3
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p—H
-1-3
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-1-3
60
_a
-4->
o3
(D
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o
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a)
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-1-3
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a
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26
MULTIPLICATION AND DIVISION.
2. Divide Sx" +1x^ - rix*-i-2x^ - 3a;3 + 18x-- 6 by z'+Zx-2.
3 +7 -12 +2 -3 +13 -6
-3
+ 2
_9 +6-0+6-9
+ 6-4+0-4+6
{x''^X'^X'>'}
8a;4_2a;3+ 0 -2x + 3
Compare this example with the secoEd example of Homer's
Multiplication, performing a step in multiplicatio.u, then the cor-
responding step in division ; then another step iu multiplication
and the second, (corresponding) step in division, and so on.
3. Divide x'' - Sx*^ + ia;-^ + 18x-^ - 7x.+ 12 bv x^ - Sx^ + 'dx - 1.
-3
+ 1
-3 +0 -4 +18 +0 -7 +12
+ 3 +0 -y -36 -27
-3 -U +9 +36+ 27 [x" - x^ =.'f4] ,
+ 1 +0 -3 -12 -9
> x-i +U -3a;3-12x- 9; (lr-'+ 8x +3
The quotient is therefore x'^ —ox'^ - IJx — 9, and the remainder
6x2+8a;+3.
4: Divide x^ - Zx' -ox'' +2a;4 + 5x3 + Ix- + 1 by xS 4. 2a;-l.
The zero coefficient in the divisor may be inserted, or it may be
omitted and allowance made for it in the 2x — iiue. See examples
4 and. 5 in multinlication.
-2
■+1
1 _3 +u _5 +'2 +.J +4 +0 +1
-2 +6 +4 -4 -f; +2
1-3-2 +2 +3 -1
I 1 _3 -2 +2 +3 1; 0+5 +0
\_x^ ~- x^ =x^] . The quotient is therefore x^ — 3x^ — 'Ix^ 4- 2a;'
+ 3a; — 1, and the remainder 5x.
5. Divide lOx^ -lla;»-3a;4+20x3 + iOa;3 + 2 by bx'^-3x^-i-
OF
MUI.TIPIiTCTI TON ANT) r)I\aSlON.
27
Arranging as in the ordinary metliod, we have
+3
-2
+2
10 -11 -3 +20
6-3-6
-4 + 2
+ 4
+ 10 +0 +2
+ 12
+ 4-8
-2-4+8
2 _i _2 + 4
Quoiieat = 2x^-x^ - 2x+4 +
24-12+10
24a;2....i2a;+10
5x^-Sx^+2x—2
We first draw a vertical h'ne with as many vertical cohimns to
the right as are less bj' unity than tlie number of terms in the
divisor. This will mark the point at v,'hich the remainder begins
to be formed. We then divide 5 into 10, and thus obtain the
first coefficient of the dividend. We next multiply the remaining
terms of the divisor by the 2 thus obtained. Adding the second
vertical column and dividing by 5, we obtain — 1 ; we multiply
by the — 1, add tlie next column and divide the sum by 5, and so
on for the others.
This method is not, however, always convenient. If the first
term of the dividend be not divisible by the first term of the divi-
sor, the work would be embarrassed with fractions. We may
then proceed as in the following exaTQples :
6. Dividea;5-3a;4+a;3+3a;2-a: + 3 hy 2x^+x^ -Bx+l.
Let 2x = y, or x = — .
^ 2
Substitute -^ for a; in the dividend and divisor, and wo have
2
y'
3,y4 y3 Sy^
V
2^
24 -r 23 ' 22 "
2
+ 3 -
%^
+
y
_ y^
23 ' 23
-2 X 82/^1 +22^/3 + 23 X 3^3 - 24?/+2s x 3
37/
— + 1
2 ^
25
y3+y3-2x3y + 23
— p '■
2/ 6 - &ij^^ +4y3 + 242/3 -16y +96
^2/3+,/2_6iy + 4.
.A.
28
MULTIPLICATION AND mviSION.
Dividing ?/» -6^/* +4»/3 +247/2 - IQy+dQ by y^+y^ - 6?/+4 by
the ordinary method, and the quotient by 2' we have
y2-7y + 17
23
1_ 39y2- 1147/ -28
23* ^3_f.,/2_6^_|_4 •
Substituting for ?/ its value 2x, and simplifying we get
a;2 7iB . 17 1 39a;2-57a;-7
-T +
8
8" 2a;3+:c3-3a;+l
B.
By comparing the dividend of A with the original question, we
find that we have multiplied the successive coefficients of the divi-
dend by 2°, 2^^, 22, &c., and. omitting the first term, we have
multiplied the successive coefficients of the divisor by tlie same
numbers. Dividing then by Horner's division we get tlie coeffi-
cients 1, —7, 17, and for coefficients of remainder, -39, 114,
and 28. The first of these divided by 2, 22, 2^ are the coeffi-
cients of a;2 &c. ; and, -39, &c., are divided by 1, 2, 22. Hence
the work will stand as follows : —
a.5 _.;:)^4 +a;3 + 8.c2- x+ 3-2a;3-fa;2-3«+l
1248 16 82 124
1-6 +4
+ 24
- 16 + 96
-1
-1 -f?
-17
+'•>
+ 6
-42
+ 102
4
- 4
+ 28-68
.1-7 +17
-39
+ 114-f28
1 -6 +4
"Quotient =
x'
X'
2
7x 17
4 "'" 8
1
§■
7x 17
4 8
1
8
S9x^
lUx
2^
28
4
2x3 +3:3 -'Sx + 1
39x2 _ 57a; - 7
2x^+x2-Sx+V
*lt will, in general, be as convenient to multiply the dividend by such a, num-
ber aa will make its first ti'vin exactly divisible by the ftrst torm of the divisor, and
afterwards divide the quotient by this multiplier.
MTJLTTPTJrjATION AND DIVISION.
7. Divide 5x^+2 by Sz^-2x+B.
29
5x-'
1
0 0 0 0 +2 -f- Sx^-2x+S
3 9 27 81 243 13
5
0 0
0
+ 2
10 +20
-50
-9
-45
-90
5 -|.io -25-140
0 + 486
—280
+225 +12G0
-2 +9
- 55 +1746.
10 25 140
Coeffs. of Quotient = ^. - -
3 32 o^
3*
Quotient =
5x^
+
lOiB^
25^
140
J^ 55 -H^
34" 3_2 + 3-
55a;- 582
9 27 81
Exercise xiii.
81 3a;2-2a;+3'
1. Divide 6ar-^ + 5^4 - 17x3 - 6x^ + 10.r- 2 by 2x- +?.x - 1.
2. (5a;«+6a;5+l)-f (a;-^+2a- + l).
4. (x5-4a;-//3 _ 8x273. „ i7a;,y4_ 127/5) ^(cc3 _2^v/_3^2).
Divide
6. 4a;'i + 8x3-3a;+lby a;3-2a-+3.
7. 10x5+ox-4-00a;3 _4i^.2+i0a;+l by a;2-9.
8. x^ —x^y-{-x^ii^—x'^ll^^xy'^-y-' hyx^—y^.
9. Multiply a;4-4a;3a + Ca;3«2 -4a;«3 4- ^4 ^y a;2 4. 2x(< + a",
ftnd divide the product by x* — 2ic3a+2a;a3 —a*.
Divide
10. x^ - ax^ + I'X^ - bx^ +r,x - 1 by a; - 1.
11. 6a;^+7.T-i+7a;3+6x3+0a;+5 by 2a;2+^ + l.
12. m{x^-\-y^)-\-S)lxy{x'^-y'-^) by 12a;3 - 13a;// + 5.y*.
•jO multiplication and division.
13. 6ae-481a;'^ + 79a;4 + 81a;3_81a:2-f86x-481 l.y«-80.
15. a(« + 2//)3_/;(2rt4-6)3 by («- A)^.
16. (x+uy^-{-3{xi-ijyz-{--d{x + y)z"-+z^hy{z+y)^ ^^
— 17. 10a;"J + ina;C + 10a:3-200 by a;' +x3 -x + l.
18. bjnx^ + {bji-}-cin)x^ +ciix~ +abx + ac hy bx+c.
19. Multiply 1+ 2^a;-18a;3 by l-L3a;3 + 3a;3 aud divide the
product by l+Va; — 8x2.
Find the remainders in the following cases :
-r 20. {xJ-\-nx^ + -i.x + 5)-{x~2).
^ 21. (x^~Sx^-]-x-S)~{x-l).
22. (a:^+4.«3 + Ga; + 8)-(a;-i-2)
23. (27a:i-?y*)^(8a;-2y).
24. {3x'' +5x^ -3x^-\-7x^ -r)x + 8)-^{x" -2x).
25. (5a;4-f90a;3 4.80a;2-100a; + 500)-^(a;+17).
Art. viii. The following are examjiles of au important use of
Horner's Division :
1. Arrange x^ —6x^-\-lx— 5 in powers of a; — 2.
11 -6 7—5
2 2 _8 -2
2
1 .
-4
2
-1
-4
-7
— 2
-1-2
-5
I 1; 0
Hence, x^ -Gx^ +lx-5 = {x—2)^ -5{x-2)-7, or as it is gen-
erally expressed, x^ - Gx^ -\-7x— 5 = y^ — 5y — 7 if y = X'-2.
MULTIPLICATION AND DIVISION.
81
a. Express x^+l2x^+'i7x^+G()x+28 in powers of x+6.
-3
12
-3
47
-27
60
-GO
2S
-18
-3
9
-3
20
-18
6;
-6
10
-3
6
-3
2-
-9
0
-3
3;
-3
-7
-*- )
0.
Hence a;4 +12x3+47x2 +66a;+28 = //'i -7y/- +10 if y=a5+3.
After a few solutions have been written out iu full, as in the
above examples, the writing may be lessened by omitting the
lines opposite the mcrements (—2 in Ex. 1, and 3 in Ex. 2), the
multiplication and addition being performed mentally. The last
example written iu this way would appear as follows :
1 12 47 66 28
1 9 20 6 (10)
1 6 2 (0)
1 3 (-7)
1 (0)
Exercise xiv.
1 Express x^ — 5x^ + 3a; - 8 in powers of x — 1.
x''-^Sx-+Gx+0 " x+1.
;c4 _ 8x3 + 2 Ix'-' - 32x + 97 in powers of x - 2. <
-3
2.
3.
4.
5.
6.
7.
8
9.
x-' + 12x3 + 5x3-7
3x'-x3 + 4x-+5x-8
x^ -7x3 + 11x3 -7x + 10
x3-2x--4x+9
x3-9x3// + (JX^2_8^3
X* —bx^ij-k-^xy^--y^
(<
x+2.
<4
x-2.
(<
x-n
((
x-h
({
x-2.y
<f
x-y.
32 SYMMETRY.
10. " 8a;S+12x2y + 10«.'/2+8?/3 •• 2z+y.
11. «' ,r3-|a;2+§a;-f^ " ia^-rV
12. *' a;'^ + 8a;3-15x-10 ♦* x+2.
CHAPTEB II
Section I. — The Principle ov Sym^metky.
Art. ix. An expression is said to be symmetrical with respect
to two of its letters when these can be interchanged without
altering the expression :
Thus if in a^+a^x + ax^ + .v^, we write x for a, and a for x, we
get x^+x-a+xa^+a^, which is identical with the given- expres-
sion. So, in a;2+i3^_f.^^_|_rt23. if ^e interchange a and 6, there
results a;- +«-iB+ai'+62a; which is identical with the given ex-
pression ; but it will be seen that the expression is not symmetrical
with respect to x and b, or x and a.
An expression is symmetrical with respect to three of its letters
a, b, c, when a can be changed into b, b into c, and c into a, without
altering the expression.
Thus a3 -^i,s^sS—Sabo remains unaltered by changing a into b,
h into c, and c into a, and is therefore symmetrical with respefct
to these letters. So, aH-\-b^a+a^c+c^a-irb^c-\-hc^ , and {a-h)'^
_|. (h — c)^ + (c — «)', are each symmetrical with respect to a, b, c.
Again (x-a) {a-by + {z-b) (b-c)^ + (x-c) {c-ay is sym-
metrical with respect to a, b, c, but not with respect to x and any
of the other letters.
Generally, an expression is symmetrical with respect to any
number of its letters a, b, c, . . . h, k, when a can be changed
into h, b into c, c into d h into Jc, and k into a, without
altering the expression.
BYMMETRY. 83
A symmetric function of several letters is frequently represented
by writiDg each type-term once, preceded by the letter 2 ; thus for
a-\-b-\-c-\- ... . +/. we write 2a, and for rt^+rtc-f atZ+ . . . .
+ 6c + k/+ . . . [i. e. the «um of the products of every pair of
the letters considered) we write "Zab.
Exercise xv.
Write the following in full :
1. 2a^h, 2(a-i)3, y_a{b—c), ^ab{x-c), ^aH^c, ^{a-\-h)
X{c-a){c—b), 2 {(a+c)--63}, and va(i-fc)3, each for a, h, c.
2. y.abe, y.aH, ^a-bc, 2 (a- i), and 2a 3 (a _jj^ each with
respect to a, b, c, d.
She"w that the following are symmetrical :
3. {x-\-a) (a +6) {b->rx)-'rabx, with respect to a and b.
4. (a+6)^+(a. — 6)3 with respect to a and b, and also with
respect to a and — b.
5. {ah -xyY^ -{a -^-h-x-y) {ab{x-{-y)—xy{a+b)} Mniih. respect
to a and b, and also with respect to x and y.
6. a'^ {b ~c) — b^{a — c) — c^ {b - a) with respect to a, b, c.
7. {ac+bd)-+{bc-arl)- with respect to a' and i-, and also
with respect to c- and (P .
8. x^ +y^ +?>x>/{x^ +xy + y^) with respect to x and y.
9. •{«^-?/3+8x^(2a;-f2/)}3 + {2/3-a;3+3a;?/(2|/ + a?)}3 with res-
pect to x and y.
10. a(a+26)3+6(6+2a)3 with respect' to a and 6, and also
with respect to a and — b.
11. ab[{{a + c) (b+c) -f 2<7(a + ft)}2 _ (a_e)2 (j_c)«] with
respect to a, b, c.
12. a'^b- -{■b'^c'^ -\^c^a- -^'2>abc{a-\-b-\-c) with respect to afc, 6c, ca
With respect to what letters are the following symmetrical ?
13. xyz-\'5xy-\-'l{x--\-y'^).
14. 2(a3a;-' + h-h/) - 2ab{xy + //z -\-zx).
15. {P - /t--^)-2 + 4//2( z'+Zi r-' 4- ( 2/7^ - 2.7-)».
34 SYMMETRY.
16. (x+y) (x—z) {7j — z) — xyz.
17. «,252^Z»2c3+.c2a2-2fl6c(a + ^-c).
18. x^ -y^ +z^ -^x^ -y-^){y^ -z^) (z^ + a;').
19. (a+ft)2+(a + c)2 + (//-(-)*-
20. (a + i)4 + (,i_^)4^(/,+c)4 + (« + f)4.
Select the type-terms in :
22. ,,2a.2ai-f/*2+2k+?^+2ca
23. a(b^ -t"~)+h[c^ -a^)+r{(r- -b"-)-\-{a + h) (h-\-c) {c-k-a).
24. «(6 + '0'+^'(''-l-«)^ + ^'(« + ^)^-l'^«/;c.
Write down the type-terms in :
25. (a;+y)5, (a;-?/)^ (x + y)^ -a^s - ^/S.
26. (x+yy + ix-yy, {x+!/y-{x-yy.
27. (a;+7/-f2)4, (a;-y-^)4.
28. (a+b + c + dy, {a-+h^-+c^+d^y.
29. (a + &)3+(z, + c)3 + (c4.«,)3.
Art. X. In reducing an algebraic expression from one form
to another, advantage may be taken of the principle of symmetry :
for, it will be necessary to calculate only the type-terms, and the
others may be written down from these.
Examples.
1. Find the expansion of {a + b-\-c + d-^e-{-&c.y
TJiis expression is symmetrical with respect to a, b, c, &c. ;
hence the expansion also must be symmetrical, and as it is a pro-
duct oi' iM;o factors, it can contain only the squares a^, b^, c^, &c.,
and the products in pairs, ab, ac, ad . , , , be, bd, &c. ; so that
rt^ and ab are type-terms.
Now {a-^by —a"^ -{-2ab-^b^ ; and the addition of terms involv-
ing a, b, c, SiC, will not alter the terms a^ -f 2fli>, but will merely
give additional terms of the' same type. Hence from symmetry
we get
SYMMETRY. 35
(a + !) + (■-{- I + e + &Q.y-' ^ a'-' +2ab-^2ac + 2ad + 2ae+
+ b- +2bc+2bd+2be +
+ 6-2 -{-2cd+2ce-\-
+ fZ2 +2de +
This may be compactly written
(2a)3=Xa2 + 2Safe.
2. Expand (a + h)'^.
This has been found by actual multiplication — see formula [5]
—but we may also proceed as follows :
(1) The expression is of three dimensions, and is symmetrical
with respect to a and b.
(2) The type-terms are a^, a-b.
■ Hence [a-^-b)^ = a^ +h^ -\-niu'^b-}-b-a), where n is numerical.
To find the value of n, puta = ft = l, and we have (1 + 1)3 =
1 + 1 + »(1 + 1); .-. n = S.
3. Expand (a;+^ +2)3.
This is of three dimensions, and is symmetrical with respect
to X, 7j, z. We have
{x-^U+z)^ = {{x+y)+z]^ = (x+^)3 +&c.
= x^ + ^x'^y-\-kQ., which are type-terms, the only other possible
type-term beiug xyz.
Now, since the expression contains ^x-y, it must also contaia
^x'-z, that is, it must contain Qx-{y-\-z). Hence
(x+y+z)^ = x^-\-dx''{y-{-z)
+2/^+3y-(z+a;)
+ z^ + -iz^{y^x)
+ n{xyz), where n is numerical, and
may be found by puttinjf x-y = z = l in the last result, giving
(l + ]+l)3 = i + i^.i+3(i4.iA4.3(i_^l)_j.3(l_^l)_^,,.
.•. n = 6,
36 SYMMETRY.
4. Similarly we may shew that
{a + b-\-c + U)^= ai-\-nn-{b+c + d) + (ibcd
+ hs+3b2{c + d+a) + Gcda
+ c^ + 3c^d-+a+b)-\-Gdab
-f d^ +'Bd^(<i + b-i-c)-\-6abc.
5. Expand (a+6+c+&c.)3.
Tho type- terms are a^, a^b, abc.
Expanding (a + b-\-c)^, we get a^ + Ba^h-\-6abc-{-&0,
Hence by symmetry we have
6. Simplify (a + & - 2c) 2 + ( fc + c - 2(y) f + (c + « - 2b) 2 .
This expression is symmetrical, involving terms of the types
a^ and ab. Now a^ occurs with 1 as a coefficient in the first
square, with 4 as a coefficient in the second square, and with 1 as
a coefficient in the third square, and hence 6a^ is one type-term
of the result : ab occurs with 2 as a coefficient in the first square,
with —4 as a coefficient in the second square, and with —4 as a
coefficient in the third square, and hence — 6ah is the second
type-term in the result: hence the total result is 6 {a^+b^+c^
•—ab — bc—ca).
7. Simplify {x-\-y+z)^-\-{x-y-z)^+{ij-~z-xy -\- (z — x-y)^.
This is symmetrical with respect to x, y, z; and the type-terms
are x^, Sx^y, (ixyz :
(1) ic* occurs in each of the first two cubes, and — os^ in each
of the second two cubes, .". there are no terms of the type x^ in
the result.
(2) 3x^y occurs in the first and third cubes, and — Bx^y in the
second and fourth, .•. there are no terms of this type in tlie
result.
(3) 6xyz occurs in each of the four cubes, .'. 24a;^z is the total
result.
8. Prove (a^ + b^+c^+d^) (w^ +x^ -^y- +z'')-
{aw -r bx-\'Cy -f- dz) - = (ax — bw) ^ +{ay — \cw) ^ + (az — dw) 2 -f
(by-cx)^ + {l)z-dxY + (cz-dy)^.
SYMMETRY. 87
The left hand member (considered as given) is symmetrical
with respect to the pairs of letters, a and iv, b and x, c and y,
d and t, that is, any two pairs may be interchanged without
affecting the expression. As the expression is only of the second
degree in these pairs, no term can involve three pairs as factors ;
hence the type-terms may be obtained by considering all the
terms involving a, b, w, x', these are a^ic'^, a-x^, b^w^, b^x^,
— a^w^, —b^x^, — 2«fettic, and are the terms of [ax — bw)'^ -which is
consequently a type-term. From (ax—bw)^ we derive the five
other terms of the second member by merely changing the
letters.
9. Prove that
(x^-yz)^-j-{y^-zxy -\-{z^ -X!jy-B{x^ -yz) (y^-zx) {z^-xy)k
a complete square.
The expression will remain symmetrical if (x^—yz) (y^—zx)-
(z^—icy), instead of being multiplied by —3, be subtracted £>-om
each of the preceding terms, thus giving
(Kg - yz) { (a;3 - yz) ^-(y^-xz) {z^ —xy) }
-f (?/2 -zx) {(y^ -zx)^ — {z^ -xy) {x^ - yz^}
-\-[^^'-xy) {{z2-xyr--(x'--:.z) (y^-zx)}
= (x^ —yz)z[X^-^y^-\-z^ — Bxyz)
+&c.
+&c.
= iX'^-j-y^+z^ — Bxyz) {x^-^y^-\-z^ - Sxyz).
Exercise xvi.
Simplify the following :
1. {a + b + G)^-\-{a-\-b-c)2 + {h+c-a)^-\-{c-\-a~-b)^
2. (a-6-c)2-f(i-a-c)3 + (c-a-6)2.
3. (a-}-6-|-c-d)2 + (6+c+cZ-a)3-|-(c-f£/-{-«_i)2_|_
(d+a+b-c)^.
4. {a-\-b-\-c)^-a(b-^c-a)-b{a+c-b)-c{a+b-c).
■ 5. (x-\-y+z+ny^+{x-y-z+7i)^+{x-y-{-z-ny-\-
(x+y-z-n)^.
6. (a+6+c)3+(a+6-c)3-|-(6+c-a)34-(c+a-Z>)3.
38 SYMMETRY.
7. (x-2>/-Bz)^-{-{y-^z-dx)2-{-{z-9x-Sy)^.
8. (ma-\-u/j-\-rc)^ — (ma-\-nb—rc)^ — (nh-j-rr — tim)^ —
' (rc-\-ma — nb)^.
9. aib-\-r)ib^-i-c.2-a2)-{-'j(c+a)(c2-{-a^ -b^) +
c(a+&)(a2+62_c2).
10. (ai + 6c-+m)3 - 2rt6c(a+6+c).
Prove t)ie following :
11. (ax-\-b;/+rz)''-{-{hx-\-cy-{-azy +{cx+a?j+hz)^ +
{nx-\-cy-]-bz)''^-\-{cx-\-l>i/-^az)^-\-{hx-{-ay-\-cz)^
= '2{ci'+b^-{-c--){x^--{-y2+z-^)-^i{ab-{-bc+ca){xy^yz-{-zx).
12. {a-^b+cy-^{b+c-a)^+{c+a-b)^-\-yu-i-b-o)^
13. {a-\-b-\-cy = 2a^ + iZa^b+Q-LaH^ + 12.Za^bc.
14. (£rt)4 = 2rt* + 42rt3/, + 62a3i2 + 122a3ic + 24SrtZ-(,-t^.
15. (^a^ -\.b^ +c2)^ + 2{'ib+bc + ca)^ -3{a^ -i-b^ +c^)X
f^ab + he + ca) 3 = (« 3 + 0 ^ _|_ c 3 _ ^abc) 3 .
16. («-/>)2(6-c)3 + (^>-c)2(c-a)2 + (c-a)3(rt-/;)2 = '
^^2 +63 +c2 -ab- ac - bc)^.
17. (2a-ft-c)-'(26-c-a)2 + (26— c-a)2(2c-a-6)2-f.
(2c-a-&)2(2(*-6-c)2=9(«s + />2+6'2-a6_6c-ci/)2
18. (rtr2+26?-s-|-cs-)((ifx2 + 26a;y + c^2)_
{ arx -{-b{ry + sx) + csy }2 = {ac — b^) (ry — sx) 2 .
19. {a^+ab+b^){c^ + cd+d^) = {aG + ad + bdy +
{ac + ad+bd) (be — ad) + {be — ad)-.
20. Sbew that there are two ways in which the given product
in the last example can be expressed in the form p- +iyq+q^, and
two ways in which it can be expressed in the form jJ^ —pg+q^-
21. 6(ti'2 +a;2 +?/2 +z2)2 = (w + x)^ + (rt'-a;)^ + (»,-+?/) * +
{iv-'y)^ + {w + z)^ + {w-z)^ + {x+i/)^-\-{x-y)^ + {x-\-z)'^ +
{x-z)^ + {y+z)^My-^y'
22. |{(a+i+c)-' + (a-fe-c)6 + (6_c-a)«+(c-a-i)5}=s .
i{(a+i + c)3-fi>-^.-e)3 + (i_c-a)3 + (c_a-i)3|x
^{{a+b + c)^ + {a-b-c)^ + {b-c~a)^-\-{c-a-b)2}.
THEORY OF BfVISORS. 39
Section II. — Theory op Divisors.
Any expression which can be reduced to the form ax"-]-hx''~'^ +
cx^~^+ . . . . -\- . . . . +Jix + k, in which n is a positive
integer and a, h, c, . . . . h, k are independent of x, is called
a PoLYNOME in X of degree n.
The ■expressions ./'(ic)", F{xy, <p(x)"', are used as general symbol.-;
for polynomes ; the index n. m, indicates the degree of the polj-
nome.
Theorem I. If the polynome/(a;)" be divided by a?— a, the
remainder will be /(«)".
Cor. 1. /(x)" —/(a)" is always exactly divisible by a; — a.
(Particular case: ic" — a"is alv/ays exactly divisible hy x—a).
Cor. 2. If /(a)" = 0, /(a;)" is exactly divisible by x—a, i.e., /{xf
is an algebraic multiple of x — a.
Cor. 3. If the polynome /(as)" on division by the polynome
^(a;)'" leave a remainder independent of x, such remainder will be
the value of /(a;)" when <f){x)"' = 0.
Examples. — Theorem 1.
1. Find the remainder when x^ —7x^-\-lSx^ — IGx^ + dx— 12 is
divided by x — 5.
The remainder will be the value of the given polynome when 5
is substituted for x. (See Art. III.).
1 -7 +13 IG +9 -12
5-10 15-5 20
1-2 3 -1 4; 8
Hence the remainder is 8.
2. Find the remainder when (x—a)^ -{■ (x — b)^ -\-(a^h)^ is
divided by x-\-a.
For a; substitute -a, then ( - 2a)3 + ( - « - h)^ + (a.+b)^ = _ 8^3. '
8. Find the remainder vfhen x^^a^-\-h^-\-{x+a){x+b)[a-\-L)
is divided by a; + a+6.
40 THEORY OF DIVISORSi
For X substitute — (« + 6) and we get
~{a+l,)^-\.a^-\.l^J^ah{ci^h) = -1ah{a-^h). See Formnla [6].
4. Find'the remainder when {x^^'-lax — '±a^Y{x^-'lax-%i-)
+ 32(a;-a)4(j;4-a)4 is divided by x'^ - 2rt.3.
x^ — 2«2 may be struck out wherever it appears.
This reduces the dividend to
{'hnxf{-1ax)->r^1{x-aY[xA-aY= - 16a4.^4 ^32(a;3 -a*)*.
In this substitute 2a3 for x'^ and it becomes
-64a«^-32a8=-32a^
which is the required remainder.
Exercise xvii.
1. Find the remainder when 3a;4+60a;3 + 54a;3 — 60.'c4-58 ig
divided by a; +19.
2. Fmd the remainder when ^^x^ — Sr/x-'-f-Sj-rc — s is divided by
3. Wliat number added to ^x^ A- 34a;4 + 58.6-3+21x3 - 123x- 41
will give a sum exactly divisible by 2x+13 ?
4. What number taken from 10a;' ° ~ 20.'c8 -lOaj^ - -SOa;* —
8*9a;3+.20will leave a remainder exactly divisible by 10a;'- — 11 ?
Find the remainders from the following divisions :
5. {x-\-\Y-x'> ~x + \, and (a;+a+3)3 - (a; + ,/ + l)3 -^a: + 2.
■ 6. a;"+y" -^ x-y; a;2"+?/2» -r- a;+// ; a;-"+i+y-:'i+i -^ ^j-f^.
7. (a; + l)3+a;3 + (a;-l)3-=-a;-2.
8. (a;-a)3(x+^f)3 + (a;3-2i2)3 -i-a;3+Z)2.
-^a;2+-2^Y2.
10. (9rt24-6a^+i/;2)(9«3_6a6+462)(81a4— 36a2i3 +16i4)-r
(3a-2i)2.
11. a2(a;-a)3+fe2(a;-i)3-i-a;-a-&.
12. (rta; + %)3-j-«3y3_|_j3jg3_3(^i;jj^^(-(2;^jy^ -^ (a+&)(a; + y)'
13. a;3 + a3 +. 63 _ 3Q^jj._i_^_^ _j,^. also -i- a; + a— 6 also-*-
THf ORY OF blVISORS.
41
H. Anr polynoine divided by a; -1 gives for lemaindev tlie
aam of tlie coefficients of the terms.
Examples. — Cor. 1.
1. ar-^-f-!/' is exactly divisible by x-\-y.
In "a;^ - n^ is exactly divisible by ic— «," siibstitnte -y for «.,
2. m.r'^ - px'^' +qx-{-v} -\-p +q is exactly divisible by x-^1.
This may be written
{v.x-- -jjx^ -{-ox} - {iii{-l)^ - p{~l)- +q{-l)} is exactly divi
sible by jc- (-1).
3. (a;3+6a,vy-I-%3)5_|_(a;2_t-2a;?/+4?/2)5 is exactly divisible by
{r+2y)^. ¥oi- (x^-hQxy-\-iy^y-{-x"-2xy-iy^)'' is exactly
divisible by (x^-\-()xy+'iy^)- ( -x-2 _ 2a;// -4?/3), which is
2(a:3-4-4a;.v+4?/3 ) = 2(a;4-27/)2.
Exercise xviii.
Prove that the following are cases of exact division :
I. a;^"+i+y2«+i ^x + y; a;^" -t/^" -f- a,-+y.
also -^ X- +?/-.
3. (rt.a;+/^^)' + {l>'^+('yy ^ (« + '')(a; + ?/)•
5. (22/-a;)"-(2a;-y)"-^3(2/-a;).
6. {2y-xf''+^-{-{'2x-y)^'"+^^y+x.
7. {my — nxY — {vix — ny)^ -=- (jh + w) (y — x).
8. (.r-{-7/)« + (a;-j/)«-^2(a;2+7/2).
9. (x2+a;//+7/2)3 + (a;2-a;2/ + 2/2)3--2(a;34-//2).
10. (7 + ?')^-(«-^)^ -^2i(Sr/3 + Z;2).
II. (-.«-' + 5ix-+/>2)'+(a;2-Z>u; + ^2)T^2(a;+i)».
12. (a+6)'i»+--^+(«-6)4"+2-- 2(^(3 _|_ia).
13. {a;3 + 3a^?/(a;-?/)-y3i3 + {a;3-9a://(a;-v/)-2/3}3^2(x-v/)3.
14. 3j;a-5x2+4a;-2-r-a;-l.
12 THEORY OP Dn'ISORS.
15. Any polynome in x is divisible by a;— 1 when the sum oi
the coefficients of the terms is zero.
16. Any polynome in x is divisible by ic+1, when the sum
of the coefficients of the even powers of x is equal to the sum of
the coefficients of the odd powers. (Tiie constant term is in-
cluded among the coefficients of the even powers).
Examples. — Cor. 2.
1. Show that a(a+2Z>)3 —h{^a-\-hY is exactly divisible bv (i-f-fc.
By Cor. 2, the substitution of —6 for a must cause the polynome
to vanish.
Substituting; o(a-2a)3+a(2.'?-fl)3= -a^^a^ = 0.
2. Show that {ah — xy^^^ — {ci-\-h — x — y)\ah{x-\-y) — X]j{a-^}))\ it.
xa,ctly divisible by {x — a)(T/ — a), also by [x — h){y-b).
For X substitute a and the expression becomes
{ah-ayY-{b-y){ah{a-\-y)-ay{a-!rb)}^
aHb-y)^ -{b-y){a^h-y)] =0.
The expression is, therefore, exactly divisible by a;— a. But it
is symmetrical with respect to x and y, hence it is divisible by
y—a, and as aj — a and j^ — « are independent factors, the expree-
sion is exactly divisible by {x — a){y — a). Again, the given
expression is symmetrical with respect to a and b, hence, making
the interchange of a and b, the expression is seen to be divisible
hj{x-b){y-h).
3. Show that %{a^+b^+c^)-b{a^+b^-^c^){a» ^-b^ + c*) is
exactly divisible by a+fe+c.
For a substitute — (6+c) and the result which would .be the
remainder were the division actually performed, must vanish.
(3|_(64-c)^+65+c5}_5{-(&+c)3+63+c3||(J^c)?^J3+c9}
= 6|_(ft+c)s + 6^+c^}+306c(6+c)(62+6c + c2). See [1] and [6] .
The expansion being of the 6tli degree, and symmetrical in h
and c, it will be sufficient to show that the coefficients oih^, h^c^
b^c* vanish, the coefficients of b^c^, be*, c* being the coefficients
THK(mY Of DIVISORS. 43
of tbe foi-mer terras in reverse order. Calculating the coefficients
of tiiese type-terms we get
6{_5/;4c-1063c2.-...}+30(64f + 263c3-l-...),
which evidently vanishes. Hence the truth of the proposition.
In the last example it lias been proved that the dilference of the
(quantities here declared to be equal, is a multiple of a + 6+c, i.e.,
in this case, a multiple of zero. Hence under the given condition
they are equal.
Exercise xix.
Prove that the following are cases of exact division :
1. (ax-b7jy + {bx-ay)^—{a^+b^){x^-y^)^a,b,x, y, a-\-\
x-y.
2. ax^ — (a2 -f 6)a;3 +52 ^ ox—b. (Substitute ax for b.)
o \ (ax+by)^ - {a- b){x+z){ax+b7j) + {a-b)^xz -^ x+y.
^' \ {ax-by)^ -[a + b){x-{-z)(ax-by) + {a + b)^xz^ x + y.
4. da^x^—iax^ — l'^^axy-~oa'^xy + 2x^y-{'Oy^ -i-2ax-y,
5. l-2a*x-16-22a^x^ +^-8a^x^ -{--dax^ - x^ -^ •Gax^2x^.
6. x^+x^y^+x^y+y^ ■^x'^^y.
7. {o-d)a^ + 6{bc-bd)a+9{b^c-b2d) -^ a + 3b.
8. itix-^^yy+yi^^r-^-yy -'^x-y.
9. a(a+26)3-6(6 + 2a)3 -^a-h, ;dso -=- a+b.
10. a5-2rt4^)+a363+a3a;3_2rtaa;3+63a;3 -^ (a-h)(x-ha).
11. a(6-cj3+6(c-a)3-f c(a-6)3 -- (a-b), {b-c), {c-a).
12. a^{b-c) + b^{c-a) + c^{a-b) -r (a-b), {b-c), (c-a).
13. a*(6-c) + 6*(c-a} + c4(a-6) -^ (a-b), (b-c), (c-a).
14. (a-6)2(c-d)3 + (6-c)3(cZ-a)2-(rf-6)3(a-.)2 ~ (a-b),
{b-c), {c-d), {d-a).
15. {{a-b)^+{b-cy' + {«-a)-'}{{a-b)^rJ + (b-c)^a^ +
(c-a)363}-{(a-6)3c-i-(6-c)2a + (c-a)2i}3 ^ (a_i), (i-c),
(e—a).
16. (:s+t/)(t/+s)(2+a;)+a;2/z-ra:+y+a.
44
THEOKY OF DIVISOKS.
18. (cib - bc-ca)l-a^b^-h2c2-c^u^-.a-\-b-c.
19. (« + 26)3 + (26-.3c)3-(3c-a)3+a34.s^3„27cS +
+2^»-3r.
20. a^b^-hb^c3^cSa3 -Sa^li^c^~<ib^bc-Jrca.
Examples. — Cors. 3 and 2.
1. Find the value of Ax'" -^-^x^ - ox^ + 23a;+6 when 1x^ = 3ar - 4.
Since 2a;2 — 3a; -{-4 = 0, we have simply to find the remainder on
division by 'Ix^-'Sx+A, and if it is iudependent of «;, it as th(
value sought. Cor, 3.
4
0
9
— 5
23
6
3
6
9
15
-3
-4
-8
-12
-20
4
2 3 5 - 1; 0 10
Hence the required value is 10.
2. What value of c will make x^ — ox^ +7a; — c exactly divisible
by a;— 2.
If 2 be substituted for x, the remainder must vanish. Cor ^.
1-0 7-6-
2-6 2
1; 2-c
1 -3
Hence 2 — c=:0, or c = 2.
3. What value of c will make 6x^ — 5x*+cx^—20x''^19x-5
vanish when 2a;2=3a;_i ?
By Cor. 8, the remainder must vanish when the given poly-
nome is divide by 2x^— dx+1. We may divide at once and find,
if possible, a value of c that will make both terms of the remainder
vanish, or we may fii'st express cx^ in lower terms in z, and
then divide and find the required value of c from the remainder.
1st. Method, (see page 28),
6-10 Ac -160 304 -160
3 18 24 12C+36 ,S6c-420
-2
-16 -8c -24
8 4c+i2 12c -140; 28c- 140
24c -{-280
•24C+12U
THEORY OF DIVISOKS. 45
Hence 28c = 140 and 24c = 120. Both of these are satisfied by
2nd Method, x^ = ix{Sx -1)= ^x^ - ix = f (3.t- 1) -^x =
2^a;-f — ia;=l|a;-i ; .-. cx^ = IJca;— ^c.
Substituting for c.v^ in the given polynome it becomes
6x^-5x*-20x^ + [lic + l9)x-ic-5.
Divide and apply Cor, 3.
6 -10 0 -160 28c + 304 -24c- 160
3 18 24 36 -420
_2 -12 -16 - 24 280
6 8 12 =ri"40; 28c -140 "^^24^+120
We thus obtain the same remainder as by the former method,
and consequently the same result. A comparison of the two
methods shews that they are but slightly different in form, but
the second method shows rather more cleaiiy tliat c need not be
introduced into the dividend at all, but the proper multiples of it
found by the preliminary reduction can be added to or taken
from the numerical remainder, and the "true remainder" be
thus found, and c determined from it.
Exercise xx.
Find the value of
1. x* -3a;3 + 4a;2 — 3a;+4, given x2 =a;_i.
2. x^-2x^-4x^ + 13x^-nx-10, given (a;-l)3 = 2.
3. 2x'' -lx'^-{-l2x^-llx" + '2x~5 given (x-l)2+2==0.
4. 3a;«+lla:5+10.'c3+7a;2+2a; + 8 given a;? + Sa:^ -2a;T5 = 0.
' 5. 6a;'+9a;*5 -16a;* -0x3- 12a;3_ 6a; + 60 given 3a;4+a;— 4 = 0.
What values of c will make the following polynomes vanish
under the given conditions.
6. a;* + 13a;3 + 26a;2+52a; + 8c, given a;+ll=0.
7. a;4-2a;3— 9a;2+2ra;-14, given 3a;+7 = 0.
8. a;* - 4a;3 - a;3 + 16a; + 6c, given a;3= a; +6.
9. 2a;*-10x2+4ea;+6, given a;2 + 3 = 3x.
10. 2a;4+a;3-7cx2 + lla;+10, given 2a; = o.
46 THBORy OF DIVISORS.
11. 4.1-4 + ra;2-f- 110a;- 105, given 2.t3 - 5a;-}- 15 = 0.
12. Sx^-lijx^-+cx--i-5x^-lUx+200, giveu x^ = 3x-4.
13. What values of /_> aud q \f ill make x'^ + ix^ -lOx^ -j'x+q
vanish, given x^ = 3(a; — 1) ?
14. What vahies of p and ^y will make a^^ -5a^" + lOrt » - 15rt '•
+29a4 -/^rt2 +,^ vanish, given (a^ -2)2 .=a2 _3 '?
Theorem II. If the i^olynome /(a;)" vanish on substituting
for a; each of the 71 (different) values a-^, a^, a^ . . . . a„
f{xY = A{x — a^){x-a^){x~a^) .... (a; — flj
in which A is independent of x aud consequently '6 the coefScient
of a;" in /(a;)".
Cor. If /(a;)" aud 9(3;)"* both vanish for the same m different
values of x, f{xY is algebraically divisible by <p{x)'^.
Examples.
1. x^+ax^+bx+c will vanish if 2, or 3, or —4 be substituted
for X, determine a, h, c.
The coefficient of the highest power of a; is 1 ;
.-. a;-'^+(/a;^+6a;+c=(a;-2)(a;-3)(a; + 4)=.a;3 -x-3 - 14a;+24.
.-. a= ~ i: b= -14: <; = 24.
2. x^+bx^+cx-\-d will vanish if —3 or 2, or 5 be substituted
for X, determine its value if 3 be substituted for z.
The given polynome =(x+3)(a; — 2)(a; — 5) ;
.-. the required value is (3 + 3)(3-2)(3- 5) = -12.
3. (ta;3+3/;a;2 + 3ca;4-rf will vanish if for a; be substituted —3,
or I;, or 1^, but it becomes 45 if for x there be substituted 3 ;
determine the values of a, b, c, d.
The coefficient of the highest power of a; is « ;
.-. aa;3 + 36.r2 + 3ra; + ^Z = rt(a;+3)(a;-i)(a;-l^)
.-. a(3 + 3)(3-i)(3-U) = 45; .-. a = 2.
.'. 2a;3 + 3/'x2 + 3ca; + <.' = 2(a; + 3)(a;-i)(a;- li)
... b = ^, c= -3i, d= 4i
THEORY OF DIVISORS. 47
4. Ir' x^ -^j'X- -\-(]x-{-r vanish for x^a or b, or c, determine p, q,
and r in terms of a, b, c.
x" -i-px- +qx-\-r={x — a){x - b){x - c)
= x^ — (a+b + c)x'^ -\-{ab + br + <-Ci)x — abc
:. p= — (rt-f /) + c) or —2".
q= ab + bc-\-ca or ^nh
r = — abc or — 2 "^c.
5. U x^ +px^ -{-qx+r vanish for x=n, or b, or c, determine the
jiolyaome that will vanish for x = b + c, or c + a, or a + b.
Since x^-{-2)x^ +qx+r vanishes for x = n or I) or c,
a;3 —px'^-{-qx — r will vanish for x= —a or —6 or — c,
and — /' -— a-\-b+c;
But the required poljmome will vanish for
x= —p—a, or —p — h, or — p— c;
that is, for x-\-p= -a, or —ft. or — c.
Hence it is (x+p)^ ~p{x+p)^+q{x+p)—r =
x^ +'lpx^ -\-{j}^ + q)x-{-pq —r.
The following is the aalculation in the last reduction. (See
page 31).
-p
9
— r
p
0
? ;
2^q-r
p
p;
p^ + q
p
1 •
2p
6. In any triangle, the square of the area expressed in terms of
the lengths of the sides, is a polyuome of four dimensions ; and
the area of the triangle, the lengths of whose sides are 3, 4, and
5, respectively, is 6. Find the polynome expressing the square
of the area.
Let a, b, and c be the lengths of the sides, and A the required
polynome.
1st. The area vanishes if any two of the sides become together
equal to the third side, hence ii a + b = c, A = 0, and consequently
A IS divisible hy a -\-b — c. Similarly it is divisible by b-\-c-a
and bv c+a — b.
48 THEOKY OF DIVISORS.
2ncl. Tha area vanishes if the three sides vanish together,
hence if a-{-b-}-i- = 0, .-1 = 0, and consequently A is divisible by
a-i-h + c.
We have .thus found four linear factors, but A is of only four
dimensions.
.-. A = vi(a + b + c){b4-c-a){c+a-b){a+b-c),
in which w is a numerical constant.
But 63 or 3G = (»(3+4+'5)(4-f 5-3)(5 + 3-4)(3+4-5)
= 576;;/ ; .•. m = ^^.
(The above includes all the ways in which the area of a triangle
can vanish, for the vanishing of only one side involves the equal-
ity of the other two, or if a = 0, b = c, and .*. a-\-h — c, which is
included in 1st. ; if two sides vanish simultaneously, the three
must vanish).
Examples on the Corollaky.
7. Prove that (x+l)'^ —j?' ^ —2«- 1 is divisible by
2x3 + dx-+x.
Factoring the latter expression we find it vanishes for x — 0, or
— 1 or — ^. Substituting these values in the former polynome,
it also vanishes. But these are different values of x, hence the
truth of the proposition.
8. (x-f ?/+ 2)"' -x^ — y^—z^ is divisible by
{x-\-}j+z)^ — x^ — y^ —z^.
The latter expression vanishes if a;= —y, so also does the former.
By symmetry they both vanish if ?/= — 2 and ii z=—x. Hence
they are both divisible by (x-\-y){y+z){z+x). But this expres-
sion is of three dimensions, as also is the latter of the given poly-
nomes, hence it is a divisor of the former.
9. Prove that { 0/4-6)^ -f(c + ri) 5} (rt- 6) (c-./) +
{{b+cy' + {a + dy}{b~c)[a-d) + {{b + dy+{c-{-ay}{b-d)(c-a)
is algebraically divisible by {a--b)(c — d)[b — c)[a ~d){b — d){c — a)
X (« + 6 + f + '0> ^^^^ ^^^ ^^1^ quotient.
Let a = b and tlie former polynome reduces to
[{a+cy+{'i-^dy'}{a-c){a-d) + {(a+d)-''-[.{c-\-n)^\{a-d){c-a)
THEORY OF DIVISORS.
49
which vanishes, the second complex term differing from the first
only in the sign of one factor, having (c — a) instead of {a - c).
Hence the former polynome is divisible by a — b, and by sym-
metry it is also divisible by a — c,hya — d,hjb — c,hyb — d, by c — d.
Again, {a + b)^ +{c-{-d)^ is divisible by (a + h) + {r -r d) ; for, on
puttinga+6= -(c-t-f/), it becomes {—{<■+<! )]■'■ +{c + d)^ which
= 0.
Similarly the other terras of the former of the given polnomes
are each divisible by a-^b-\-G+d, and consequently the whole is
so divisible.
Now all these factors are different from each other, hence the
former of the given polyuomes is divisible by the product of these
factors, i.e., by the latter of the given polynomes.
Both of these polynomes are of seven dimensions, hence their
quotient must be a number, the same for all values of a, b, c, d.
Put.< = 2, 6=1, c = 0, d=-l, and divide. The quotient will
be found to be —5.
... i^(a + hy'-{.{c + d)^}{a-b){<,--d)-\-{{h + r)^ + {a + d)^} X
{h_.c){a-d) + {{h-^d)-''+{c + a)^}{l>-d){c-o)= -o{a-b]{c-d)
X{b-c){a-d)il'-d){c~a}{a -{-!>-{-,,• -h'l).
N.B. — It is not always necessary to find the factors
of the divisor, as the following examples show.
10. Prove that x^ +x+l is a factor of 3;^ 4+j-'' + 1.
.f 2 4-x+l will be a factor o? z^'^-\-x'' +1 provided
a;i 4 _}.a;7 _f. 1 = 0 if x2 +X + 1 = 0.
Ifa;2+a;+l =0
.-. x^+x-^x =0
.-. x^+x--{-x+l = l
.'. «3 =1
.. .* = landa;i2 = l
x^ =x and x'^'^ =x'^
.. ^i4+a;'+l =.x^+x + l = ^
,■ , x'^ 4-a;-f 1 is a factor of x^ * +j;' -{-1 .
50 THEORY Od" divisors.
Art. XII. Two other methods of proving this proposition
are worthy of notice,
1st. x^+x + 1 will be a factor of x^^-\-x'' +1 provided it is a
factor of {{x'^'^+x'' +1) ± a multiple of {x^+x + l)\.
x''- "^ -\-x' -{-I differs by a multiple of x'^ +x-4-l from
x^^+x^'{x^-+x + l)+x^{x''+x-hl)-\-x'+x'^{x'-^+x + l) +
x[x'^+x + l) + l
= x^-'[x^ +x-rl)+x^{x^-\-x-\- i)-\-x'-{x-^-\-x-{-l)^ x^[x-' +x-rl) -{-
(x^+x+l)
= {x^^+x^+x''+x^-\-l){x-^-tx-^l).
Hence x^+x + 1 is a factor of «*'*+.<;• -fl.
2nd. — = . —
(x^i-l){[xi^-l)-x{x^^-'l)}
(a;^^-l)(a;'^'-l) _ x{x^^ -l){x^i -I)
{x^-l}{x^^ {x^-l}{x'^'^^;
Bitt we see at once that on reduction both of these fractions
give an integral quotient, hence {x^'*^-\-x'' -\-l) ~x^ -{-x + l gives
an integral quotient.
11. x^-\-x+l is a factorof (x + 1)'' -x^ -1.
If a;-+.« + l = 0, {x + iy -x'' — 1 will vanish also, for in such
case aj+l = —x^.
.-. {x-\-iy -X' -1 = {-X^)' -X' -1= -x-14_^7 _l^
which by the last example vanishes if x''-^-\-x+ 1=1);
.-. a;-+.i + l is a factor of [x-[-iy —x'' 1.
For X substitute — and multiply by i/^ and ?/^ respective! v
y
a.nd this example becomes
^^-\-X!J+i/^ is a factor of {x+y)'' —x'' —i/'.
THEORY OF DIVISORS. 61
Exercise xxi.
Determine tlie values of n, b, c, d, e, iu the lollowing cases : —
1. a;3 + 36.u2 4-3(M—|-// vanishes for x = 1, or 3, or 4.
2. x*^ -it-cx^ +dx + e " " a;=l^or —3 or 4^.
^. x^+hx-+cx + 'i^ " " r/; = 2or-3.
4. .^6-">-f/;a;2+rx+90. " " a; = 3or-5or2.
5. ax^+>-jyi -?>Qx + e. " " x= 1|- or -4, or 2^.
Q. Qlx'^ + iScx'+Ux + e " " a;=l| or -81- or 1^.
I. iix'^+1>x^-\-cx'^-'dl " " .^• = for|or3.
8. ax'^+cx^ + ilx+e ♦* " x = 2 or 1^ or -1 and be-
comes 14 for x=l.
9. ax^-\-cx-\-d vanishes for x=l\., or 2|, and becomes 49 for
aj = 3, determine its value for x= —Q.
Given that x^ - px^ +qx — r vanishes for x = a, or b, or c, deter-
termine the polj'nome that vanishes for
10. x = tt-[-l, ovb + 1, or t' + l.
II. .f = a — 1, or i— 1, or c— 1.
Ill
12. x=l - — , or 1 — -T", or 1 — — .
13. x = ((b, or be, or ca.
lA. x = a^, or b^, or c^.
( r )
15. x = a{b + c), or b(e + a), or c(a+b). Ui{b-\-c) = q 1.
( " )
a4-b b4-c c-\-a («+^
16. a;- — or — - or -f— . ' ^
c a b
Prove that the following are cases of exact division :
17. (a;-l)i3_a;G4-(a;2-x-i-l)2 -x^ -2:k^ +2x- 1.
18. (x-l)i«--a;« + (:^2 -a;+l)«^a;3-2a;2 + 2.c-l.
19. (a;-2)i"(2a;-5)'0-a;i" + 2iO(x2-4a; + 5)5^
a;3_6«2 + 13x-10.
20. (x2 + 4.« + 8)i«-ic'^-3;--5x-3-a;3 + Ga;2+8.f- + 3.
21. (9x-4)2 i (a;- 1)2 '-•a;2 i - {^x"- - 14a;+4)2 ^ ^(;c_ i) x
(9x-4)(9a;'^-Ma; + 4).
22. {a(.'K-l)f '3_(.2a;3 + 3a;-4)'3 + (2uj2_3;c + 2)i3^
(2a;2 +3x - 4)(2a;2 - 3a; + 2)(a;- 1).
52 THEORY OF DIVISORS,
23. {2(x+l)(x-2)}'^''-\-{x^-Sx+dy^-(3x^-ox-iy-^
{X'^lrx-2){x^ -Sx+3){dx^ -5x^1).
24. {6(a;-l)]i6-(2a;3+3x-4)i«-(2x2— 3.c+2)i« +
2{2x^+Sx-A)^2x^ -'3x+2)^-^{x-l)(2x^+dx-A)[2x'^ -Sx+2)
25. {2(a;+l)(a;- 2)[ 20 _ ,,.3 _Sx+B)^'' -('Sx^ - 5x- 1^ +
2(x^' -3x + d)\3x^-5x-iy^'^{x+l){x-2){x^ -3x-^'3) x
(3a;2-5a;-l).
26. l+x'^+x^ -^ l-f.r4-a;8.
27. a;io +0:5^5 + //! 0 -^ a;2+a;?/ + ?/2.
28. l + a;3+a;«+a;9+x'2 -^ l + x+.t:2+«3*+a:4.
29. l4-x4+ic«+.fi24-a;ie -^ i+x+x^+x^+x"^.
80. a;i^+s;iO//-5+a;5iyio^7is ^ ^3_|.;^2_,^_j.a.^2_}_y3;
31. x^' +x'>'+x^-\-x+l ^ x^+x^+x^+x + 1.
32. l+x+x'+x^+x^'+x^'+x^^ ^
l+x+x^-\-x'^+.c'^+x'-+x^.
Find tbe quotient of the following divisions in which D denotes
the product
{b-c){c-a)ia-b)(a-ri){b-d)ic-d) ;
33. (62c2 + «V2)(6-c)(a-(/) + (c3a2 4.62(Z3)(c_a)(6-rf)4-
{a^b^ +c^d'-'){a-b){c-d) -f- D.
34. (fcc+ad)(62 _c2)(«2_^3)+(ca + 6cZ)(c2-a2){62— d2) +
(ab+cd){a^-b^){c^-d^) H- D.
35. (i + c)(a+'Z)(i2_c2)(fl2_t?2)^the two similar terms -=- D.
86. (/>2+c2)(a2+r/2)(i-c)(a_</)+ ". '* -i- D.
37. {bc{b + c)^+ad{a + d)^}{b-c){a-d)+ " -^ D.
38. {bc{b + c)+ad{a + d)}{b^-c^){a^-d^]+ " -7- i*.
39. {bc(b^+c^)+ad{a^-{-d^)}{b-c){a-d)-{- " -^7).
40. (Z;+c-a-rf)^(&-c)(«-(/)+ " -^ D.
41. The sum of the fractions |, |, i, \, increased by the
sum of their products two by two, increased by the sum of their
products three by three, increased by their product is
equal to n.
^3
THEORT OF DIVISORS. O
42. Ill any trapezium the square of the area expressed in terras
of the lengths of the parallel sides and the diagonals, is a poly-
nome of foiu* dimensions, determine that polynome.
43. In any quadrilateral inscribed in a circle, the square of the
area expressed in terms of the lengths of the sides, is a polynome
of four dimensions, find that polynome.
Theorem III. If the polynome /(a:)" vanish for more than
n different values of x, it vanishes identically, the coefficient of
every term being zero.
Cor. If a rational integral expression of n dimensions be divi-
sible by more than n linear factors, the expression is identically
zero.
Examples.
{x-a){x-h) (x-b){x-c) {z-c){:v-u)
1- ^c_a)(c-/>) + {a-b){a-c) "^ (b-c){b-a) "r^-"' ^^ «'
b, and c are unequal ; for this is a j^olynome of two dimensions in
X, but it vanishes for x = a, aud, therefore, by symmetry for x=b,
and for a; = c, that is, for three different values of x, hence it
vanishes identically.
2. \{a + h)^-h{'-\-<l)^](^-b){c-d) + {(c-hh)2 + {h + d)^
{b-c){a-d) + {{c-{-'()-+('' + dy}{c-a){b-d) = 0.
Substitute b for a p.nd the expression becomes
{{b+cy^Mb+dr^}{b-~c){b-d)-h{{c+by' + {b+dy\[c-b){b-d)
which vanishes, hence the given expression is divisible by a— 6,
and consequently by symmetry it is divisible by («—?;), (b-c),
(^c — d), {a-c), (b — d), and (a — d), But the given expression isof
only four ditaensions, while it appears to have six linear factors,
hence it vanishes identically.
Exercise xxii.
Verify the following :
54
THEORY OF nwiSOBS,
1
{x+a)[x-\-h){x^-c)'
Q o+x a-\-y a-]-z ^ a^
x{x-y)(x—z) "^ y{v-x) {y - z) z{z - x) {z - y) ~ xyz
a'{h-c)+b-[c-a]+c-{a-h)
8. {iiclf+h.-f-\-he(l-acc)^-\-{hce-\-aed-^acf—hilj)- =3
{a-b){i,-c){c~<i) ~
10. (-a:+?/+z)(x — 7/+2)(a:+y— ■• -^xix — y+z^ix-Vy-z)-^
il{x+y-z){-x + y+z)+z{-x + y\-::)\x-y-\-z) = 'ixyz.
(a3-/;S)3-|-(/;3-c2)3_|_^C^_^2)3
• (a-+T)(6+c)(c+a) "" ; -
(a-i)3 + (/y-c)3 + (c-a)3,.
12. x-{y + zY-{-y-[z^xy'-^z^{x+yY+2xyz{x+y-\-z) =
2{xy+yz-\-zx)-.
Theorem IV, If the polynomes /(a;)", f (a;)'" (n not less than
m) are equal for move than n different vAliics of x, they are equal
for a// values, and the coefficients of equal powers of x in each
are equal to one another.
THEORY OF DIVISORS. 55
(This is called the Principle of Indeterminate Coefficients, The
full use of it cannot be exhibited till the student is able to work
simultaneous equations.)
Examples.
+ 71— :^7T— a71— ,T> +
{a-h){a-c){a-d) ^ {b-a){b- c){b- d)
{G-a){G-b){c-d) "*" (7i"(iy(7Z-6)(rf^j ^ ^'
Assume
{x-a){x — b)(x-c){x—d)
A B CD , >
j^ . I 1. (a)
x — a x-b x—c x—d
in •which A, i>, C, D are independent of x.
Mnltiplv by {x-a){;x-b){x-c){x-d).
:. x^ = [A 4- B-{-C-\- L')a;2 +terms m lower powers of x.
Now this equality holds for more than three values of x, hold-
ing in fact for all finite values of x.
Again multiply both sifes of (or) by a;—;6
x^ ,, I B C D \ , .
(x — b){x — c){x — d) \x — b x — c x — d
Put x = «
(a — b){a — c){a -d)
By symmetry — ^ = b, &c.
{u—,a)[b — c)[b — d)
Adding
«2 62 c2
+ 7Y. :Y7, x-71 T^ +
{a-b)\^a-c){a-d) ^ {b-a){b- c)[b -d) ^ {c — a){c-b){c-d)
+ id^:aji^b)ld-c) = A^B+C+D = Ohyi&).
2 a^(a + b)ia+c) b^{h+c)ib+a) c^{c+a){c + b)
{a-b){a-c) "^ (b-c){b-a) "^ (c-aXc-6)
;=(a-f 6 + c)-,
58
THEOHY OF DIVI90ES.
17 a(^ + h){a+c){a + d) ■ -, .
*■ ' • 7 7w w T\ + three similar terms*
a-(a+b)(a+c){a^d) ^^ ^^
•'°' {a-b){a-c)(a-d) "*"
aS(a+b)(a+c){a + d) ^^
^^- {a-b)[a-c){a-d) "^
bc{b-\-c)
20. , Y^, - , +two similar terms.
[For numerator use x^-\-2px--\-ip^-^q^x-\-('jiq - r).]
(2a+6:(2a + c) . '
21. -7 fw r- + two similar terms.
(a— f>)(a— c;
[For numerator use x^ —'lpx^^^^(J)^■\^q)x — (yq — r^^
+ two similar terms.
" {(.t — b){a—c)
[For numerator use a;(a; + ;j).]
h-\-c.-\-d ■
23. 7 Tw \7 j\ + three similar terms.
{a — o){a — c){a —a)
a^(hc-\-cd + db)
"^- (a-6)(a-c)(a-rf) ^
hc-i^cd-\-db
"^^^ (fl--6)(a-c)(a-J) ^
Extract the square-root of (to 4 terms) : »
26. l+x. .1 27. 1-a;. I 28. l + 2.^+3a;*+4z3 -t &c.
29. l-4a;+10x2-20a;3 + 35a;4-56ic5+84a;«.
30. Extract the cube-root oil+x. (To 4 terms).
Art. XI. 1. Find the condition that px^' + ^qx+r andi p'x^
+ 2q'x+r' shall have a common factor.
Multiply the polynomes hj p' and p respectively, and take the
difference of the products, also by r' and r respectively, and
divide the difference of the products by x.
p 'px^ -{- 2p 'qx 4-/J 'r
pp'x^ 4-2pq'x+pr'
2( pq'—p'q)x+{pr' —p'r)
p7-'x^ +2qr'x-\-7r'
p'rx^ + 2q'rx-{-r'r
(J)r'-p'r)x + 2{qr' -r'q).
Multiply the former of these remainders by (pr'—p'r) and th$
latter by ^{pq'—p'q), ^'^'^ the difference of the products is
^^r'-p'r)^-^pq'-p'q){(p--r'q).
THEORY OF DIVISORS. C9
But if the given polynomes have a linear factor this remainder
must vanish, or
ipr' —p'r)^ =4:(pq' —p'g){qr' — r'q).
If the given polynomes have a quadratic factor, the linear re-
mainders must vanish identically, or (Th. III.)
pq'—p'q — O, pr' —p'r = Q, and qr< — r'q = 0,
par
ox, ±- = J- = —
p' q' r'
2. Find the condition that px^ -\-3qx^ +Srx-^s shall have a
square factor.
Assume the square factor to be (x — a)^. On division, the
remainder must be zero for every finite value of x,- and conse-
quently (Th. III.) the co-efficient of each term of the remainder
must be zero. Divide by (x — a)^, neglecting the first remamder.
P Sq 3r $
a * pa pa^ -\-dqa
p pa + 3q pa^-{-Sqa-\-3r ; R
a pa. ^pa"^ +dqa
\~~p ^a-\-3q ; 3{pa^ -\-2qa+r)
.•. pa^-\-2qa-\-r = 0;
:. joaj^+^aj+z- is divisible by a; — a (Th. I. Cor. 2),
or, px^ +3qx'^ -\-3rx-\-s and jDa;2+25a;+r have a common divi-
sor. Multiply the latter polynome by x and subtract the product
from the former, and the proposition reduces to
lipx'^-\-3qx^ -\-orx-{-s have a square factor, ^;a;--f-2g'a;-|-r and
qx--\-^lrx-\-s will have the square-root of that factor for a com-
mon divisor.
3. If joa;3 + 3ga;^4-3ra;-|-s vanish for a; = a, or h, or c, find in
terms of x, p, q, r the value of
X — a x — b X — c
Eeduce to a common denominator and add the numerators
• 3x^ -1{a + h + r)x + {ab -irhc+ca)
[x — a}{x — h){x — c)
60 THEORY OF DIVISOES.
Multiply both numerator and denomiuator by p and reduce by
Tii. II., and Ex. 4 of Th. 11.
px^ -\-''dqz'^ -\-'6rx-\-s
a.m+1 y,m+\ rj,m+l ^ 'd{px'"-+^ + 2qx''^+'^-\-rx:^+'^)
' ' X — a x — b X — (■ px^ -\-'iqx^ -\-or.c-\-s
4. li 2}x^ -VQqx^ -\-^rx-\-s vanish foi- x = a., or h, or c, express in
terms of/>, q, r, s, the following, a + 6-}- e, a~-\-h~ +c^, a^ + h^ -\-c^
, a"* + 6'" +<;'«.
Divide a;"'+^ bv x — a.
1
a
a ft' a^ ft™ a"^+i
1 a a2 ^s ^m . a"^+^
Similarly divide x''"'+'^ by x—b and also by x—e.
add together the quotients
/^m+l r^m+1 ajW+l •
1- ■ 7 + =3x"' + (a + b + r)x'"-' + [a- + b^ + e^hf-^
X— a .V— h X — G ^ ' ^ '
+(rt.34-63+c3)a;'"--rf- ,1'c.
Hence, by tlie last example, the required expressions are the
coefficients taicen in order, beginning with the second, of the
terms in the quotient of 3(/>a;'"'+3 +29'a;""''+^+rx"'+^) -=- {iix^ -^-^qx^
+ 3/-a;4-«). These may now be found by Horner's Division.
5. Writing s^ for a+/'+c, s^ for ii~-\-h"-^c-, &c., express
(a— 6)4+(6— c)4 + (c — fl')4 in terms of s^, .Sg, Sg, s^.
By actual expansion
{x-a:)^-\-{x - 6)4 + (a;-r)4 =
3x4-4(a+/. + c)a;S+G(«2+i2_j.c3)a;2_4(„s_|.^3^c3)a-4-
<j4 ^ ^4 _i_ (:4 = ga.4 _ 4s^a;3 -f Gsgic^ - 4.v3rc + s^.
Puta; = «, = i, =c in succession.
(a_6)4 _|_(c-fl)4 =3^?4 _4s^a^+6s2fl2_.4.<,^,,^,,_^
(ft_c)4 _[_((._rt)4 =3r4 -4SiC3+6s2c2 -4s3r+S^
... 2{f«-&)4 + {i-r')4 +(f:_a)4}=3.S4-4.s,.^3 + 6.s:--4s3S,+3.v^ .
in which s^ is written for 3 or 1 + 1 + 1, i.e., a^ + Z^o+'j".
THEORY OF DIVISOBS.
61
Exercise xxiii. (a).
1. Determine the condition necessary in order that x^ frV^-\-9.
and x^-{-p'x+q may have a common divisor.
2. The expression x^-\-Za^x^+Zbx*-{-cx^-\-Mx--\-^e^x-\-P
will be a complete cube if
e d c — a^
•^ ~ a ~ b ~ 6a2 ~ ~
3. Prove that ax^+bx + c and a+6a;*-f ca;* will have a common
quadratic faekir if
Z)2c2 = (c2 - a2 +&2)(c2 - a^+ah).
4. Prove that ax^+bx^+c and a+bx^+cx^ will have a com-
mon quadratic factor if
«362 = (rt2 _ cS)(«2 _ c3 +5c).
6. Prove that ax^+bx^-^cx+d and a+bx+cx^-^dx^ wiilhwe
a common quadratic factor if
{a-\-d) 3 = {b-c){bd-ac).
6. x^ +px' +qx+r will be divisible by x'^+a^' + b if
a^ -2pa^-\-{p^+q)a-}-r—pq = 0, and b^ —qb^ -t rpb — r^ = 0.
7. a;*4-pa;-4-5' will be divisible by x^ 4-ax-\-b if
rt6 - 4?a3 =^3 and {b^- -¥ q){b^ - qY =JJ-bK ^
8. Determine the condition necessary in order that x* +4pa;3
-{•Qqx^+Arx+t may have a square factor.
JI x^+Apx^ + Qqx^ -jr^Li-x+t vanish for a; = a, or b, or c, ovd,
find in terms of x, p, q, r, t, the value of
a;" x" a;" a;"
x — a x — o x—c x — d
10. 2 a, 2a^ ^a^, S«*, 2 «^ 2 a".
11. 2(a-&)S S(a-fc)*.
12. Determine the values of the expressions in Ex. 9, 10, 11, for
the poiynome a;* — 14a;^ +a; — 38.
62 rAOTORING,
CHAPTER in.
Section I. — Dieect Application of thk Fundamental Formulas
Formulas [1] and [21. {x±yy =x^±1xy-\-y^, &c.
Art. XII. From this it appears that a trinomial of which the
extremes are squares, is itself a square if four times the product
of the extremes is equal to the square of the mean, and that to
factor such a trinomial, we have simply to connect the square
root of each of the squares by the sign of the other term, and
write the result twice as a factor. •
Examples.
1. 4a;4-80a;3//2 + 400?/4 = (2a;2-20y2)(2a;3_!20?/3)
2. l-12cc2?/24-36a;42/4^(l_6a;22/3)(i_6ic2y2).
3. (a_6)2_|.(5_^)24.2(a_6)(&-c). This equals (a- i + ^-c)
X (a — 6+fe — c) = («— c)(a— c).
4. x^ ^xj^ -li-z^ -\-'ixy -1xz--%jz.
Here the three squares and the three double products suggest
that the expression is the square of a linear tnnowial in x, y, z.
An inspection of the signs of the double products enables us
to determine the signs which are to connect x, y, z: we see that
ist. The signs of x and y must be alike,
2nd. The signs of x and z must be different.
3rd. The signs of y and z must be different. Hence we have
^^y—z, or —x—y-\-z= - i^x-k-y-z), and the factors are
{x+y-z){x + y-ri).
Exercise xxiv.
1. 9m 2 + 12m + 4; c2"'-2c"' + l.
2. ?/6_2?/323+z«; 16a;2v/2 + -iGa:i/3 + 47/4.
i). 9^262 4.i2«^c-i-4c^ ; 'di5x'y^-'lixy^'\-iii'^
FACTORING. 63
0. (a4 6)2+c3_2e(a4-6) ; S)^'^ - fx4y2 4-^j^y4.
/ a \ -'" / 6 \ ^^
G. ^2^.^^_,^)3_2^(x-y); jy) + (-] -2-
8. (rc^— x'?/)2— 2(cc2 —xi/)(x)/ — y^) + {x}j — y')^.
0. (a + ft+c)2-2c(rt+/)-fc)-trc2 ; ff./?'^-2/>3(^2^.y.^4.
10. (8.c-4/y)3 + (2a;-3//)- -2(3a;-4)/)(2ic-37/).
1.1. (a;3 -.,;y-i-y2)3 + (a;3 +^^ + i/2)2 +2(s;* +.-^2^3+7/4).
12. (o.r2 + 2a;?/+:.73)34.(4x-24-6//2)8 _2(4a;'^+6?/3)x
■5x^J^2xy-h7y--^).
13- (t) +(t) -Mt) •
14. a? + 63_j.c2_2fl6-26c+2rtc.
15. a4 + i4+c4-2a-53_2«3,;3+262c2.
16. {a- i)2 + (t - <')^ + (f; ~ '0^ -f- ^(« -- &)(6 -c)) - 2(a-5)(c-a)
+ 2(?>-c)(«-c).
17. 4^,4 _ 12^26+ :)/;3 +.16rt3c + 16o3 -24&C.
ForjiutjA [4]. a;3 — ?/3 = (cc-}-2/)(a; — ?/),
Art. XIII. In this case we liave merely to take the square -
foot of each ox the squares, and couueet the results with the sign
4- for one of the factors, and with the sign — for the other.
Examples.
1. {a-\-b)''-{c->rd)^.
This={(a+6) + (^+(0}{(«+6)-(c+fO}
= (,',+6 + 'j + d)(a + 6-c— d).
2. Factor {x- +^rij-\-i/^)^ -{x^ —xy-^y^)^.
Here we hav^e
{{x^- + oxy+y^) + {x^-x!i+y^-)]{{x"-+5xi,-^y^)-{''^-xy+y^-)\
This = a2 - (i> - cf^ = (rt + 6 -c)(«— fc+c).
64
FACTORING.
4. EeSolvG (a2 4-53)2_(<j8_52)2_(„2+ftS_c3)2.
This = 4rt2/,2_(a2 + /,2_,.2')2
The former of these factoi-s = (a + i)"-^ — c^ = (^a + h-\-<:)(a-}'h — t') ,
and the latter = c2 — (a — b)^) = (c+a — b){c — c(,-{-h).
.•. the giveu expression
= {a+b+c){a+h — c){c-\-a—b){c--a+b).
Exercise xxv.
1.
49^8 _452^
9.
81a4-l.
2.
9a2_ife3.
10.
ai-16b*.
3.
81a4_i664.
11.
a»e-6i6.
4.
100a;2-36y2.
12.
a2_52^26c-c*.
5.
5rt.2&-20&a;22/a.
13.
(rt+26)2-(3a;-4^)-=.
6.
9x<5- 162/4. »
14.
(0,2+^3)2 _ 43,2^2.'
7.
9 ^2 1
y^C — i.
15.
{x + yf-iz^.
6.
4]/4-|a;2z3.
16.
{Sx+5y-{6;e-i'd)^ -
17. 4a;22/2-(a;2+y2-z2)2.
18. (^x-2+xy-y-'y -{x^- -ary-7r-)\
19. (»2_,y2+22)2_4.,.223.
20. {a+b-\-c+d)^ -{a-b+c-dy.
21. (2+ya;+4a;3)3_(2-3a:+4:c-')».
22. («2+62+4a6)2-(a2+i3)2.
23. {a^-b^+c^~d2)^-{2ac—2hd)^.
24. (a;3_y2_ 23)3 _ 4-^223.
25. (a6-a3t3^iC)3_(^C_5fi3/;3_j_i8)2.
•20. ai2_/>i2 + Ga9/>3_6i9a3 + SZ?»fi3-8r/S6».
27. (a;24-?/2+23 -xy — ijz.—zx)'^ — {xy-^yz+zx)*.
!i8. (ic^ +7/2 +22 - 2x,y + 2x2 - 2?/^)-(;?/+s)2.
29. 2a2i2 4.2&2^.2+26-2a3_^4_i4_,;4.
SO. .S4 +^4 _i.24 _ 2::2^2 _ 2^223 - 2»3a;2.
PACTOEING. 66 ,
FoxiMULA A. (x->rr)(x+s)=X' + {r + .i)x+r$,
Examples
2. {x^ij)^+x-y-110 = {x-7j-^h){x-y-10).
^(aS _ ab + b2)-^{2a + Sb)} {(fl3 _rt?;_{-62) _ (2«- 3&)}.
4. (a;2 - 5a;) 2 - 6(x3 - 6x) - 40 = (a;3 - 5a;+4)(a;2 -ox- 10).
5 . («a; + i?/ + c) 2 — (77?, — w) («.r ■+by-\-c)—mn
= [ax-i- by -\-c —ni){ax+by + c-{-n).
Art. XIV. It will be seen that the first (or common) term ot
khe required factors, is obtained by extracting the square root of
the first term of the given expression, and tlfat tlie other terms
are determined by observing two conditions :
(1) Their product must equal the third term of tlie given ex-
pression.
(2) Their sum (algebraic) mnltiplied into the common term
already found, must equal the middle term of the given expres-
sion. Hence, to make a systematic search for integral factors of
an expression of the iormx^±bx + c, we may proceed as follows :
Ist. Write down every pair of factors whose product is c.
2nd. If the sign before <; is +, select the pair of factors whose
sitm is b, and write both factors x+,ii the sign before 6 is + ; a; — ,
if the sign before 6 is — .
3rd. But if the sigh before c is — , select the pair of factors
whose difference is b, and write before the larger factor x+ or a; — ,
and before the other factor a;- or a;+, according as the sign be-
fore 6 is + or — .
Examples.
1. a;3 + g.x-l-20. The factors of 20 in pairs are 1 and 20, 2 and
10, 4 and 0. 'The sign before 20 is +, hence select the lactors
whose sum is 9.. These are 4 and 6. The sign before 9 is +,
hence the required factors are (a;+4)(a;-f5).
66 FACTORING.
2. a!2_8a;^.i2. Pairs of factors of 12 ore 1 and 12, 2 and 6,
3 and 4. Sign before 12 is +, therefore take pair whose sum is-
8. These are 2 and 6. Sign before 8 is — , heiice the factore
are (x — 2)(x— 6).
3. a;2-21a;-100. Pairs of factors of 100 are 1 and 100,
•2 and 50, 4 and 25, 6 and 20, 10 and 10. Sign before 100 is - .
therefore take the pair whose difference is 21. These are 4 and
25. The sign before 21 is — , therefore x— goes before 25, the
larger factor, and the factors are (a;+4)(a; — 25.)
4. a;2+12a;-108. Pairs of factors of 108 are 1 and 108,
2 and 64, 3 and 36, 4 and 27, 6 and 18, 9 and 12. Sign before
108 is - , therefore take the pair whose difference is 12. These
are 6 and 18. Sign before 12 is +, therefore x+ goes before 18,
the larger factor, and x — before 6, the other factor ; hence the
factors are {x — Q)){x-\-lQ).
Note. — It will be found convenient to write the factors in two
columns, separated by a short space. Taking Ex. 2 above, pro
ceed thus : Since the sign of the third term is + , write the sign
of the second term (in this case — ) above both columns.
1 12
(a!-2) (x-Q)
Ex. 3 above. Since the sign of the third term is — , write tbe
sign of the 2nd term (in this case — ) above the column of larger
factors, and the other sign of the pair +, above the other column.
+
1 100
2 50
(a;+4) (a; -25)
6. a;'»-84a;4-64.
Here we have the factors
1, 64
x-% a;-32
4, 16
and since the last term has the sign -+•, and the middle term has
the sign — , we write — over both columns.
FACTORINOc 57
6. a;3+12a;-64.
+
1, 64
2, 32
X-4:, x + lQ.
Here, since the last term has the sign - , we write the sign
( + ) of the middle term, over the column ol larger factors, and
the sign — over the other column.
7. jc*- 10x3 -144.
Here we have the pairs of factors :
+ -
1, 144
2, 72
4, 36
x + S, u;-18.
And since the sign of the third term is — , we write the sign ol
the second term (in this ease -) above the column of larger
tactors, and the other sign (of the pair +) atove the otiier
column.
Exercise xxvi.
1. x-^-^5x-U; x^-Qx+U; a;2+7;B+12.
2. a;2 -8x4-15 ; x^ - 19x4-84 ; x^ -7x-60.
3. 4x3-2x-20; 9x3-150x4-600.
4. ix3 4-4*x- 36 : 25x'^ 4-10x4-15; 9x«-27x34-20.
y5. ^^x^ + Hx+12; 16x4-4x3-20.
^-^6. ^4_u,34-63)x3+.y2/,2; 4(x4-2/)2-4(x4-^)-09.
7. (a;2+.v-^)3-(a2_fe3)(a;2+y2)_a2i2.
'^7'(:;rHp4ij[#+2/3)(x+2^
10. {a+bf-iab{a + b)-{a^-b^f.
11. (x3 4-x//4-!/2)2+x3-?y3_5xi/-2^3-2x».
12. a-^-<ia{b-c)-^{b-c)\
68 FACTOKINa.
y^ 3:3. (x^+y^)^-h^a^{x^-\-y^) + a^~b^.
14. (a;2-10aj)3-4(a;3-10a;)— 96.
15. (a;2_l4a; + 40)2-25(a;2-14a;+40)-150.
16. {x^ -x>j + :i^)^+2xy{x-^ -xij-\-y^) ~3x^y^.
17. z4-3z2 + 2; x^ -2x^-3: 9x^+9x^y^ -lOy^.
18. c^'" + c™ - 2 ; a;« - a;3 — 2 ; a;^'" - 2a;"'2/» - 8^/'".
19. a;^"— (a — b)x"Y - « V-
Art. XV. Trinomials of the form ax^ + bx+c (a not a squai e)
may sometimes be easily factored from the following couaiJera-
tions : —
The product of two binomials consists of
1st. The product of the Jirst terms.
2nd. " " second '«
3rd. The sum (algebraic) of the products of the terms taken dia-
'^oxially.
Tnese three conditions guide us in the converse process ol
resolving a trinomial into its binomial factors.
EXAIIPLES.
1. Resolve 6ic2-13ar7/+ 6^/2.
Here the factors of the first term are x and 6x, or 1x and 3a; ;
those of the third term are y and Qy, or 2y and oy. These
pairs of factors may be arranged
(i) (2) (3) (4)
» 2» y 2.i/
6a; 3x 6?/ dy
Now, we may take (1) with (3) or (4), or (2) with (3) or (4) ;
but none of these combinations will satisfy the third condition.
If, however, in (4) we interchange the coefficients 2 and 3, then
(2) and (4) give
2a; 'dy, and
3x 'ly, where we can combine the " diagonal"
products to make 13, and the factors are
FACTORING.
69
2a; — Sy, and
Sx - 2y.
The coefficients of (2), instead of those of (4), might have been
iuterchauged, giving the same result.
2. ea;2-15a;?/+6y2.
Here, comparing (2) and (3), Ex. 1, we see that their diagonal
products may be corabined to give 15, and the factors are
Ix—y, and dx—Qy.
3. Qx--'iOxy+%y^.
Here, again referring to Ex. 1, we see at once that it is useless
to try both (2) and (4), since the diagonal products cannot be
combined in any vv'ay to give a higher result than IQxy. But com-
paring (1) and (4), we obtaiB by interchanging the coefficients
in (4) x—oy, and
6a;— 2^, which satisfy the third condition.
Or, v/e might interchange the coefficients of (3), and take the
resulting terms with (2), getting 2x—Gy, and
8x- y.
4. iJx^ -i-Sbxy — 6y^ .
Here the large coefficient of the middle term snows ai once
that we must take (1) and (3) together. Interclmnging the co-
efficients of (1) we have
6x— y, and
ai + 6?/. The same result will be obtained by inter-
changing the coefficients of (3).
Exercise xxvii.
1. 6a;2_37a;?/ + 6?/2.
2. 6x2-f9x(/-6;/-.
3. 56a;2-76a;?/+20?/2.
•i. 56a;2-36x2/-20y2.
5. 56x2-1121a:*/+207/2,
6. 56a;2-68£c?/ -1-201/2.
7. 56a;2-558.-c(/-20?/2.
8. 5Qx^ + 'dQxy-2Qy-.
9. 56a;2 -67x^+207/2.
10. 5e:i;2+3.c^-20i/2.
11. Qx^ -IQanj — Qy'^.
12. 6a;3+5a;//-67/2.
13. 56a;-^-H562a;//+ 207/3.
14. 56a;--122ic/y + 20?/^
15: 56a;-'-102a;i/-20?/2.
16. 56x2 -229a;?/ +20^/2.'
17. 5 Sa;3 -94x^+207/3.
18. 56x2- 276a;)/ -20//3.
19. 3Gx- — 33x^ — 15?/2.
20. 72x3- 19x1/ -4U(/2.
70 FACTOBINO.
Art. XVI. More fjeneralbj, trinomials of the form ax^-\-hx-rO
[a not a square) iaay be resolved by Formula A, thus
Multiplying by a we get a^x^ +bax+ac. Writing z for ax this
becomes 2=^+ '!/z+'i';. Factor thid trinoaiial, restore the value of
z and divide the result by a.
Examples.
1. 6.t3 -\-5x-4:. Multiplying by 6, we get (6a-)2 +5(Ga;) - 24 or
2^ + 02 -2i. Factoring, we get (-2;-3){c + 3), hence tlio required
factors are ^(6x-'d)(Qx+'S) = {2x—l){\ix+-±}.
2. 6a;2 - 13x1/ + 6»/3 . Factoring z^ - Idzy + 36?/3 we get {z - 4t/)
{z — 9y), hence the required factors are i«(Da; — -i^j(6a; — %) =
{3x-2y){2x-3y).
3. 33-14a;-40a;3. Factoring 1320- Ue-g^ we get
(^30 — 5;)(444-z), hence the required lacfcors are ^^^(30- 40a;) x
(44+40^) = (3- ^^(ll + 10.'-c).
NoxB. — The factors may conveniently be arranged in two col-
umns, each with its appropriate sign above it.
+
Ex. 1, above 1 24
2 12
^(6a;'-3)(6a:+8) = (2a:-l)(Sx-f4).
Ex." 2, above 1 3G
2 18
3 12
^ (6a; - 4) (6x - 9) = (3j; - '2 ) (2» - 3).
[Another method of factoring trinomials of the form ax'^~{-lx-\'e
is as follows :
Multiply by 4a, thus obtaining 4:a^x^ +4ff&^-+4ac. Add 6* - 6*,
vhich will not change the value, Aa^x^ + 4:abx+b' —b^ + iac ; by
[1] this may be written {2ax+by~-—{h^ — 'l:ac). Factor this by
[4] and divide the result by 4a.
FACTORIXO.
71
Ex. Factor 56x» + 137a; -27885. Multiply by 4x56 or
2x112, 1122ic3 + 2.137.1120;— 6216240. Add 1873-1372, then
il2-'.«3 + 2.137.112.^+1373- (1373 +P.246240) = (112^- + 137)3 -
6265009 = {{112a; + 137)+2503}{(112.r+ 137) -2503} =
(112a;+2640)(112.c-23GG).
Y/e multiplied by 4 x 56, we must, therefore, now divide by that
number. Doing so, wc obtain as factors (7a; + 165)(8.£— lCij).j
Exercise xxviii.
1. 10a;3+a5-21
2. 10.c3 - 29.e - 21.
3. 10.«2 + 29a;-21.
;. 6a;3-37a;+55.
5. 12«3_5«_2.
6. 12a;2-37a;+21.
12*2 + 37a; + 21.
ISaS + lSaS^S-OOS*
7.
9. 12.r^-a;-l
10. 'dxhj^ -'dxy^ -e,y<^,
11. 4x3+8.v^+3//-.
12. 662x2 -7ia;3-3.r*.
13. &x^-x^y^
14. 2a;4+a;3_45.
•35//*.
■%*.
15. Ax^~'61x-y'^-
10. 4(2; + 2)* -87a;2(:/j+ 2)2-1- 9a;*.
17. 6(2a; + 37/)3 + 5(Ga;3 + 5.r//-6//-)-6(3a;-2?/)3.
18. 6(2a;+3?/)* + 5(6.j;3 + 5is,//-6//-)2 _G(3a;-2,v)*,
19. 6(a;3+a;//+y2)3+i3(a.4+.^2^^2+y4)_385(a;^^-a;'/+?/2)3.
20. 21(a;2 + 2a:^+2j/2)3-6(ic-^-2x^+2?/2)3-5(a;4+-i^-i').
Section il. — Extended Application of the Formulas.
Art. XVII. The methods of factoring just explained may be
a-pplied to tind the rational factors, where such exist, of quadratic
multinomials.
Examples.
1. Eesolve 12u;2_a;//-20?/2+8a;+41y-20.
In the first place we find the factors of the first three terms,
!7hich are
Ax+5y, and
Bx — Ay.
Now, to find the re^naining terms of the required factors, we
must observe the following conditions :
72 fACTORINd.
1st. Their product must = — 20.
2ad. The sum (ah/ebraic) of the products obtained "by xuulti-
plyiug them diagoually iuto the y's, must ==41?/.
3rd. The sum of the products obtained by multiplyiug them
di.agonally into the x's, must =8a;.
We see at once that —4 with the first pair ah'eady found, and
+ 5 with the second pair, satisfy the required conditions, and .".
tlie factors are
4:X+5y -4, and
8a; — 4?/ 4- 5.
Here the factors oi p^+ V'] —^q^, are
p+2^, and
p — q.
Now find two factors which will give - 3r-, and which multi-
plied diagonally iuto the p% and qs, respectively, will give 2pr.
and Iqr ; these are found to be — r taken with the first pair, and
+ 3r taken with the second pair. Hence the required factors are
j9 + 2g— r, and
■ ■ . /
Art. XVIII. But the following examples illustrate a surer
method.
3. ic2+a;?/-2?/3+2a:3 + 7?/*-32».
Reject 1st the terms involvings,
2nd. " " 2/,
3rd. " " jc, •
and factor the expression that remains in each case.
1st. x^-{-xu-2y^ = {x~y){x+2y).
2nd. x^+2xz-Sz^ = {x+Sz)[x-z).
8rd. -2//'^+7//2-3z3 = (-y+Sz){2y-z).
Arrange these three pair of factors in two sets of three factors
each, by so selecting one factor froin each pair that two of each
act of three may have the same coefficient of x, two may have the
?ACTORIN(J. 73
game coefficieut of y, and two the same coefficiant of z {mefficimi
including su/n). In this example there are
X— y, x + 3z, — y-\-?iZ,
and x-\-1y, x— z, ^y—z.
From the first set select the common terms (inchuling signs^
and form therewith a trinomial, x—y + Bz.
Eepeat with the second set, and we get x-\-2y—z.
:.x^+xy-2y^ -\-2xz-\-7yz~Sz^ = {x-y-{-3:){x-\-2y-z).
4. 3a;2-acy-8//2-i-30a;+27.
1st. Sx^-8xy-3y^ ={Bx + i/)(x--By).
2nd. 8a;2-L30x + 27 = (3a;+3)(a;+9).
8rd. -%2 ^_27 =(j,^-3)(_3y + 9).
.'. the factors are {3x+y+'d){x — 3y-\-2).
6. 6a2-7a6 + 2rtc-2062 + 646c-48c3.
1st. 6«3_ 7«6-20Z)3 ={2a~5b)(3a-i-'ib).
2nd. 6rt3+ 2rfc-48c3 =(2r«+6c)(3a-8c).
8rd. -2062+ 646c -48c3 = (_.56 + Ge){46-Sc>,
/. the factors are (2a - u6+Gc)(3a -j- 46 — 6c).
Exercise xxix.
1. 7x^~xy-6y9-6x-20y-16.
2. 20x2-15a;?/-5t/S_68a;-427/-88.
3. 3«4+a;27/3_4y4 + i0a;2_i7^2_i3,
4. 20a;2- 20^3 _^9.i-?/ + 28^4-35?/.
5. 72a;2-8?/3 + 55a;?/-}-12!/— 169a; + 20.
6. x^ —xy— 12y^ — 5x-~15y.
' 7. 8a;3 + 18a;(/+9^3_^2a;2— s*.
8. 6x^+6y^-13xy-8z^-2yz-{-Sxz.
9. 6a;*- 10?/^ + lla;2;/2 -2532 -[-107/2 +25!/2z2-15:K2^10a;2z3
10. 16.c4 -16^4 - 22a;3j/2 j. 1524 _[. 14^222 ^. 50a;222,
11. 4rt3_i552_4aj_21c2 — 366o— 8ac.
12. «* + &4+c*-2rt»62-263c2-2c2a».
OFTHc
lift.ii<<___ .
74 FACTORING.
Art. XIX, Trinomials of the form ax^ + bx^ + c can always be
broken up into real factors.
If a and c have different signs, the expression may be factored
hj Art. XVI.
If a and c are of the same sign, three cases have to be consid-
ered : i. 6 = 2v/(ac), ii. 6>2v/(ac), iii. 6<2v/(rtc)
Case I, b = 2y^{ac). This case falls under Art XII., formula
[1] . where examples will be found.
Case II. i>2y(ac). This case falls under Art XVI., where
examples will be found. The following additional examples are
resolved by the second method of that article.
Examples.
Here we see that (^y^)^ will make, with the first two terms,
a perfect square, and we therefore add to the given expression
(fy^)'~(f!/^)'- T^^® expression then becomes
= (2x3+12/2 + 12/ =^)(2a:»+|-y»-f?/2)
= (2a;= + 22/3)(2a;' + ij/^') = {x^ + r/^){4x^ +y").
2. 3a;* + 6x2 +2.
Here multiplying by 4x3, and completing the square as in
Ex. 1, we have
86x* +72^2 + 62 + 24 -62 = (6x2 + 6)2 -12 .
= (6x3 4-6-i/12)(Gx=+6 + i/12), which divided by 4x3 give
the required factors.
8. ax^+hx^+c.
Proceeding as in Ex. 2 we have, by multiplying by 4a,
ax^^bx- +c = {ia^x^-^iabx" +63 - b^ +4ac} -• 4a
=:{2ax^+b + y/{b^-Uc)}{2ax^+b--i/{bi-4.ac)}^U.
FACTORINO. 75
Exercise, xxx.
1. ic^+TxS+l; 4a;4 + 14a;3+l.
8. 40:4+10x3+3; S{xA-y)^ + oz^{x+yy +2'^.
4. a;4 + 7cc32/2+3J?/4; x^ +7x2^/3+8^7/4.
5. 4x4+9xV+tI^*; 4(rt+6)4+10c2(a+i)2 +-3c*.
6. 3x^+8x2i/= + 4-T\jr*; 36x^+96x3+55.
7. 5x*+20x3+2; 4a* + 12rt3 + l.
8. 4(x+?/)* + 12(x+?/)^z3+24; 5.r4+20x3j/2+2T/*.
9. 9x4 + 14x2+4; 2x* + 12x3(f/+2)2 + 15(t/+z)4.
10. 2x4+12x3 + 15; 7x4+40x3+45.
n. 8x4 + 36x3?/2+29?/4: 7x4+20x-^2 -20?/*.
12. 7(a-i)4 + 16(a-5)36-2+5c4; ^a^ + SaH^+b*.
13. 8x4 + 6x^vM-2//4; S(a+by+Q(a^-b^y +2(a-b)K
U. 49a4_84a2i'^+2264 ; 25TO4-{-60w2n3 4.27«4.
15. 49(m+7i)4-84(»i2 -»43)3+22(m-M)4.
Case III. &<2;/(ac). This case may be brought under
Art. XIII. The following examples illustrate the process oi re
duction and resolution.
ExAJfPLSS.
1. x4 -7x3+1.
"We have to throw this into the form a* — 6' :
a;4_7a;3 + l = (^2 + 1)3 -9x3 = (x3+H-3x)(x- +1 -Sa;).
2. 9x4 + 3x3?/2 +42/4 = (3x3 + 22/3)2 -9x3y3
= (3x= +2^/2 -8x>/)(3x3 + 2^/2 +3xy/).
3. x4 +2/4 = (a;S +2/3)2 _2x3|/S
= (X3 +J/2+XV ,/2)(x3+2/3_x2/-/2).
4. x4 -ixs^/^* +1/4 = (x^ +2/3)3 -la; V •
= (x-^+y3 -^%xy){x^ +.v2-i^?/)-
5. «x4+ix3+c = (,/«. a;3 + |/c)3-{2v/(ac)-6}a;»
= {ya. x3+|/6--|/(2;/^-i)x}x
il/a. x2+i/c+v'(2v^^— ^)a;}.
76 FACTORING.
Art. XX. It is seen from these examples that we have merely
to add to the given expression what will make with the first and
last terms (arranged as in Ex. 5) a perfect square, and to subtract
the same quantity. In Ex. 2, e. g., the square root of 9z'^ = 3x^,
the square root of 4t/* = 2v/ 3, /. 3a3^+2?/2 is the binomial whose
square is requii'ed ; we need .". 12x^1/' ; but the expression con-
tains Sx^y'-^ ; .". we have to add and subtract 12x^y'^ - ox^'y^ =
Hence we derive a practical rule for factoring such expressions.
(1). Take the square roots of the two extreme terms and con-
nect them by the proper sign ; xhis gives the first two terms of
the required factors.
(2) Subtract the middle term of the given expression from
twice the product of these two roots, and the square roots of the
difference will be the third terms of the required factors.
6. x*+^\x-y^-\-y'^. Here ^/a;* =a;^, l/y* = i/2, and the first
two terms of the required factors are x^-\-y^ ; twice the product
of these is -\-2x^y^, from which subtracting the middle term,
Y^x^i/^, we get xe^^i/^ j *^^ square roots of this are +izy.
Hence the factors are x--\-y^±:-lxy.
Note that since s/y^= -^u'^, ov -y^, it may sometimes hap-
pen that while the former sign will give irrational factors, the
latter will give rational factors, and conversely.
7. x'^ — lix^y^+y*. Here, taking -{-y^, we have
x^+tj^+xy /13, and a-» +y-^ -xy v/13.
But taking —y^, we have
a;3 ^yZj^Zxy, and x"^ -y^ — 3xy,
Sometimes both signs wiU give rational factors.
8. lQx'*' — l'lx^y^-\-y^. Here we have
(^ix^Jf-y^J^'dxy){4.x'^+y^ -'dxy, and also
{^x"^ ~y^-Yoxy){Ax" -y^ -oxy).
FACTORING. 77
Exercise xxxi.
4. a;4— 7a:-+l, x-4-f9, Ja;4+^4 _ ^3a-.'^2.
6. 4a;4+y4_8i^.3^,/3^ ^4 +^4_^7^^3_,/-2^ 4;t;44 1.
7. a;^"' + 64i/*'», a;*"' + %*'", ia;*+y\v/4-5fx37/2.
9. m^x^ + n^i/^~{2tiin-\-ij)x''^ij2, .t^"' + 2^"'-y".
10. 16a;'i-2o.62 + 9, 4:;;4 _ le^s 4.4^ ISx'^y^ -dx^ - ii/-'.
11. 4a;'t-12^|x-2//3-l-()?/4, x4+6.x2+25.
12. a^~\-h^^(a + b)^, l+«4+(l_|_a)4.
Vl3. (;«4-Z/)'^-722(^-+t/)2+24.
■\ 14. (rt-f i)4+7(.2(«_|_ft)3+c4,
15. 16</4-j_4(i_c-)4_9«2(i_,.)3.
16. 4(a + Z))4 + 9(a-i)4-21(./3_/,3)3,
17. (a;2+y2_^^)4_7(^34.^3)24.(^.+^)4.
18. (rt2_|_rti + /;3j4^7(^,3_63)3_|_(a-i)*.
19. 16rt4 + 4rt2 4-l, ;f;4-41a;2+16.
20. x4+8l7/8-63a;2i/4, 14-24+2528^
21. (a24-l)4 + 4(rt24.i)2^2_|.iea4^ (.«+l)* + 2(a;2 _ 1)24.
{»(a;-l)4.
Art. XXI. We can apply [4] , Art. XIII., to factor expres-
sions of the form ax'*'-\-bx^+rbx—r^a. This may be written
a{x*-r'^)+bx{x''-\-r)={a{x^-r)-\-bx}{x^+r).
Examples.
1. 6x-4 + 4a;3 + i2x-54. This
= 6(a;4 - 9) + 4a;(x3-f-3) = {x^-+S){6{x^ - 3) + 4a}
= (a;2 4-3)(6a;2+4a;-18).
rs
FACTORING.
2. 11^4 + 10a;' -40a; -176. This
= n{x^-lQ) + 10x(x^-4:) = {x''-i){ll{x^+.i)+10x}
= (x^-A){nx^+10x+U).
3. 40a;4 + 30x3+ 60a; -160. This
= 10(4a;4 - 16) + 1.5a;(2a;2+4) = (2a;2 +4){10(2.t;3 -4)4-15a:}
= (2a;2 +4)(20a;^ + lya;- 40).
Note. — To determine r, take the ratio of the coefficient of x^
to the coefficient of x.
Exercise xxxii.
Resolve into factors
1. a;4+2a;3 + 6a;'-9.
2. 2.r* + 2a;3-f6a;-18.
.3. a;4 + 3a;-^+12a;-16.
4. 3a;^+a;3_4a;-48.
5. 5a;4 + 4a;3-12a;-45.
>lf>. 10a;4 + 5a;3+30a;-360.
3 _ S ^_ 4
7. ix^-\-20x^+4x-j^^.
8. 2;"^a;4-40a;3+8a;-l.
9. 37ia;'i-30.^;3+48x-96.
>il0. 63a;4- 39x3 + 52a; -112.
11. 810x4 + Va;3+|a;-2i.
12. 242a;4-33a;2-3a;-2.
13. ix^ + ^y^x^-^2^x
14. 80a;'' - 32a;37/+64a;?/- 320;/*.
15. 24a;4 - 12.x-3y+30a;/y3 - ISO*/*.
16. 2x^ + ^z^i/-8xy^-512ij*.
17. lla;* + 10a;3-12a;-15fi
18. 40a;* + 30a;3 + 60a;-160.
19. 13a;4-12a;3y+72a;?/3-468?,'4.
20. 3a;* + 3a;3^+12a;//3-48?/4.
21. oa;4+4a;3//-12a;</3_45(/4.
22. 4x4 - 14x3/7+28x^3 -16i/*.
23. x4+80x3?/+16x//3-^i3.y*-
24. 2x4 -x3?/+6x^3_ 72^/4.
Art. XXII. Formulas [1] and [4] may sometimes be ap-
plied to factor expressions of the form
ax^-\-bx^-\-cx^-\-7-bx-{-r^a.
This may be put under the form
afx* +r2) + 6x(x2 +/-)+cx8 = a{x^ +r)3 + bx{x^+r) +
(e — 2ar)a;3, which can sometimes be factored.
Examples.
1. x*+6x3+27x2+162x+729.
We "have x4+729 + 6x(a;2+27)+27x2.
= (x2+27)3 4-6x(x2+27) + 9x3-36x2
= {x3+27+3x} 3 - 3Gx3, which gives the factors
X
2-3X+27, and x2+9x+27.
FACTORING. 79
2. x^ + ix^+ix' +20x^-25. This
= (a;2 +5)2 + 4a.-(a;3 +5) - Ga;3
= (a;3 + 5)3 +4a;(a;2 +5) + 4a;3 - 10a;3
= {a;2 + 5 + 2x-xVlO}{a;3+5+2u;+aVlO>.
Exercise xxxiiu
Eesolve into factors :
1. a;4_6cc3 + 27a;3- 162a; +729.
2. a;H-2a;3+3a;2+8x+16.
3. a;4+^3 + a;3+a;-j_i.
4. a:4-4x3+a;3-4a; + l.
6. 4a;4_i2a;3_6a;2-12a;+4.
6. a;4 + 14a;3-25a;2-70a;+2?.
7. 16.T*-24a;3-16x3 + 12a;+4.
V8. a;* + 5:«3-16x2 + 2Ux+16.
9. a;4 + 6a;3 _ 11x3 -12x+4.
10. a;*+4a;3?/+ic22/2+12a;y3_|.9^^4.
11. x^+6x^-9z^-6x-\-l.
12. a;4+-4^•32/-19a;2J/2_}_4,^.y3^,^4.
13. 4a;4 +4a;3|/- 65x2^2 _io.ri!/34_25,2/*
14. x*^ +6x^y-9x^y^ -Gxij^-^y*.
15. a;*-+6.c3// + 10a;22/2+12ic^3^4^4.
16. ^x^-\-18x^y - 52x^1/- -12xy^ + i7j*.
17. lla;4+10x37/+39/^V-22/3+2Ua;?/3 + 442,'«./
Section III. — Factoking by Parts.
Art. XXIII. To factor an expression which can be reduced
to tlie form a.F{x)+b.f{x).
When the expression is thus arranged, any factor common to
a and b, or to F{x) and f{x), will be a factor of the whole ex-
pression. The method about to be illustrated will be found use-
ful in cases where only 07ie power of some letter is found.
80 rACTORiNa.
' Examples.
1- Factor acx" —nbz — hc^x + b^c.
Here we see that only oae power of a occurs, and we therefore
group together the terms involving tliis letter, and those not in
volving it, getting
a{cx^-hx)-bc^x+h^c
= ax{cx — b) — bc(cx — b) -- (ax — bc)(ex — b).
2. Factor m^x^ -mna^x — tnnx + n^a^.
Here we observe that a occurs in only one power («^).
Therefore we have
— a^ [^)imx~ 71^ ) + m^ x^ — 7nnx
= —na^[mx—n)-\-nix{mx — n)
— {mx — n)(^wx — na^).
3. 2x^+4:ax + Sbx+Gab.
Here we observe that the expression contains only one power
of both a and b. W" may, therefore, collect the coeflficieuts in
either of the following ways :
a(4x-{-(jb)+.{2x^+3bx),
or, b{3x + Ga)-\-{2x^+Aax).
Now the expressions in the brackets ought to have a common
factor, and we see that this is the case. Hence,
a{ix + 6b)-\-{2x^+Sbx)
= 2a{'2.x + Sb)+x{2x + Sb) = (2a; +36) (a; + 2a).
4. abxi/ + b^ij^-\-acx — c^
= a(bx!/-[-cx)-rb^y^ —c^
= «x(%+5) + (%+c)(%-c) =(by + c){ax+bi/-c).
5. yS - {2u + b)ij^ +{2ab-\-a^)y - a^b
= _f,(y2 ^2ay-}-a^) + y^ -2mj^+a^y
u: - b{y^-2ay+t(2) + y(y2 _ 2((// + a3)
= {y-b)iy-a)^'
6. 2x^y-\-2bx^-bx^y + 4:abx2y -x^y^+iaxy- - 2abxy^-2ay^.
= b{2x^-x^y+4:ax^y-2axy^) + 2xhj-x^y^ + 4:axy^-2ay^
=zbx{2x^-x'-^y + iaxy-2ay'^) + y{2x^-x-y+iaxy-2ay-^)
= (y + bx){2x^ -z^y+iaxy - 2ay^).
FACTORINQ,
81
And 2a?« — a;2 }/ 4- 4aa;y — 2a?/»
= a{4x!j - 2?/2 ) +2x3 — x^?/
= 2ay{2x - ly) +x2(2x - y) = {2aij+x^){2x - y).
7, x^ + {2a-b)x^ -{2ab-a^)x-a^b
^h{-x^~2ax-o^)+x^ + 2ax-+a^x
= -b{x + a)^+x{x + a)^ ={x-b){x+a)^.
8. px^-(;:p — q)x^-\-{p-q)x+q
= q{x^ —x-^l)+px^ —px"^ -\-}>x
= q{x^-X+l)+px{x^-X + l) ={pX + q){x^-X-il).
Exercise xxxiv.
6. x^ —b^x- ~a^x + a^b^.
7. x'
-a^x-
•62a;3 + a362
8. 8a;3 + 12aa;+ lOhx 4- 15^(6.
9. a^^{ac-h^)x^ ■\-bcx^.
10. a^+{ac-b'^)x^-hcx^.
1. a;2y— iK^z— "y^ -\-yz.
2. abxi/-\-b'^y^ -{-acx — c^.
8. ic322^.flja;2_^322_a3. ,
4. 2a;2— «a; — 46a;+2a6.
6. a;2+26a; + 8a^+6rt&.
12. ;?a;3_(p+y);j.3_j_(^^^)aj_j,
13. a2-\-ab + 2ac-2b^+7bc-Bc^.
\14:. x^+{a + l)x^ + {a + l)x+a.
15. ynpx^ + {mq — n]))x^ — {))i;r+7iq)x-\-nr.
16. x^ — {a-\-b-\-c)x'^ -\-{cLb-\-bc + ac)x-abc.
17. x^ + (a — b — c)x^ — [ab — bG-\-ca)x+abc.
18. x^ + {a-\-b — c)x^ —(be — ca —ab)x — abc.
"VlD. a^x^ — a^x^y — a^Xy + a^y^ —ax^yz-{-x^z — xyz-\-ciy^z,
20. a^bx^-{-nb^xy + acdxy -f hcdy 2 — at'/icz — befyz.
^1. a2a;3-a(6-e)a;2-5-c(a-6U- + c2.
22. 7Ha;3 —nx^y-\-rx^z — mxy^-\-ny^—ry^z.
23. amx^ -{-{mby — nay-\-incz)x — nby^ —ncyz.
24. (aw. — bcm)x- -{-{mn — bcn)x-^an-\-nax.
25. a262c3-6=c2a;</-a2c2?/2 + 6-2a;^32_a2i22a; + ft2^.3^2^rt222^
26. a;5 -m^x^ -{n-n')x^ + {m''-n-m'^n^)x^ -a{x--^n^ -n).
27. l-(a-l)j;-(a-/; + l).«2_^(a+6-c)a;^-(6 + c)a;4+ric5.
28. a3a;3 -a^{b- c+d)x^y - {abc-abd+acd)xy^ +bcdy^.
29. m^npx^ — {n-p — m"}i^ - m^j.q)x^ — (rr' + n] q — m^nq)x — n^q.
82
FACTOEING.
— [n^q- -\-n'^q^z)y'^.
Art. XX IV. Sometimes an expression which does not come
directly under the preceding form, may be resolved by first find-
ing the factors of its parts.
Examples.
1. ahx^ -{-ahy^ —a-xy — h^xy. '
Here, taking ax out of the first and third terms, and by out of
the second and fourth terms, we hav£
ax{bx — ay) — b>j(bx — ay), and hence
{ax — by){bx — ay) .
2. x^-{a+b)x^ + {a^b+ab2)x-a^b^.
Here, taking the first and last terms together, and the two
middle terms together, we have
{x^+ab){x^-ab)-{a+b)x^+ab{'i-{-b)x
= (x^ i-ab)[X' — ab)- (a + b)x{x^ ab\
= [z- - ab) [x- +ab - {a-\-b)K} = {x* —ab){x-a)(x - h)»
3. a:3»*-4a;"»+3. This eq lals
3^m _ a;m _, s(x^i -i) - x"'{x""' - 1) - o(a;"' — 1)
_ x^(x"'' + 1 ) (a;'"- - 1 ) - 3 (a''" - 1 )
= (a;"^-l){a;"^(a;^'^+l) — 3}-.
Exercise xxxv.
1. a^ — ab-^ax — bx.
2. abx^ + b^zy --a-xy - aby^ .
3. x^ +ax^ —a^z- «"*.
L a^x + 'la^x^^^ax^+x"-.
5. acx^ + {ad — bc)x — Ixl.
0. 'I'ux'*' -5x^+x -1.
7. a~ — A- +ax — ac — bx + br.
8. a^ + {l+a)ab + b^.
9. z^ + 2xy{x^-y-) -y^.
'.0. x^-y^+x-'+xy+y^.
A. 2b+{h^-4:)x-2bx^.
12. a;3-4-3x3-4.
13.
P^
.p2q
2pq^ + 2q^.
U. a3 4-a2_2.
15. '3an^-2ab^ -1.
16. //3-3f/+2.
17. 2a^-a''-b -aV--\-2b^.
18. &:^'» + i2,7i_2.
1 0 7/3n _ 2_,y2»2v _ 2i/"2':^" H-2"-\
20. a3_4rt52_j_3i3.
21. a'^"i-3ft'"f"+2c^".
22. ax^-{a^+b)z^+b^.
23. 35.-«^»-6a2a;»_9a4.
24. ^^3^2_|.2rt/,r3-rt3<---^)-'.a
25. aia^ —iib^ -\-b^m — )ii^.
2ti. i - tta' -4-27::''
FACTORING. 83
27. (x-y)^ + {l-x+y){z-y)z-t^.
28. 2Am9 -28in^n+Gmn^ -7n^»
29 . a;"*+" + a;"?/" + a;"'|/"' + ?/'" +" .
80. x^ + 2x^y-a''x^+xhj^ -2axy^ -y*.
Section IV. — Application of the Theory of DiVvSORS.
Art. XXV. By Theorem I. we.prove that
a;" — «" is divisible by x— a ahcays
jcR — a" *' " " X + a -whenn is even
af + a''" " " x-\-a yfhen n is odd.
By actual division we find, in the above cases ; —
= a;"-^+a:"-^«+ . . . Jca^-'+a"-' (1).
x —a
af — a"
X +a
= a;"-'-a;"--<^+ . . +«rt"-^-a"-^ (2).
'^^'^ = a;'-^-a^'-V(+ . •. -xa^-^+a^' (3).
Examples.
1. Eesolve into factors x^ — y^ ; here a;—?/ is one factor and by
(1) the other is a;- +a;?/ 4-2/^ •
2. Eesolve n'^-\-{h — c)^ ; here a + (b—c) is one factor; and by
(3) the other is a2_^a(6-c) + (i-r) 2.
3. Eesolvo a; '5 -f-1024?/^o. This =(x''y + {{2y)^}^, one factoi
of which is a:^ + (2_!/)2, and by (3) the otlier factor is
(a;?)4-(x3i^(4?/2) + (a;3)2(4.v2)3_a;3(4?/2)3 + (47/3)4.
= a:i2_4a;9y2^1Ga;'^2/4-64a;3v'5 + 256i/8.
4. Eesolve (x—2y)^-\-{2x — y)^ into factors.
Here by (3) we lia-.e
:. the factors are
'dix-y)(7x'^-ldxy+7y^).
S4 FACTORING.
0. Piesoh-e x^+x^y+x^ij^ +x^ 11^ +x}/^ + If ^ :
x^ — y^ (x" -{-i/^){x^ —y^)
-By (!) we see that this = ^ — =
■^ ^ ^ x —y x—y
=^{x->ry){x^ -xy + y'^)(x^ ■\-xy+y^).
6. Eesolve x^'^ -x^^'a+x^a^ — x^a^+x'' a*' - x^a^+x^a*
-x^fl'' +x^a^ -x^a^-\-xa''' -a^K This = r—
_ {x^+a^){x^-a^) __ {x^ +a^){x^ - a^){x^ -\-d^)
~ x-{-a ~ x-j-a
Exercise xxxvi.
Factor the following : —
1. x«-.vS x^-1, x^+S, 8a^-27x'^, S-{-a^x^.
2. j;5-(i>o, 27a3_G4, ai2_i8^ 3.io_32_y6,
3. find a factor which, multiplied into
a*+a3/,_|_rt362^„/,3_f./,4^ will give a^-b^.
4. By wiiat factor must x^ — ix'^ y-{-16xy^ — My^ be multiplied
to give a;* - 250//* ?
5. Fa.Gtor x^ +x'''y+x-'''y^ +x*yS-\-x^y^-{-x-y^ i-xy~^ fy'.
Find the factors of the follow! iv
6. (32/2 -2a;2)3_(3a;3 -22/2)3, a«~16i*.
7. x^ —y^ —x{x^—y^)+y{x — y).
10. a;6-t/6+2x2/(a;4+x22/^+2/'').
11. (a2_ftp)3 + 8&3c3, a;^/»_a-t'».
12. x^-dax-^+Ba^x-a^ + bs.
13. a;3 4-8y3+4a;?/(a;2-2xv + 4?/2j.
14. 8x^-Gxy{2x+Sy) + 21y3.
15. l-2a;4-4x2-8.c3.
IG. t/5+u4it-|-f/3i2c2_Ua253c3^«i4c*+6''r».
FACTORING. 85
Art. XXVI. The principles iilnstrated *in Section II., chap.
II., may be applied to factor various algebraic expressions, ap in
the following cases : ^
Examples.
1, Find the factors of
{a-{-b-\-c){ab+bc-i-ca)~~{a + h){b+e){c+a).
1st. Observe that the expression is symmetrical with respect
to a, b, c.
2nd. If there be any monomial factor a must be one. Put-
ting a = 0, the expression vanishes. .". a is a factor,
and, by symmetry, b and e are also factors. .". abc
is a factor.
•8rd. There can be no other literal factor, because tlie given
expression is of only three dimensions, and ahc is of
three dimensions.
4th. But there may be a numerical factor, vi suppose, so that
we have
{a-\-b+c){ab-{-bc-^ca)— {a + b){b-\-c){c + a) =:mabc.
To find m, put a = h = c = l in this equation, and w = l,
.". the expression = aic.
2. Eesolve a^{h-c)+b^{c-a)+c^{a-b).
1st. For rt = 0 this does not vanish. .*. a is not a factor,
and by symmetry neither is b nor c.
2ad. Try a binomial factor ; this will likely be of the foi ni
b — c ; put 0 — 0 = 0, i.e., h = c in the given expression,
and there results
a.2[c-c)+c^{c—a)+c^{a-c), which = 0,
.'. b~c is a, factor, and by symmetry c — a and a — b are fac-
tors. Since the given expression is only of three
dimensions, there can be no other literal factor ; but
there may be a numerical factor, m (say), sq that
a^ib-c)-^r'{c-a,)-{-c^{a-b) = m{a-b)(b-c){c-a).
To find the value of m, give a, b, c, in this equation, any values
which will not reduce either side to zero; leta=l, b = 2, c = 0,
8G FACTORING.
and we iiave 2 = m( — 2), or to= — 1 : so that the given expres-
sioii= — (a — b)(J) — c){c — a), or {a-b)[b — c){a — c).
3. Eesolve aS{h-[-c^)+b\{c + a^)+c^{a+b^)+abc{abc+l).
Here we see at once that there is no vwnomial factor :
put &+6'2 = 0, i.e., h— —c^, and the expression becomes
«3(-r3+c2)-c6(c + rt2)+c3(a+c4)-f3«(_c3a+l) which = 0;
.*. i + c2 is a factor, and by symmetry c + a^ and a+b^ ars also
factors ; and proceeding as in former 'examples we find m = l ; /.
the expression = (6 + c2)(c-f-«^)(« + i^).
4. Eesolve into factors the expression
(a-i)3+(6-c)3 + (c-a)3.
As before, we find that there are no monomial factors.
Let a — b = 0, or a = b, and substituting b for a the expression
becomes zero ; hence
a — b is a factor.
By symmetry, b— c "
and c—a "
Hence the factors arc
vi(a — b)(b — c){c~a).
To find TO let a— 0, 6 = 1, r, = 2, and we hav«
6 = 2m, or «i=3.
The factors are, therefore,
S{a-b){b-c){c-a).
5. Eesolve into factors
a^{b - c) -f-63(c--a)+c3(a - b).
A-s before, we find that there are no monomial factors.
Let a~b = 0, or a = b ', substituting b for a, the expression be^
comes zero ;
therefore a—b is a factor.
By symmetry b — e "
and c — a "
Now the product of these three factors is of tJiree dimensions,
while the expression itself is of four dimensions. There must,
therefore, be another factor of one dimension. It cannot be a
FACTORING. 87
monomial factor, for the expression has no such factors. It can-
not be a binomial factor, such as a+b, for then, by symmetry,
6+c and c+a wonlii also be factors, which would give an
expression of six dimensions. It cannot be a trinomial factor,
unless a, b, and c are similarly involved. For instance, if a — b+c
were a factor, then, by symmetry, 6 — c-f a and c — a-{-h would also
be factors, and the dimensions would be six instead oifour. The
other factor must, therefore, be a-\-h-{-c. Hence,
a^{h-c)+b^{c-a)-^c^{a — b)=m{a—b){h — c){c-a){a-\-o + r).
To find m, put a = 0, 6 = 1, and c = 2, and we have
— 6 = 6v//, ;
.*. m = — 1.
Hence the factors are
— (a — b){b — c){c — a)(a -\-h+c),
or, {a — b){a — c)(b — c){(i-^b + c).
6. Prove that «•
a3+b3 + c^-]-S(a-hh){b + c){c + a)
is exactly divisible by a-j-6 + c, and find all the factors.
Let a4-ft4-c = 0, or a= —{b-{-c); substituting this value for n,
we have
- (b+c)^ + b^ -\-c^-^Sbc{b-^c), or
-(6 + c)3 + (i + (;)3 which = 0, and
therefore a-\-b-^c is a factor.
As before, we find that there are no monomial factors. Since
a-{-b-{-c, the factor already obtained, is of one dimension, the
other factor must be of two dimensions, and cannot, therefore, be
a binomial ; for if a-\-b were a factor, by symmetry i + c, and c + a
must also be factors. The factors in that case vv'ould give a
qi;antity of four dimensions, while the expression itself is only
oi three dimensions. Nor can a^-\-b^-\-c^ be a factor. For
if so, the other factor must involve a numerical multiple of the
first power of a, and, therefore, on taking the first power of a out
of terms involving first and third powers, we should have left
some numerical multiple of a^-\-b^+c^, instead of wiiich we get
58 FACTOBINa.
a2_|-3(6-]_e)». Nor can af-i-{h+cy be a factor, for symmetry
would require two other factors, viz. : b- -{-{c-i-a)-, awl c- +(« +h)^,
tiius giving a quantity of seven dimensions.
The only factor admissible is, therefore, {a-^h+c)^.
Hence
a^-i-h'-^ -i 9^--hB{a-*-b){b+c){c + a) = ?n{a-\-h + i:\a-Jrb + :)'
= m{a+b-![-c)'^.
To find m, let a = l, 6 = 0, and c = 0, and we nave 1 = »«.
Hence the factors are
(^a + h + '^)(a + h + c){a+b+c).
7. Simplify
a{b+c)^ +b{a + cy -^c{a + b)^ — {a-'rb)(a-c){b-c)
-{a-b){a-c){b + c) + {a — b){b — c){a + c).
Let a = 0, and the expression becomes
bc"-^cb^-{-bc{b—c) — bc(b-^c) -bc{b- c), vriiich equals zf;ro ; there-
fore a is a factor ; by symmetry b and c are also factors.
The expression is of three dimensions, and abc is of three
dimensions, there cannot therefore be any other hteral ff,ctoi\
Hence the expression =viabc.
To find m, let a = b = c=l, and v/e have
m = 12.
.-. the expression =12abc.
In the preceding examples the factors have been linear, but the
principle applies equally well to those of higher dimensions. (See
Th. ii. Cor.)
8. Examine whether «" + ! is a factor of a;3" + 2a;-"+3.f" + 2.
Let a;" + 1 = 0, or x-"=— 1, and substituting, the eiipvessiou
vanishes, therefore, x"-\-l is a factor.
9. Examine whether a^ + b^ is a factor of
2a4+a36+2a2t/2+a^3,
Let a^ + b- =0, or a^ — —b", substituting, we have
2^-i_a63_2i4+a63 which = 0, and
therefore <^^-\- 6^ is a factor.
FACTORING. 89
10. Prove that a^+h^ is a factor of
a^-^a^^b+a^h^ + a-b-'^ab^ + b'^.
Lei a3_}./,3_o, or a^ = -b^; substituting, we have
-a?b^ -ab^-b^+a^b^-^ab^-{-b^, wliich = 0, and
therefore a^-{-b^ is a factor.
Exercise xxxvii.
Eesolve into factors
1. {x-i-i/+z)^ — {x^+y^+z'^)'
2. bc(b — c)—ca{a~(;) — db{b~ti).
8. (a3-63)3 + (62-f=^)3+(c2-aa)3.
4. x{y+z)^+7j{z+xy+z{x + ijy-4.xijz.
5. (^a+b)^ ^{b-\-c)^ + {c- a)^.
6. a(6-c)3+6(c-«)3 + c-(«-6)^
7. (a-i-b-\-c){ab-{-bc-{-ca)—abc.
8. a^{c-b^) + b^{a-c'^)+c^{b~-a^)+abc{abc-l).
9. «3(6 + (;)4.i2(f + a) + c2(a + /^)4-2(/ic.
10. (a -b){c-h){c-k)-{-{b-c){a-h){a-k) + {c- a){b- 'i){b-Ic).
11. a;4^3+,c2^4+a;4s3_^a;2z4+^423^y3244.2a;2^223.
12. (a-^)5 + (fe-6-)5+(c-a)s
13. afc(a+6)+M^+c)+«a(f+rt)+(a3 + 6»+c-3).
14. a4(c-63)+64(rt_c3)+c'i(6-a3)4-a66-(a262c3-l).
15. a;4(y3_23)+^4(23_^2)4.24(a;2__,/3).
IG. a;* + ^4 ^24 _2aj22/3_ 2^223 _ 222^3.
17. (6-c)(a;-fc)(x-c)+(c-a)(a;-c)(a;-a)4-(«-^)(^-«)(^-6).
18. {a+by^ + {h + c)^ + {c + a)s +
B(rt+26 + 6-)(6+2c+rt)(6-+2a+6).
19. Shew that a^ +(t^b^-ab^ - b^ has a^ -b for a factor.
20. Shew that {x + yY -x' -y' =lxy{x+y){.v^ +.'-V/ + ^/-)^
21. Examine whether x^ — 5x-\-iJ is a factor of
^3_9^2_^26a;-24,
90 fACTOEINO,
22. Shew that a — b-\-c is a factor of
23. Shew that a" +Sh is a factor of
and find the other factor .
24. Find the factors of W^ib - c)-i-b^{c ~ a)-{-c^(a—b').
Section V.. — Factoring a Poltnome by Teial Divisors.
Art. X,XVII. To find, if possible, a rational linear factor of
the poiynome.
a;»^6a;"-i+ca;"-'+ ^^nx+k.
Substitute successively for x every measure (both positive and
negative) of the term k, till one is found, say m, that maT\es the
polynome vanish, then x - m will be a factor of the polynome.
Examples.
1. Factor a;3+9a;2+16a;+4.
The measures of 4 are +1, +2 and +4. Since every coefS-
cient of the given polynome is positive, the positive measures of
4 need not be tried. Using the others <• it will be found that — 2
makos the polynome vanish ; thus
1 9 16 4
-2 -14 -4-
-2
17 2; 0
Hence the factors are (a3 + 2)(x3-|-7a;-f2).
The labour of .substitution may often be lessened by arrang-
ing the polynome in ascending powers of x, and using 1 -h
(measure of k) instead of the measures of k. (This is really
substituting 1 -r measure of k, for l^x). Should a fraction
occur during the course of the work, further trial of that measure
of k will be needless.
FACTORING.
91
Examples.
2. Factor x^ - lOz^ - 63a;+60.
The measures of 60 are +1, +2, ±3, ±4, +5, &c. Neither
4-1 nor - 1 will make the polynome vanish. Try 2 ; thus
60 -63 -10 1
1^
2
63
30
30 -16^
A fraction occurring we need go no further. — 5 will also give
a fraction, as may easily be seen. Next try 3 ; thus
60 -63 -10 1
•63
20
20
-14i
A fraction again occuring, we may stop.
fwaotion. Next try 4 ; thus
— 3 will also give a
1
T
60
-63
15
^10
-12
Next try —4.
-1^
T
15
60
-12
-63
-15
5*
-10
15 -19i
Next ti:yiu^ 5 we find it fails, then try — 5, thus
60
-1
-63
-12
-10
15
1
-1
12 -15 1; 0
The remainder vanishes as required ; the factors are, therefore,
(a;+5)(a;3-15a;+12).
Art, XXVIII. When k has a large number of factors, the
number that need actually be tried can often be considerably
lessened by the following means.
Add together all the coefficients o£ x (including the constant
term k) ; let the sum be called k^.
92 FACTORING.
From the sum of the coefficients of the even powere of x
(including k) take the sum of the coefficients of the odd powers of
x; let the remainder be called Jc^. (In the coefficients are in-
cluded the signs of the terms).
1st. If k^ vanish, x — 1 will be a factor of the polynome.
2nd. If /Cg vanish, x+1 will be a factor of the polynome.
3rd. If both k^ and Ag vanish, x^ —1 will be a factor of the
polynome.
4th. If neither ky nor k^ vanish, (writing p for " a positive
measure of ^ greater than 1 ") ;
(a) We need not try the substitution of p for x unless ^ — 1 be
a measure of ky, and ^-fl a measure of k^.
{0) Nor need we try the substitution of —p for x unless p-^-l
be a measure oi k^, and p — 1 a measure of kc^.
(In trying for measures, the signs of k^ ky, and k^ may be
neglected.
Examples.
1. Find the factors of x^ - lOx^ - 63a;+G0. (^ee Ex. 2 above).
Here ^ = 60 ; ^1= 1 -10-63 + 60= -12,
A;2 = -1-10+63+60 = 112.
Tabulating the trial-measures we get
12
1,'
2,
3,
4,
60
2,
3,
4,
5, 6,
10, 12,
112
4,
7,
12
3,
4,
6,
60
2,
3,
5, 6,
10,
112
1,
2,
4,
(It is evident that 12 is the highest measure of 60 we need try
in the upper table, for the next measure, 15, would give 14 as a
trial-measure of 12, and higher measures of 60 would give higher
trial-measures. Similarly, 10 is the highest measure that need
be tried in the lower table.)
FACTORING.
93
In the upper table, 8 is the only measure of 60 that gives a
full column ; hence of the positive measm-es of 60 we need try
only the substitution of 3 for x.
In the lower table, 2, 3, and 5 give full columns, hence we
must try the substitutions —2, —3, —5 for x.
On trying the four substitutions to which we are thus restricted
we find — 5 is the only one for which the polynome vanishes.
(See Ex. 2 above).
- 2. Find the factors of x^+12x^ -4:0x^ +Q7x-120.
&=-120; /.-i=H-12--40+67-120= -80;
A, = 1 - 12 -40 - 67 - 120 = - 238.
80
120
238
1,
2,
2, 4, 5,
3, 4, 5, 6,
7,
8,
10.
12,
15, &c.
80
120
238
2,
1.
4, 5,
3, 4, 6, 6,
2,
8.
7,
10,
i
16,
15, 20,
14, 21,
24,
&c
The upper table gives us 6 as a trial-measure, and the lower
gives us —3 and —15.
Trying these
) we get
-120
67
-40
12
1
1
-20
6
- 20
U
-120
67
-40
12
1
-1
40 -
3
-40
36f
-120
67
-40
12
1
-1
8
— 5
3
-1
16
- 8
6
- 3
1:
0
94
FACTOEINO.
Hence a;+ 15 and x^ — Sx^ + 5x - 8 aie the factors. The latter
cannot be resolved, for our tables above tell us we need try only
x—6, x+3, and a;+15. The first two have been found not to be
factors, and 15 will not measure 8.
4. Factor a;" _ 27a;2 + Ux+ 120.
k=120] ^'i =1-27+14+120 = 108
/.•2 = 1-27-14 + 120= 80.
108
1,
2,
3,
4,
9
120
2,
3,
4,
5,
6,
8,
10,
12,
15, &c
80
4,
5,
16,
108
3,
4,
6,
9
120
2,
3,
4,
5,
6,
8,
10,
12,
15, &c
80
1,
2,
4,
5.
The upper table gives us 3 and 4, the lower table gives us - 2,
—3, and —5. Using these in order we get
Hence x — 3 is a factor.
Hence a; — 4 is a factor.
Hence ar+2 is a factor,
and there remains x-\-o, a factor.
Hence the factors are {x-B){x—4:){x-\-2,){x + 5).
5. ¥&ctor x*-px^-\-{q-l)x^+px q.
k=-q; k^ = l-p + {q-l)+p-q = 0;
k„ = l +p+{q-l) -p-q = 0.
Since both ki and /c, vanish, the polynome is divisible by both
x—1 and x-\-l.
1
120
14
40
-27 0 1
18 -3 -1
3
1
40
18
10
-3-1; 0.
7 1
4
-1
10
7
-5
1; 0
-1
2
5
1;
0
1
1
-V
1
q-1
-p + 1
P
q-p
1
1
1
-;,+l
-1
q-p
+P
9;
-9
0
1
-P
9;
0
FACTORINa.
95
Hence the othei- factor is x^ —px-^q.
6. Factor x^+2ax^+{a^ + a)x-'-{-2a^x+a^.
k^ = 1 -2a+(a2 +a) - 2a^ +a^ ^a^-a^ -a-1.
The positive measures of k are 1, a, a^, a^. Of these 1 may
be rejected at once, since neither k^ nor k^ vanish, and a^ and a^
may also he rejected since k^ or (a + 1)3 is not divisible by eithei
a^ + l or a' + l. But k^ is divisible by a-\-l, and k^ is divisible
bya— 1; thus we need only try the substitution of -aforx.
(See 4 j5, page 92)
1
2a
a2+a
2a3
flS
— a
— a
-a2
-a3
-a?
1
a
a
a3;
0
—a
— a
0
-a2
1
0
a;
0
Hence the factors are {x-{-ay^{x^-\-a).
7. Factor a;3_(rt+c)a;2 +(&+ac)a; -6c.
^: = — 6c ;
4j = l — {a+c)-^{b+ac) — bc = l—a+b—c + ac — bc
fe, = - 1 - (a+c) — (6+ac) - 5c = — (1 +a+i+c + ac+6c).
The factors oi k^, other than 1, are b and c. k^is not divisible
by either 6+1 nor by c+-l. However, Zr^ is divisible byc-1,
and k^ is at the same time divisible by c+1, /. we need only try
the substitution of c for x. (See 4 «, page 86).
(a+c) (6 + ac) —6c
c — ac ic
1 —a 6;
Hence the factors are {x— c){x^ — ax+b).
96
FACTORING.
Exercise xxxviii.
1. a3_9„2_^l(5a_4.
2. a;3-9a;2+26a;-24.
3. a;3-7a;2+lGx-12.
4. x^-Ux+16.
5. x^ + Sx^ + 5x-\-B.
7. x^-dx+2.
8. a;4 + 2a;2+9. 22
9. m^-Sm^-n+^mn^-27i^. 23
10. x^-\-2x^-\-2. 24
11. jftS _ 5m2n-\-8t7in^ — 4n^.
12. 63+62c+76c2 + 39c3.
13. m^ — 4mn^ + 3n'^.
14. a4_7a36 + 28rti3_i664.
29. x^-18x^ + UBx^-288x+252.
30. a;4-9a;3i/4-29x22/2_3ga;i^3T-i8y4,
15. a;3_llx2+39a;-45.
16. x^-\-5x2+lx-\-2.
17. «3_3o,2_i93rt_|.i95.
18. p^-Sp^-6p-8.
19. a4+3,t3_3rt2_7fl_|_a;
20. a'''-6a*"+lla"'-6.
21. aA-Ua^b~ + lGb^.
rt4_rt3/,2_2a63H-264.
^,3_4^;2^67;-'4.
a;'"+4a;"" - 5.
25. y4-5?/M'8i/2-8.
26 a4-2a3 + 3a3-2«+l.
27. a3+a253^a62_363.
28. 2a''' -a'" -a" 4-2,
Art. XXIX. To find, if possible, a rational linear factor of
the polynome
aaf+bx''-^-\-car-^+ +hpB + k.
First Method. Multiply the polynome by a"~^
(axy+b{ax)'^^+ac{azy-^' + -^a^-'hiax) + a'^-'k ;
or writing y for aXy
ynj^lyn-i ^ acy"-^ + +«"-'*% + a^-^k.
Factor this polynome by the method of the last article, replace
y by ax, and divide the result by a"~\
Example.
Factor 3j;4-|-5a;3 -33a;2+43a;-20.
Multiply by 33 and express in terms of Sx.
(3a;)4 + 5(3a;)3-99(8a;)2+387(3a5)-540 ;
or, 2/4+52/3-99^/3 + 387// -540.
FACTORING.
97
Here /t=: -540; i5:i = 1 + 5-99+387 -540== -246:
5 - 99 - 387 -^540 = - 1030.
k, = l
246
640
1030
246
640
1030
1, 2, 3, 6, 41, 82, 123, 246.
2. 3, 4,
5,
3,
2,
1,
6, 41, &c. (Trying by factors of 246
5, instead of by factors of 640,
for convenience).
The only factors of 540 in full columns are 4 in the upper
table and 2 in the lower one ; hence we need try only the substi-
tutions 4 and —2.
1
-540
387
-135
-99
63
5
-9
1
-1
4
-135
63
- 9
-1;
0
Hence ?/- 4 is a factor. The substitution —2 need not now
be tried, since we see that 135 is not a multiple of 2. The other
factor is therefore y^-\-2y^ —Q^y-^1^6.
Replacing y by 3a; and dividing by 27 ;
^V(3a; - 4)(27a;3+-81aj8-189a;-}-135)
= (3a; - 4)(a;3 +3a;2 -7a;+5),
wnich are the factors.
Art. XXX. Second Method. Writing m for "a measure of
a," and p for a " measure of k, positive or negative ;"
For aj substitute every value of p-wtill one, say;?'-r-m' bb
found which makes the polynome vanish ; then m'x—p' will be
a factor. Should a fraction be met with in the course of substi-
tution, further trial of that value oi p-rvi will be useless.
Should k have more factors than a, it will generally be better
to arrange the polynome in ascending powers of x and use values
of m^p instead of p-i-m, making 2^ positive and ?« positive or
negative.
98 FACTORn<[G.
To reduce the number of trial-measures, calculate k^ and k^, as
directed on page 92, tken 1, 2, 3 hold as on that page, but in 4
read^ — m for^ — 1 andp-j-m for^j+l. ,
1^^ Examples.
1. Factor 36a;3 + 171x2 -22X+480.
;t = 480, yti= 36+171-22-1-480 = 665
;;;2= -364-171+22+480 = 637.
m may have any of the values +1, ±2, +3, +4, ±6, +9,
+ 12, +18, +36.
In forming the table write out the measures of ^j ; take each
measure in succession and add to it each value of m separately,
should the sum measure 480, i.e., k, add to it the same value of
m, and should the new sum measure 637, i.e., k^, keep the mea-
sure of 480, writing above it the value of m used. Should the
sum in either case iiot be a measure, another value of ni must be
tried ; "^hen all the values of m have been tried, another measure
of 665, i.e., k^ must be tried till all have been tested. (Measures
of yfc, or 665 have been used in this instance because they are
much fewer than those of 480 ; measures of k^ or 637 would have
done equally well).
•
m=+3, +1, +3 -2 -8 -9 -3
665 1, 5, 7 5 7 19 19
480 4, 6, 10 3 4 10 16
637 7, 7, 13 1 1 1 18
Hence the only substitutions that need be tried are
8J_8 z^ zl Zl Z^ iox L
T' 6' To' 3' 4' 10\ 16' x'
Arrangement in ascending powers of x.
By actual trial, as below, we find ^1 is the only one of these
giving a zero remainder.
FACTORING.
99
4
1
3
10
-2
8
-8
4
-9
480
- 22
360
171
86
10
-3
120
84i
80 -
80
144
48
12-2
-320
228
-266
160
-114
360
133;
-230
120
- 95i
-432
48
-45-4
- 90 .
21
-36
16 30 - 7 12; 0
(The coefficients are written only once, and understood for the
other hnes of substitution.)
Hence the factors are 3a; + 16 and 12a;2 _ 735+30,
■ The latter factor cannot be resolved, for 16 will not measure
30, and all the other factors left iox trial by the tables above,
have been tried and have failed.
2. Factor 10a;* -a;3(15|/+4z) -a;3(40y» -6y«) +
Here m= ±1, +2, ±5, or ±10. k= -2.4t/»z.
k^ = l0-{15y+iz)-(4:0y^-eyz) + (G0y^-\-16y^z)-2iyH
= 10- 151/ - 402/2-1-602/3 -2;s(2-3y- 82/2 -M2?/S)
= (5-2z)(2-3y-82/24-12.y5).
A-- = (5 + 2z)(2+3?/—8y2 — 127/3), as may easily be found
by making the calculation.
We get at a glance 2z a factor of k, 2z — 5 a factor of Ic^, and
2z+5 a factor of k,^ ; hence taking w = 5, we are directed to trj'
2z
the substitution -_ for x.
5
10 -{15y+iz) -{iOy^-6yz) {my^ + 16y^z) -24^/3-
4z —Qyz . -16y'-z 24?/ "^2;
^%2 12^^3'; 0
6
2 -3y
100
FACTORING.
Hence 5x~2z is a factor, tlie other being
2x^-3x^y-8xy2-i-12y3.
• The latter factor being homogeneous, the method of this article
may be appHed to it.
'^i=+lor±2, k = 12, ki=8, k^ = 15.
m = l, 2, 1, -1
1, 3, 3 The other columns
3, 4, 2 are not full.
5, 5, .1
Hence the trial- substitutions (arrangement m ascending powers
of x) are i, |, ^, :±.
3
1,
12
2,
15
3,
12
-8
-3
2
1
6
-1
-2
2
6
-1
-2;
0
2
4
2
3
2
1;
0
Pinal factor is 2>j+x»
Hence the factors are {z—2i/){2x-3i/){x+2y), and these, with
the factor 5x — 20 already found, give the complete resolution of
the polynome proposed.
(The factor 5x — 2z, might easily have been got by the method of
Art. XXIII., page 79, but the present solution shows we are inde-
pendent of that article. It may also be obtained by rearranging
the polynome in terms oiy).
Exercise xxxix.
Factor
l.aa;3_20fl;»+38a;-20; x^ -Ix^y + lQxy^ -I2y^.
2. 12x^ + 5x^y-\-xy^+dy^ ; 8a;3_i4a;+6.
3. Bx^-15ax+a^x-5a^; 2x^-\-9x^y-\-7xy^ -Sy^.
4. 2^*4_7J3c_462c2 + 6c3-4c4 ; 15a3+47a36 + 13a6»- 1263.
6. UOx'^ -725a;3y+98ia;3y2 ^Q20xy^ - 1152^4,
7. d6x^-6{9 -7y)x^ - 7{9 + Uy)x^y+3{4:9-40y)xy^ + 180y3.
8. lOx* -x^l5y + iz) + x^{4:0y^-\-Gijz)+x{60y^ -Hj^-^z)-24.y^z
DIVISION. 101
CHAPTEE IV.
Section I. — Division. Measures and Multiples.
Art. XXXI. When one quantity is to be divided by another
the quotient can often be readily obtained by resolving the divisor
or dividend, or both, into factors.
Examples.
1. Divide a'i-2,ab-{-b^ -c^-{-2cd-d^ hy a— \b+c—d. Here
we see at once that the dividend ={a—b)^-{c — d)^, and .•. quo-
tient = a-b — {c-d) = a — b — c-^d.
2. Divide the product of a^-\-axi-z^ and a^-\-x^ by a*'-{-a^x'^
•{■z^. Here a^-[-x^ = {a-\-x){a'^ — ax-\-x^), and the divisor =
(a^-^ax+x^){a^ — ax+x^) :. the quotient is a -fx.
3.*Divide a^+a^b+a^c—abc-b^c-bc^ by a^-bc. The divi-
dend is a{a^ —bc)-\-b{a^ -6c)4-c(a* —be) /.the quotient =a+b-Jr-c,
4. (a3+b^-c^-^3abc)^{a-{.b-c).
Dividend =a^-\-b^+Sab{a+b)-G^ -3ab{a+b)-\-SabG={a'^b)^
— c^ — Bab{a -{-b — c) which is exactly divisible by a +6— c; quotient
=a^-^b^ -\-c^ —ab-\-bc-\-ca.
5. Divide x^ —x^y-{-x^y^ — x^y^ +xy^—y^ bjx^ — y^.
The dividend is (Art. XXV.) evidently (x^ —y^) -^ (a'+*')j and
this divided by x^ —y^ = (^x^-\-y^) -h (x+y) = x^ — xy-\-y^ .
6. Divide b{x^ +a.3)-{-ax{x^ - a^)+a^{x-{-a) by {a-^b){x^-a).
Striking the factor x + a out of dividend and divisor we have
b{x^ — ax-\-a^)-\-ax{x — a)-^a^ = b(x^ — ax + a^y-\-a{x^ — ax-\'0-)
= {a-\-b){x'^ — ax+a^) .'. quotient =x^ -ax-\-a^.
7. Divide apx^ +x^(aq+bp)-\-x^{ar-\-bq-\-pc)-\-x{qc-\-br) +cr by
ax'-^bx-i-''
102 DIVISION.
Factoring the dividend (Art. XXIII.) we have
(ax^ + bx+c){px^ -^qx+r).
:. the quotient = the latter factoi".
S. Divide Gx^ - ISax^ + ISa^x^ - 13a^x~ 5«4 by 2a;^ - Sax- a''.
This can be done by Art. XVII. The divisor is 2x^ ~a^- Sax,
and we see at once that 3x'+5a'^ must be two terms of the quo-
tient.
Multiplying diagonally into the first two terms of the divisor,
and adding the products, we get -{-7a^x^ ; but -^ISa^x'^ is re-
quired. .'. -\-Qa-x'^ is still required, and as this must come from
the third term multiplied into — 3ax, that third term must be
— lax; :. the quotient is 3x^ -^-ba^ —2ax.
jJoTE. — By multiplying the terms - lax, — 3ax, diagonally into
the ic^'s and a^'s respectively, we get the remaining terms of the
dividend ; it is, of course, necessary to test whether the division
is exact.
9. Divide 2«* - a^J - 120252 _ s^js _{.4J4 by «« - ja - 2a6.
Here, as before, one factor is a^ — h'—1ah; :. ft<7o terms of
the other factor are 2a2-462. Multiplying, as in the last
example, we get -&a^h^; but -lla^^b^ is required. .•. — Sa^fts
is still needed, and -\-3ah is the third term of the requii-ed quo-
tient, which is therefore 2a2_463_|.3a/,.
Prove that
10. (l-fa;-|-a;2+ - - - -^x-'-^){l-x+x^ - .... -fa;"-^)
= l-fa;2+a;'i+ .... +x'
1 - X" 1 -1- a;"
^i»— 2
Product =
1—x l-\-x
l — a?^
z ^ = 1+X^+X*'i .... +^-»-».
1 —X^
11. Divide (a^ -bc-y -{-Sb^c^ by a^-\-hc.
= (a^ -bc)^ +{2bc)^ hy {a^ -bc) + 2bc
« (a3 _ic)3 -(«2 _ be) X 2k-f-(26c)»
= a4^.4a3ic-f763ca.
DIVISON. 103
12. Divide l+23579-i7691a;9 by l-llx+121a;3
Dividend =l + (lla;)^ ^
= [1 -(llx)3+(ll.'c)«}{H-(lla:)3}
Divisor= {H-(lla;)3} ^(1+lla;).
.-. quotient =a-(ll£c)3+(llx)«}(l + lla:).
Exercise xl.
Find the quotients in tlie follo\ving cases :
1. l-x-\-x^-z^-^l-x.
2. l-2a;4+a;»H-a;4+2a;3 + l.
4. x4+4x3j/3-32?/4^a;_2?/.
5. l-4:X^+12x^-9x'^-^l + 2x-Sx'^.
6. (rt2 -2ax+x^-)(a^+3a^x+3ax^-{-x^) ^a^ -x».
7. x^—y^+z^+dxyz^x — y-rZ.
8. 6a4-a36 + 2,'i262 4-13«63_f.4j4 ^ 2rt» -3a6-f 4fc*.
9. 4«4-a;32/2+6a;;/3-9^4 ^ 2a;2+3»/3-arj/.
10. rt*+/)4_c4_2«252 -^rt3_&2_c2.
11. 21a4-16rt3J4-16a363-5a63 + 264 -r- 3«2 -a&^-J*.
12.-2rt3_7a3_4Grt_21 ^ 2a2+7a+3.
13. {a^{b~c)+b^{c-a)-\-c^{a-b)} -r a-{-i+c.
14. x^-Sax-+Sa^x-a^+b^ -^x-a+b,
15. a;* - ^4+4 + 2a;2z2- 2^3-1 -7- a;3 -1/8+22-1.
16. a:* — (a+c)a;3 + (6 + ac)x3— ica; H- a;-c.
17. x^-^x^y +xy^-{-y^ -i- a:+?/.
18. x'' —x^y+x^y^ — x'^y^-\-x^y*-x^y^+xy^—y'' -^z^+y*,
19. (j44.;,4_c4_2a2i2_2c3_l -H a2-62-c2_l.
20. a4-rti3_rtc3_2a3&+264 + 26c3 + 3a3c-363c-3c4
-5-a + 8c-26.
21. a^b-bx~+a^x-x^ H- (a: + 6)(a-a;).
22. a(6-c)3 + 6(c--a)3+c(a-6)3 ^a^-ai-ac+Sc,
104 MEASUKES AND MULTIPLES.
23 aH^ + 'lnhc--a^c^ -ir-c^ ^ab+ac-bc.
2A. x^+y^ + -d.v.i/-l -y x + y-1.
25. x^-x^-2 ~ x^-x + 1.
26. a^-2da2-50a-21 -^ a^~5a-l.
27. (2x-'y)^a^-{x+y)^a^x^+2{x+'i/)ax'^-x^ -^
{2x — y)a--t-(x + y)ax — x^.
28. {x^-l)a^-{x^+ao^-2)a^+{4:X^+dx+2)a-d{x+l) *
-=^ {x-l)a^~{x-l)a-{-d.
Art. XXXII. The Highest Common t actor of two algebraic
quantities majj in general, be readily found by factoring. The
H. C. F. is often discovered by taking the sum or difference (or
sum and difference) of the given expressions, or of some multiples
of them.
Examples.
1. Find the H. C. F. of {h-c)x--\-{2ah-2ac)x + a"-h-a'^c, and
iah — ac-\-b'''^ — bc)x-\-a^ c -\- ah^ —a^b- abc.
Taking out the common factor b — c we get (b — c){x^ +2ax-\-ub}
and {b — c)\(a — b)x-a^-i-ab} ;
.'. b — c is the H. C. F. of the i^iven expressions.
2. Find the H. G. F. of
1 — x+y + z — xy + >iz —zx- xyz, and
\ — z—y—z^xy-\-yz-\-zx—xijz.
Their difference is %j-\-2z — 2xy-'lzx = 2(1 — x){}j-{-z).
Their sum is 2 - 2x-Y2yz — 2xyz = 2(1 — a;)(l +^2).
.'. theH. C. F. is (1-a;).
3. Find the H. C. F. of a;"' + Sa;^ -8^2 -9a;-3, and
jc-^ -2a;4-6a;3+4a;2 + 13a;+6.
The annexed method of finding tlie H. C. F. depends on the
principle, that if a quantity measures two other quantities, it will
measure any multijjle of thek sum or difference.
MEASURES AND MULTIPLES.
1
1
+ 3
- 2
0-8-9-3 (a)
-6 + 4 4-13 + 6 {h)
0
+ 6 -12 -22-9 (c)
2
1
+ 6
- 2
0 -16 -18 - 6
_ 6 + 4 +13 + 6
3
+ 4
-6-12-5 (f/)
15
16
+ 18 -36 -66 -27
+20 -30 -60 -25
-2-6-6-2
1 + 3 + 3 + 1 (f)
25 +30 -60 -110 -45
27 +36 -54 -108 -45
-2 - 6 - 6 - '2
(a)x2
(6)
(c)x3
(£i)x6
(r)x5
(ti)x9
1 + 3 + 3 +~1 (gr)
E. C. F. = (a;+l)3.
The coefficients are written in two lines, (a) and (6). They
are then subtracted so as to cancel the first terms, (a) is next
multiplied by 2, and added to cancel the last terms. K (c) and
(d) had been the same their terms would hav.e been the coefficients
of the H. C. F. Since they are not, we proceed with them as
with (a) and (6) till they become the same. When {a) and (b)
do not contain the same number of terms it is more convenient
to find only (r), aud then use this with the quantity containing the
same number of terms. The general rule is to operate on hues
containing the same, or nearly the same number of terms.
4. Find the H. 0. F. of Sx^ + 2x^ -Ux + 8, and
6a;3-lla;3 + 13a;-12.
8 + 2-14 + 8 (a)
6 -11 +13 -12 (6)
6~+ 4 -28 +16 (a) x 2
15 -41 +28 (c) (6)-(«),
(5_7)(3-4)
H. C. F. = 3a; -4. (d)
If (a) and (b) have a common factor its first term must measure
8 and 6, and its last term must measure 8 and 12. (c) is not
106 ME4SURES AND MULTIPLES.
therefore, the H. C. F. Eesolve (c) into factors. 5a;-7isnota
factor of (a) and (b). If, therefore, (a) and {b) have a common
factor it is 3a; -4. On trial 3a; -4 is found to be a factor of (a)
and .-. it is the H. C. F. of (a) and (6).
6. 1{ X- -{-px+q, and a;2 + >'a;+s have a common factor, prove
that this factor is
x"+ . If x—a be the common factor then the remainders
p — r
on dividing; the given expressions by x—a, must be zero, i. e.,
a^+pa+q = 0, and a--\-ra+s = 0, or
s—q
i (^ - r)a = *-?,.-. a = ^3^, and
s —q q— s
x-a = x— =x+ r~~Z'
p —r P — ^
6. What value of a will make a^x^+{n-{-2)x+l, and
a^a;2-fa2— 5, have a common measure.
They cannot have a monomial factor. Neither can they have
one of two dimensions unless {a + 2} vanishes, i.e., unless a= — 2,
in which case the expressions become Ax^ + l, and 4a;'^ —1, which
have no C. F. Hence if the given quantities have a C. F., it
must be of the form a;+»i; dividing a3a;2+a2 — 5 by x-\-m, we
have for remainder,
5 — a' 1
fi2„i2 + fl,2_5 = o, orm3= — ~^- ; .-. w= — ^/'(S-a^), in which
l/{5 — a^) must be possible and integral, .-. a2=4, («2 = 1 gives
values to m which on trial fail) and a=± 2, of which the positive
^ alue must be taken, and .-. 2a;+ 1 is the C. F.
7, If the H. C. F. of a and b be c, the L, C. M. of
(a+6)(a3-63),and(a-6)(a3+fe3)is ^^— •
Lei a = mc, b = 'nc, and .-. a^=Tn^c^. b^=n^c^. Thus
(a +b ) = c {tn +n ); (« -b ) = c {m —n ), and
,«. (a + 6)(a3 _ 53) _ c4^TO+n)(wi3 -n^),. and
(a-6Va3+63) = c4(w-n)(m3+?i3).
MEASURES AND MULTIPIiBS.
107
The H. C. F. of the last expressions is c'^(m*—n^), .-. the
h. C. M. = c4(m6-»6\^ _v — -^_ —
8. If (x—a)^ measures x^ + qx+r, find the relation between q
and r.
Let a; + m be the other factor, then
x^+qx + r=^-^c)"{x+m)=x'^ + im — 2a)x%+(a^ - 2,ou>n)x-\-ma^
equating coefficients,
m — 'ia = Q,a^—2am — q,ma^=r
.*.»» = 2a, and .'. a^ — 4a2 =q, 2a3 = ,-, and
q q'^ r r'
a* = - Y' or «« = - 2^ ; and a* = -^ or a^ = -^
rS ^3 ,.8 qZ
''' T=~ 27' °^* ^"^ 27 =^
Or thus : —
Dividing x^ + qx+r by (a;— a)' we find the remainder
\ {q + 3a^)x+r-2a^
and as this vfiil be the same for all values of x, we have, by equat
ing eoefOciente,
2+3a»=0,
and r— 2a3 = o,
or 3-3=— 27a«
and r3 = 4a® ;
therefore ^ + 27 = Of as before.
Exercise xli.
Find the H. C. F. of the followmg :
1. 2x^+dx^ + ox^+9x-d, dx^-2x^ + 10x^ -6x^3.
a. a;^ + (a + l)x^+{a+l)x-fa, x^ + {a-l)x^ --{a-l)x+a.
3. px^-{p + q)x^+{p-q)x + q, px^ -{p+q)x^ +(p+q)x-^q.
4. ax^-{a-b)x^-{b~c)x-G, 2ax^t+{a+2b)x^-\-{b + 2c)x+o.
5. l-3|a;-3ia;3+ia;3_a.4, i^i^i^a;_3a;3-[.l^a;3+a;4.
103 MEASt)KE5 AND MUL,TIPLES.
7. a^x^ i-a'' -2abx^ + b'-'x^ +a5b'* -2a^b, and
8. {ax-\-hyY — {a— b)[x-\-z){ax->rby)-^{a - b)^xz, and
(ax — hy)^ — {a-\- h) {x-]rz) {ax — by) + (a +, b) '^xz.
9. «(i2-c3) + fe(c2-rt3)+c*(a2-62) and
a{b^-G'') + h{c^-a^) + c\a^-b^).
10. a^"'+a-"'-{-a^-Jfl, and a^m „«'■*'» ^-a^-l.
11. If x^ +ax^ +bx+c, and x^+a'x+b', have a common factoi
of one dimension in x, it must be one the factors of
{a — a')x^ + {b-b')x+c.
12 Determine the H. C. F. of (a-fe)^+(6— c)6+(c-a)», and
18. Find the H. C. F. of
2(7/3 _2?/2-y + 2)a;3+3(y2_i)a;a_(2?/3_j,2_2y + i^, and.
3(2/*-4?/3+5y-2)a;3+7(2/2-2y+l):B-(3i/3-5v=5+2/-fl).
14. If a;2 4-i^a;4-9, and x2+OTa;+" have a common hnear factor,
shew that
{n—q)^-\- n{m—p)^=m(m—p){n — q).
15. Find the L. C. M. of x3-3a;2 + 3a;_l.^ a:3-a;8-a!+l,
a;4_2a;S + 2a;-l, and x^ -2a;3 + 2a;3 -2a;+l.
16. Find the L. C. M. of
a;3 + 6a;2 + lla;+6, a;3 + 7a;3 + 14a:+8.
a;3+8a;3 + 19a;+12, and x^+dx^ + 2Gx-^24^.
17. Find the value of y which will make
2(y^-{-i/)x^ + illy-2)x+4: and
2(^/3^-2/•^)x3+(lV-2^/)a:2 + (2/3^-5y)a; + 5^y-l, have a
common measure.
18. The product of the H, C. F. and L. C. M. of two quantities
is equal to half the sum of their squares, one of them is
2a;3-lla;3+17a;-6 ; find the other.
1.9. If a;+a and x — a are both measures of x^+pz^ +qx+>\
shew that pq = r.
FRACTIONS.
109
20. IS x^+qx+r and x^ -j-mx+n kaye a common measure
[x—a)^, show chat q^n^=m^r^.
21. II' the H. C. F. of x^ +px-^q and x^ +mx+n, he x-\-a, their
L. C. M. is
x^ + {m — a)x^-\-px^+{a^ + mp)x-\-a{m — a){a^+p).
22. If x--{-qx + l, and i3^jr,^3_|_^aj^l^ liave a common factor
of the form x+a, shew that (jb-1)3 — ^(jo — 1) + 1 = 0.
23. 'iix^-\-px''-^q, and x^ +mx+n, have x+a for theii- H. C.
F., shew that their L. C. M. is
x'^-\-{ni — a-\-p)x^-{-p(in — a)x'^-\-a^{a—p)x + a^{a-'p){m — a).
24. If x^-\-p)x + l, 2i,udiX^+px"-\-qx-^l, have x — a iox a com-
mon factor, shew that a =
1-q
25. Find the H. C. F. of {a^ ~b^Y + {b^- -c^yJr{o^ -a^Y,
ai,ndia^{b — c)-\-b^{c — a)-\-c^{a-b).
26. If a. be the H. C. F. of b and c, & the H. C. F. of c and a,
y the H. C. F. of a and b, and X the H. G. F. of a,.b. and «, then
a6c5
the L. C. M. of a, b, c, is — ^t"*
27. If a;+c be the H. C. F. of x^ +ax-i-b, and x^+a'x+b', their
L. C. M. will be x^ + {a+a'-c)x^ + {aa' -G^)x+{a-cJ{a' -c)c.
28. Shew that the L. C. M. of the quantities in Ex. 2 (solved
above) will be a complete square if x = y^-^z^ -y^z^.
29. Kind the H. C. F. of x^+^x^+dx"^ -2a;3 + l, and
ex^+ x'+nx'--7x^-%
Section II. — Feactions.
Art. XXXIII, When required to reduce a fraction to its
lowest terms, we can often apply some of the preceding methods
of factoring to discover the H. C. F. of the numerator and de-
nominator.
110 feaotions.
Examples.
^ ac+by+a7j-{.be _ c{a+b)+y{a+b) _ c-\-y
aJ-{-2bx+2ax+bf " /(a+5)+2a;(a+6) " /+2a;*
a(«+i)(a-6)2 a
g z^+x^y-\-x^y^+z^y^+xy*+y^
x^—x^y+x^y^—x^y^-\-xy^-'y^
Here the numerator is eyidently (x^ —y^) -^ (x-y), and the
denominator is ^. The result is .'. --I^-^.
a+y x-y
{x+y)^—x^—y^ 5x^y+Wx^y^ -{-lOx^y^ + 5xy*
{x+yy+x^-^y^ = ■(cc+^)4-a;V + (^^+2/l)f-aJ,V
_ 5a;?/-{a;^+y^ + 2a;j/(a;+y)}
~ (a;S+2/2+a;'/){(a;+?/)^+a;«/+a;2+?/2-iC2/}
5a;y(a;+y)(a;^ +xy+y^) _ 5.ry(a;+y)
2{x^+xy-\-y^y ^X'+xy+y^)
V *2-12a;+35
a;3_i0a;3+31a;-30
Here we see at onee that the numerator=(a;-5)(a; — 7) ; and
it is plain that x— 7 is not a factor of the denominator; we .•. try
x — 5 (Horner's division), and find the quotient to be x^ —5a; +6.
x-7
. the result =
6.
x^-5x+Q
a.4+2a;2+9
a;* -4x3 + 8a; -21
The factors of the numerator are at once seen to be a;'+2a;+3,
and x^ — 2x+B, of which the latter is one factor of the denomin-
ator, the other being (Horner's division) ?:" - 2x— 7 : .'. the result
x^+2x-\-B
z^-2x-7
FRACTIONS. Ill
«
Exercise xlii
Reduce the following to their lowest terms :
a.2_7a; + 6 Sm/'' -ldxy-\-Ux
2.
8.
5.
x^-2x"--8x-96 72/3-172/2^-62/
x^+ax^ -a^x — a^' x^ —5x^ -{-Ix-S'
a;3-3a;+2 x^ +2x^+9 ^
^ + 4a;2_5' ^ _ 4a;3 -I- 4:X^ - 9*
2+J.r xl + 2x^+^
26+(62_4)a; - 26a;2' a;5+4a:
5a^+10a4.r+5ff.3a;^ 20a;4+a;--l
a3jc+2^^3 + 2rta;3 +a;4 ' 25a;4 + 5a;3 - a; - l'
a;' — iv^jf +x^y^ —x'^y^ +x^y^ —x^;/^ +^y^ — y'
^' x'' +x^'y+x^y^ +a;*2/H- jc^z/^ ^a;2,/5 _|_ ^yG~^y7'
I a b
Sa'^x^ - 2ax^ - 1
^'+ r + ^h»z/+2/*
8.
abc{a — h)(b — c){c - a)
^- aS(6-c) + &3(c-a)+c3(a-6)
10. From Ex. 4 (solved above) show that
. (a-6)5 + (6-c)5+(c-a)« ~ 5(a-6)(6-c)(c-a)
(a;+y)^-a;''-2/^
(a;+2/)^-a;^-2/'*
12. Shew that
\a-hy-^[h-cy. + {c-a)'' 7 ,, ^^, ^, ^
112 FRACTIONS.
Art. XXXIV. In reducing complex fractions it is often
convenient to multiply both terms of tiie eompiex fraction by the
L. C. M.. of all the denominators involved.
EXAMPIiES,
1. Simplify K^-Mi)-t(l-t:«)..
Here the L. C. M. of all the denominators involved is 12 ;
.-. multiplying both terms of the complex fraction by 12, and
removing brackets, we have
6a;-f8-8+6a; _ 12a; _ ^x
21~4a;-17 ~ 4-4* '" 1-x
a — b
II. a-
l+ah
^^aia-b)^ Here multiplying both terms by 1 ^-ah, wa get
l-\-ab
a{l+ah)-a+o &(a2 4-l) .
3.
Here multiplying both terms of the frac-
tion which follows x-1 by 4-a;, the given fraction becomes at
gjjQg , and now multiplying both terms by 4, we
4 — X
«-l+
have
4
4 4
4x— 4+4 — a; 3a; .
It may be observed that when the fraction is reduced to the
form -^ -^ — , we may strike out any factor common to the two
b d
denominators, and also any factor common to the two numerators ;
it is sometimes more convenient to do this than to multiply
du-ectly by the L. C. M. of all the denominators.
FE ACTIONS. 113
la+b a-b\ la^ + b^ a* -h^\
4. Simplify [^^ + ^ ^ [^^^-ZTbi - ^s+li' '
Here the numerator of the first fraction is {a + by^ + {a — b)^
and the denominator is a^ -b^ ; tlie numerator of second fraction
is (a2+Z,3)a_(«2_j2)3^ and the denominator is a-^-b*", the
former denominator cancels this to a^-i-b'^, which, of course, be-
comes a multiplier of the first numerator :
.-. Wehave — ^2_j_^3^2_^^3_^2X2 - 2.(3/yJ
Occagionally, we ai once discover a common com^^iex factor,
strike this out, and sin^plify the result.
a "^ /' "^ c M 1 \ 8 1
5. -:; : ^T ^ : here the den. = V ~t\ —~^
^ "^ iis" - "^ + ^
/I 1 IWl 1 1\
ss 1- ■-; — I — + -7- , and cancellmg the eom-
\a b c I \a b c j °
mon factor we have
1
1,1 1 , and muItiDlying by tibc, this = .
■ r ~r — — " bc-^ca — ub
a o c '
Exercise xiiii.
Simplify the following :
l-^{l-^(l-aj^} a-b '^ a+6
a-\-b a — h
a~b g+ft
^ - ^{l-i(l -^)T <»+6 a^)
a — b ~ a + b
X jc_ _3_ 1
x-i-y x—y 1—a \-\-a
2.f
^ - + '-
8. 1 +
aj*- y* 1 -a 1 -\-a
1
a x—y
1+a x-\-y
114
4.
FHACTIOWS.
a'+h^ 2^>2 1 1
a+b a~h h^
c-\-d ^ c-d
a+b a~h'
+ :r—i «+&+-
a
6.
a;— 1 ?/.— 1 T*:*?*
7.
yz~ zx — xu 1 1 1
— + —--{- —
— 4. — A g'^ + fe^-f c*
^2 + ^2 + ,.3 + ~7ii3,.3
a /) r;
-r— -U -I-
DC ac ' ab
/«+6 a^+fegy /„^5 a"--b^\
\a-b "*■ rt2-62 I ^ \^+^ ~ a^^^j
1 1
— 4-
'■1 11 ^^ 26c [
a b
■c
\x-\-i' I ' .c- a i
tr^^T" m ranMiTrf-iTi — n — fwn
(1— .)+! WJ+^+t-J
2(1 -a;) (l-a;)2
— >^ 4. > 4.1
14-x ^ (1+a;)- ^^
l-a;
a;
+ 1 +
J/_
1
—
y3
0
7;
a;
x-^
^«
a;
7'
•
V '
—
-1 +
1
-p
.7
X
/
X •
FKACTIONS.
]15
/a~bY (a--bY {ci~h\
^[^nl -[-v-J - 3(— -7/4-1
13.
x^ +yAi/-'rx^y^+x^'^y^-{-xy^+y^ ' \x+y/
' [l^^ + l~x+xV "^ [l+x-hx^ ~ 1+xy'
16. Find the value of
Zi^^^v + ^b^^2^x ^l^enc. = i(a+6).
17. Find the Talue of y'{l -i/{l -x)}
\l + b} \l+bl
18. Find value of
l/ia+hx) + i/{a-hx) ^^^^ ^ ^ 2ae ^
l/{a'{-bx) — i/{a-bx} 6(ln-c2)
Art. XXXV. When the sum of several fractions is to be
found, it is generally best, instead of reducing at once aU the
fractions to a common denominator, to take two (or more) of
them together, and combine the results.
Examples.
1, Find the sum of
x+y y-x ' x^-y*
2x-2y 2x+-2y x^+y^
Here taking the firgt two together we have
i^+y)' + i^-y)' _ t±yi ; now add this to _ -^IH^'.
2(a;3-?/3) x"-y' x^+y"
and we get
a;*— 2/* a;*— 2/*
116
FRACTIONS.
2. Find the sum of
l-\-x 4x 8x 1—x
r^ "^ 1+x^ + T+^ ~ l+i'
Here, taking the first and the last together, we have
(l+^)^-(l-a;)3 _ Ax
l-a;2 - i_a;3'
taking this result with ihe second fraction, we have
8a;
W-^- 4- -^—] - -
\l+x^ ^ 1-xV - 1
.+x^ ' l-aV 1-x^
now take tliis result with the remaining fraction and we get
^\i-x^ + iT^j = r::^'
ar'" a;-" 11
8. ^;rri - ^r:^ - ^^rzi + ^jqH' Taking in pair.
those whose denominators are alike, we have
ic"^ — l a;^" — 1
• ^iTZl - ^T^ =x"""+x" + l-(a;"-l)=ar"'+2:
The work is often m zde easier by completing the divisions repre-
sented by the fractions.
2a;+l 4ic+5
4. Pmd the sum of 1+ oT^ITTv — o^lTo' By dividing num-
erators into denominators, this
3 1 3 1
= 1 + 1+ sr^ -2-
2a;-2 ~" 2a;+2 " 2a;-2 ~ 2a;+2
3a;+3-a; + l _ a;+2
- 2x3-2 ^ a;2-l'
x x—9 x-\-l x—S
2 2 2' 2
1+ ^32 +1- ^=7 -^- ^:ri -1+ ^THS'^^
2_ ^_ ^_ 2 2(2a;-8) 2(2a;-8)
fl;-2 ^ a;-6 a;-7 a;-l ~(a;-2)(x-6) (a:-l)(a:-7)
f _1 1 )
= (4a;-16) |a.2_8a.^l2 " a;3 - 8a;+7/
= (80 -20a;)-=-(a;4-16a;3 + 83a;3 - 152.r+84).
[denommator=(a;2 -8a;)2+19(x3 8x)+84].
FRACTIONS. 117
6, Find the value of
TT' + Kl when X = — r \
x-2a ' x—2b a + b
Bydmsion,l+^32;^-h 1+^32^
= 2 + 4|-~|j- + ^,\; but the quantity in the brackets
(a-\-b)x~4:ah
= ~{^^2a)i^~2h = 0 «^^^^ {a + b)x = Aab
:. the value of the given expression is 2.
Exercise xliv.
Simplify the following :
, ct — a x^+ax+a^ x^ — a^
5 x+a x^—a^
^ a^+bs a^-Sa^b + 3ab^ -b^ a(a - h) - b(a - h)
/ 1 _1_ 2a X
\a -\- X a — x a^+x^J
I 1 J. 2a; \
\a + a; a — x a^ +x^l'
a —
a;
1
a —
«
h
a —
b
2 -
Bx
, a h ab ah
4. _ J_ — -f -.
a •\- h a — b ab—h^ a'-+ab
. 3+2a; 2 - 3a; IQx—x^
^' 2-a; ~ 2 +a; "*" a;2-4
1 1,1
4a3(«+«) 4a3(a-a;) ^ 2a2(a2+a;3)
. 1 /3^+2y)\ _ 1 /3a;-2yl
'• 2 \3x-2i/)/ 2 l3a;+22/i
a;+l a;-l l-3a; , x
'^' '- .1 ~ 2.C+1 a;(l-2a;) "*" a;(4a;2-l) "^ a;(l'6a;*-l) •
4 9 x-\
-I-
.2x+2 a;+2 ^ 2(a;+3) (a,-+2)(a;+3)
12.
H8 FRACTIONS.
10. ^(^+y) _ %-^) _ 4(a;'^-y") _^ Hx*+1/^)
x — y x+y x^-{-y^ «^ — 2/*
( 1 1 )
\x -\- a X + b f
I a+x 4:ax 8a^x a-x \
\ a^x "*" a^x^ "*" 'oM^ ~ «+« J "*"
I ffl2_^2 + a4 + a;* ~ a^+x^\'
^o 5a;-4 12a;+2 lO.r+17
13. -I- — — •
9 ^ llic-8 18
,^ 12a: + 10a , 117a+28x _
Jo. + fc H — — lo.
3a;+a 9a + 2a!
4a; -17 8a; -30 10a; -3 5a;-4
^^- ~^1T " 2a; -7 + ^^T ~ ^T
17. Find the value of ^:^:^32Z- + ^+b^rd
nhena-\-h= --_r^.
ar'" ^"a;"* a;"" «'"
a;"-?/" af+r/" a;"-)/" ^ a;"H-J^"'
19.
{a-bf- (a - by- _ 1 1
1 1 _ 1 _
1+a; 1 -^ 2 _ 2a;3_
^-^- l-a;3 + i+a;3 ~ l-a;3 a;« + l*
FRACTIONS. 119
Art. XXXVI. The following are additional examples in
which a knowledge of factoring and o. the principle of symmetry
is of advantaofe.
Examples.
Cancelling the common factor z — y-{-z in the two terms of the
first fraction, there results ■^- , hence by symmetry, the
denominators of the other two fractious will be x-\-y+z, and the
numerators will be y-\-z-x, z+x-y; .'. sum of the three
numerators = 33 +?/+0, and the result =1.
^ „. ,., ab be ca
2. Simplify 7 ., .. + ,, .,. — s +
{c-a){c~h) ^ {a-b){a-c) ^ {b-c){b-a)
The L. C. M. of denominators is evidently {a—b){b — c){c — a).
This gives for numerator of first fraction —ab{a — b), and by sym-
metry the other numerators are —bc{b-c), —ca(c — a),
ab(a-b)-i-bc(b — c)-^ca(c-a)
. : we have — -. txtt — w ^^ '
(a — b)[b — c){c — a)
_ {a — b){b — c){a — c)
{a — b)[b — c){c — a) ~
2. Eeduce the following to a single fraction :
a h c
+ -nr—xn — z^7:7-^^ +
{a-b){a-c){x-a) ^ {b-a){b-c){x-b) "^ {c- a){c-b){x-c)'
Here the L. C. M. is {a-b){b — c){c-a){x—a){x-b){x—c) ; the
numerator of the first fraction is
~a{b — c){x—b){x — c), and .*. by symmetry that of
second is —b{c—a){x — c){x — a), and that of third is
— c{a — h){x — a){x — b); and their sum is
— {a[b — c){x-b){x — c)-^b{G-a){x — c){x — a) +
c{a — b){x — a){x~h)).
This vanishes if a = 6, hence « — & is a factor, and .*. by sym-
metry b — c and c—a are also factors. Now* the product of these
i'^i) FRACTIONS.
is of the third degree, while the whole expression rises only to
the fourth, hence x^ cannot be involved. The other factor must
therefore be of the form nx+n, in which mis a number.
To determine n put a; = 0, and the expression becomes
abc{a — h-¥b—c-\-G — a]=Q; .-. n = 0, or the other factor is mx.
To determine m put a = 0, h = l, c= -1, and m will be found tc
be 1. The numerator is .-. x{a - lj){b — c){c — a), and the result is
X
lx — a){x—b){x—c)
o cj- Tf a + h b+c c + a
8. Simplify ^ _j_ +
{b — c){c-a) ^ {c-a){a~b) {a — b){b-c)
L. C. M. of denominators is {a — b)[b — c){c—a) ;
.'. first numerator is a^ —b^, and By symmetry
second " b^—c^,s.ndL
thkd " c2-a2;
the sum of these = 0, which is the required result.
4. Eeduce
2 2 2 (x-2/)2+(7/-z)2 + (2-a;)«
+ :-- + - .. +
x-y y-z ^ z-x ^ {x -y){y - z)(z-x)
Here the numerator becomes
^y-z){z-x)^-^x-y){z-x) + '>.{x-y)(:y-z) +
{x—yY-¥{y—z)--\-{z-x)-, which is evidently
{(•■-2/)+(^y-^)+C--^)p=o.
ffl3 + 263]3 (2a3+£3,3
Observe that the denominators become the same by changing
the sign between the fractions, and that the expression is sym-
metrical with respect to a and b. The numerator of the first
fraction is a^* + 6a^i3_^.l2a^6^^-8a369, and by symmetry that
of the other is -b^^ -Qb^a^ -l'2.b^a^ -Qb^a^ . Their sum is .".
ai 3 _ &i 2 _j_6a3/;3(rt6 _ ^fi) _ 8a3i3((^e _ ^6)_
— ^^^li^l)S'^(^a'^-.^J^^i^ and since the denoniinatur of the given
expression is (a'^—b^)'^ :. the result is a^-\-b'^.
FRACTIONS.
Exercise xlv.
Simplify tht! following :
121
1- ^U + //j -^^ \x+y
\ a — b I \0 — a I
\
a-{-b O-^-r __*' + *
^- (ir-7)(c-a) + (7-</)(.i-6) "^ (^-b)(b-c)'
_ 1 1 1
*• (a-/;)(a-c) + {b-a)(b-c) + (6--aj(f - i)'
((-6 /'~V- c-a (a — b){b — c){c-u)
(^a:^'}j)^a + c){x + ay(a-{-b){b - c)(a; + b) {a+c){b - c){x-\-c)
'^' {x-y){x-z} + {>J-x){y- z) "^ {z~x){z-y)
,,3 /,3 C3
11. ^. ' ^, + ,
(b^.~S2^)(^c+a-2b) ^' (c + a-26)(a-i-6-2c)
1
(a + b-2c){b-}-c-2a)
b2-c^ c^a^ a^-l]^
1Q " I I
■ {a-b){a-c){x-a) "^ {b - a){b - c){x- b) ^
c2
{c — a]{c - b){z — c)
122
RATIOS.
14. ^iy+^) , y(z+x) , z{x+y)
+ iT.'^'z^nrh. +
(x-y){z-x) ^ {)/-z}{x—y) {z-x){y-z)
15. [(^ + ir-+{(^-o)'+{a+cy^ _ -^ _ ._!_ a. ---^.
(a4-6)(6 — c)(a + c) a + c b — c ' a-\-b
1ft 1 1 1
x[x — a){x — b) a(b — a){x — a) b{b-a){x—b)
Section III. — Eatios.
Art. XXXVII. If -^ = -j ••• ad = hc. Now,
b d
dividing ad -he by ca we have - ....
(1)
a b
" ad = bchvcd " — = —....
i'2)
, , 1 ^' c •
•^ da
(3).
ma + nc -
Also , , , — each of the given fractions . . .
(4)
, mb[ , )+«(/( , ) Imb+nd) ,
_ wa-\- nc \ b / ' \d / ^ ' b o r
mb-\-nd mo +nd imi + nd b d
A very important case of this is m = 1, n= ±1, hence
a c n -{-(■, a — c
b ~ d ~ b+d ~ b — d
(5).
a — h e — d.
Also . — ;
(6).
■^^s" (,^b cH-ci
For by (2) and (5)
(f b a—b a+h a — b c — d
c ~ d ~ c — d ~ c -\-d " a-j-b ~ c-\-d
^ X, a-b _ b _ d c-d
a+b « , 1 '^ , c+d
d d
RATIOS.
123
Generally, to prove that, if — = — , anv fraction whose nu-
merator and denominator are homogeneous functions of a and b,
and are of the same degree, will be equal to a similar fraction
formed with c instead of a and d instead of h : — Express the first
fraction in terms of — , and for — substitute its equivalent —
b b W
and reduce the result.
By (2) the fractions may be formed of a and c, and b and d.
If — ■ - — = — , . LxL _ — or — or — (7)
b d J mb-{-nil-\-pj b d f
I I w
ma-+-'nc-\-pe
'ij) + ""(t) + "At)
mb-{-nd-f-pf mb+nd-l-jif
a
{mh-{-nd-\-2)f)Y a
~ mb+nd-^pf b
U± = ^ and ^ = A
b d n q
ma + pr pan^mc ma pa
nb + gd qb + nd ~ nb qb
■n ma DC ma-\-pc ■, ,_.
For — = :^ = -r^=-^, by (5)
nb qd nb + qd
pa mc pa + mc
or &c . . . . C8)
qb ')ld qb±nd
But !!^ ^ -^, hence the equality stated in (8).'
nb qb
If ^ = _1 = _!. and !!L = Z. = ^,
b d f n q s
rr c-{-pc + re pa -^ re + me „ ma
-irr-ij—e = 7.7 . , ,- = &c., = — = &c. . (9).
nb±_qd+sf qb+sd + nf no ^ '
If an upper sign be taken in a numerator, the corresponding
upper sign must be taken in the denominator ; if a lower sign,
the corresponding lower sign, otherwise all the signs are inde-
pendent of each other.
124
a e
1- I^T = T
The given fraction =
EA'iiOS.
EXAMPI^ES,
i.1. i ^^ ~ 46 Be — Ad
a ^ c ^
Fjc - Ad
a c
7-r- + 5 7-7 + 5
h d
lc+5d
2- I^T = -J '^'^^ ^^''^' S^h^Ah^ = Bc^d-Ad^'
Dividing the given fraction by b^ we have
, and this becomes, on substituting for -^ its equal --7- >
2p- + 37J ^
2c3 + 3r2^
^3 i.;,R i^s \
3. If 3a = 2&, findthe vahie of ^^^^^^^. This = (rj + l) r^
— _ — -) [by dividing both numerator and denominator by
b^ b I
a 2
o^j. But from the given relation — = -3- we have, by substi-
a
tuting for -t-»
4- I^ T = T- ^^'^^^ *^^* 03 + ^3 X -J = (^;:p;/j •
a b a+h
We liave ^ = T = c^TT/- ^^'^^
— ^- — = -TTTT + 1M--^+ M= Ti-, and this muUiphed
by y gives ^=1^-/;.
RATIOS. *■-'*'
„ r^ x^+rix^ -bx + c x^ + ax—h , . , ^
o. if _- -„-— 7 — -— = -;; --, shew that x = — ■-.
ic* -fl,r- + to+t' x- — ax-\-b' a
Multiplying both terms of second fraction by x, it becomes
x^-\-ax^ — bx
Zz — n^2~L.h^'^ ^<^^ 68,ch of the given fractions =
difference of numerators
difference of denominators ;
=
c
^. =1/. x^+ax-b -x^ -ax+h
or lax =
6
2b .-. «= —
a -
6. If
ace ac + ce-\-ea a^ + c' +«•
For
ac c<; ea ac-\-ce-\-ea
bd-df^fb-hd+df+fb' ^y (7)makingm=r.=i, = l.
Also
a2 c2 ^2 a,2^.c2_j.^2
fcs -</2 -yg- ^2 4.(73+^2- By (7).
afl 0.2
But 73" = T^ hence the required equality.
The problem is a particular case of (9), with all the signs +
and a for m, b for n, c for p, &c,
(If the fractions given equal to one another have not monomial
terms, instead of seeking to express the proposed quantity in
terms of one fraction and then substituting an equivalent frac-
tion, it is often better to assume a single letter to represent the
common value of the fractions given equal, and to work in terms
of this assumed letter.)
„ J. a+b h-\-c c4-a
prove that 32a+85i+27c = 0.
Assume each of the given fractions = «, so that a-{-b = 3(a — b)x,
b-\'C=i{b~c)x. c-\-a = 5{c-a)x,
126
RATIOS.
01' -^- + --J- + -^^ = x{a-b-\-h-c-^c-a) = 0.
/. adding these fractions we have 32rt+356+27c = 0.
This example might also be worked as a particular case of (7),
thus ■
a+b
h + c
B{a-b) Mb-c]
5(c — a).
20(a+Z)) + 15(6+c) + 12(c+a) _ 32a + 35b + 27c
60(«-//) + 6U(6-c) + 60(c-a)
a-\-b
0
32a + B5b+27c = 0 x
3{a-b)
= 0.
8.1f^ + ^ - '^i
a + c +
a
d I 17
c
If
e
+ 7
prove that
«2 4.^24., 2
b + d-hjj "^ W-^Ij2~J^}^'
Transposing terms, &c., we have
a2 2ac . c2 t'S 2ce
fe2
C2 t'2
i(Z "*" ;^ "^ 7^
^/■^"^ £f2 "^ ^'
or
la P \3 / /? c \2
that is, the sum of two essentially positive quantises = 0 j
.'. each of them must=:0 ; heuce we have
a c e c
-J- — -y = 0, and -— — — = 0;
b d ' J (jf
c
~d
e
7
Also
«2 ^ a2_|.c2+g2
/a+c+e\-.
RATIOS. 127
Exercise xlvi.
, Tf « c a"'^ —ab-\-b^ c^ — cd + cl^
1. 11 — = — , prove ^ — = — ; — - .
b d ab-W^ cd-U^
2. It — = — , prove = , — , _ I— ^|.
b d^ b-'-d^ \b-dj - \b+dl
3. Given the same, shew that each of these fractions
= yXb-^ + d^l'
i. U 2x = Sy, write down the value of
x^:/ + xj/-^ + 2ii^' [x^-ifi)^
5. 11 — = — = — , shew that — = —.
b d J b mb — nd - pf
?}. From the same relations prove that — = ( '■ j •
^3 Kb-md—n/J
7. If J±^ = AfJ±^+^\ then 2.3 = (6_«)-(6+«).
1 — x a\l—x-t-x'-'/
o- ^ /,' , \ ■// \ = «, prove that a; = ^i — o'
f. Tr rnx-\-a-\-h mx—c—d b — c
9. 11 ^ = , prove x = .
nx-{-a-T-c nx — b — d n—ni
in 1" '^■^^ b — c c—a a-\-b-{-c
ay-j-bx bz-\-cx cy + az ax + by-\-cz
then each of these fractions = , «-f-/*+c not being zero.
x+y+x
11. K«t^ = Jp^^ = -^±^,then8a + 9b+5c = 0.
a-b 2[b-c) S[c-a}
VC-\- \/{a- x) 1 a — x /I— a\*
12. If \ , r = — > shew that = hn-
ya — Y{a — x) a ■ a \l+a/
x^ — yz 7/2 —xz
13. If .. _"'T = ."{iTZ — V ^^" ^' ^' ^ "® unequal, shew
x[i yz) y^i. • xz)
that each of these fractions is equal to x+y-rz.
128
RATIOS.
x^-\-2x + l //-+2// + 1 , . 1
14. If ^ „ , o = .. b . o' shew that each of theso
x-—2x + o y^—2,i/-{-^
fractions = {xy — 1 ) -f {xy — 8 ) .
25x2-16 8(a;2J,4) , , a;-4 8
15. If —,(. — ^~Q = o r~' shew that - ~p ■
10a; +8 2a; — 4 x + O; 5
ibc V + ^b y+2c
IG. If 7/ = T~-r~ shew that ' ^ + ~o =2.
25rt2+2762 4-22t;2=o.
18. If -^~ = -T, = -^ — , shewthat((2^_|./,2^+c32,
x^ -yz y- -zx z^-xy
19. n -r^ — = ,-— — = ■ t .' then wiU (.( - h)x +
[b — c)?/ + (c — a)z = 0.
^°- -^ T = T "= 7 *^®^ (//^ +c/3 + /•2/ - M -4- ,/4 4-/4 ■
hx + ay cy-\-hz az-+cx , ,, ,
21. If --~ = 4-^ = ' shew that
a—b b—c c — a
{a-'(-b-{-c){x-\-y-\-z) = ax-\-by-{-cz.
22. If --0 , 2 , 2-r-r = ^r:.' shew that each of these
expressions =1,
23. If I (^^3 = j(,'~-^) - M^)^ and a, /., o he
different, shew that ]6« + 116 + 15 =0.
24..If(^-^'f = ^-C pit)vethatx2+^2^22+2a:i/. = l.
\y-\-zx) l-x- ^
as. If ^* = ^^ = _^, show that a. + /^ + c = 0.
x — y ?/-2 z — «
2G. If _ = — , prove that — ^-- :^ ' .— f- /Vrfx-
RATIOS. 129
to
27. ii — - - — , then each is equivalent
h - d f
./'x'^rL^.*, hence shew that
a h <? 1
. _ _ when
2z + 2a;-?/ 2.c+2?/-2 2?/ + 2z-a;
X y z
2a + 21)'^ ~ 26 + 26'- a ~ 2c + 2a^*
c.r. Tx- <^ c ,1 , /a-6V //a^" + 6''"\
28. li _ = — , prove that — - = a(- ^ , . .
29. If ^^^ = 77-^ = ^-^' PJ^ove that
^{y-z)-{-JL{z-x)-^I^{x-y)^0.
a 0 c
30. If - " - '= ,/ , = - * , ,. then wiU
/x(?i^ — ///2) w?/(/z — nx) nz[mx — iy)
31. If e ^ y ^ -^ •', aua y = *^-^ ', shew that
y ^
X
z
»2
32. If ^--^ = ^q^ = '^ = 1, shew that
a3 0^ c^
38. If !^ = -!^ = ^, and ^ = ^ . ^1 . 1,
prove that __+_ + - =3^:.^y3+^-
34. If -^ = ^ = 7 = &«•' ^^'^«°
J
130 COMPLETE SQUARES.
a-, a a^ a„
as. If T^ = 7^ = 7^ - . . . = 1^. then
*] ^o t'3 ^„
CTlffg - g^ rtg + . . . ( - ly-^an-ian ^ «i v/fl,a3 -^-a-iV'a^a^ +&C
&i ^2 - ^2^3 + • • • ( - 1)"~^^»-A ^l/i'X + ^2T/^3^" + &<^-
A+B+C A ^ B ^ C
36. If 1 = h y- + — ,
abc a b c
and (J + 5+C)(a4-& + ^) = ^4a+£&+Cc,
1+^2 + i^^2 + i+c
then -will -, , ,„- + i— rr^ + T~m ~ 0-
and also — - — + + = 0.
1 11
'a b ' c
xh yk zl x^ ?/- ?2
37. If -;■ = 77 = -v' and -^ = y^ = -r =1)
a2 6^ c^ • rH 6^ c'
I X If 2 \2 a2 1,2 c2
thenwill^ly + y + vj =,)^ + 7^ + 7^
7
Section IV. — Complete Squakss, &c.
1. What quantity must be added to x^ +r>x to make it a com-
plete square ?
Let r be the quantity.
Then fl;2+;>a; 4- r = complete square = (.c + y7~)^
Equating coefficients we have
2|/r= r
7'" / /. \ 3
" = T - \2
Or thus: Since {a+x)" =a- -^ 'lax + x^ ; we observe, (See
Art. XII), thsit four times the product of the extremes is equal to the
square of the mean,
.'. Ax^r=p^x^ ;
r z= l — \ » as before.
COMPLETE SQUARES. > 131
Or we may extract the square root and equate the remainder
to zero . thus
P
X"
/>
4
p2
ISTow, if the expression be a complete square, this remainder
must vanish ; hence we have
jr)3 I P ^' ^
4 \ 2
2. Find the relation connecting a, b, c, if ax^-\-hx+c is a com-
plete square.
Assume ax'-^ +bx+c = (^ a.x+ -i/c )^ =ax^ +2-i/(ac).x+c.
Now, since this holds for all values of x, we have 2 ^/ac — h, or
6- = iac, the relation required,
3. Determine the relation amongst a, b, c, in order that
a'^x^+bx+bo+b- may be a perfect square.
As in Ex. 1, we have 4:a'^x-{bc + b-) = h^x-^ ;
• i _ -1 - 1
4«3 h ~
Or thus :
Assume a^x^+bx+bc+b^ = (ax+l/'oc + b■^)''
= a^x2+2ayb^b^ + bc+b^ .
Equating coefficients, we have b = 1a^bc-\-b'^ ;
.♦. — — , — = 1, as before.
. . 4a" b
The same result may also be obtained by extracting the square
root and equating the remainder to zero.
132 COMPLETE SQUARES.
4. Show that i{ x*-\-ax^+hx^+cx-\-d be a complete square,
the coefficients satisfy the equation c^ —a^d = 0.
Is it necessary that the coefficients satisfy any other equation ?
Extracting the square root of x'^+ax^+hx^ Tcx+d in the
usual manner, we have for the final remainder
Now, if the expression be a complete square, this remainder
muist vanish ; and, that it ma^ vanish for general values of x, we
must have
a I a^\
1 / a*\
(2);
^3
Eliminating * - x' we have <;- -«3f/ = 0 . . . (3).
The coefficients must satisfy the equations (1) and (2), an J
therefore either of these equations, together with the equation (3),
which results from them.
The same result may be obtained by assuming
x^+ax'^ -^bx"-\-cx + d = {x--if^ax-\- y/d)^
= x^+ax^ + 2x^y/d
+ ^a^x^ -taxs/d + d.
Equating coefficients, we have 2 v/d+i<*2= 6 ... (1)
and a^/d=c . . . (2).
From (2) we have c^ — a-d = 0, as before.
5. What must be the value of in and n if
4x* — 12a;3 +25a;2 — 4???a;+8w is a perfect square ?
Assume the expression = {(2aj2 - 3x4- \/{Sii)}^
= 4:x^-12x3+4x" y{8n)-\-9x-' -6xy/ {8n)+Sn.
Equating coefficients, we have Gs/{Sn)=Am .... (1),
and4N/(8w) + 9 = 25 . . .' . (2);
.-. »=2,
»i = 6.
COMPLETE SQUARES.
133
Or thus : Extracting the square root in the ordinary way, the
remainder is found to be (— 4m + 24)a; + 87i-16 ; /. -we must
have 4?/; + 24 = 0, or m = 6,
and 8m- 16 = 0, or ?i= 2.
6. If ax^-\-hx^+cx+d be a complete cube, shew that ac^ = db^f
and b^ = 'dac.
Assume ax^ +bx^ +cx+d = (x+d\). '
Equating coefficients,
b = 3a^d^ , (1)
c = 3Jd' (2);
b ^3
dividing (l)by(2),_ = — ;
c d' ■<
Also, h^=^aU^ (3);
dividing (3) by (2),^ = 3a.;
c
.-. b^ = 3ac.
7. Find the relations subsisting between a, b, c, d, e, when
QX* + bx^ + cx- +dx+e is a complete fouich power.
Assume ax* + bx^+cx^+dx-\-e = (a^x-\-^)'*'
= ax'^+4:a^e^x^ + Ga^e^x^ + 4^a*e'x-[-e.
Equating coefficients, we have
b = 4aT cT,
d = Aai ei ;
whence fc(/ = 16«t' (1).
bc = 2iaie^ = QaAa^ei = Qad (2).
€d = 1ia^ei = QeAa^e^ = Qhe (3).
8. Shew that x^^-fx^ + qx^ +rx+s can be so resolved into two
rational quadratic factors if ^ be a perfect square, negative, a,nd
,.2
equal to —- —.
134 COMPLETE SQUARES.
Since — s is a perfect square, let it be n^.
Assume x^ +px^-\-qx^-'rrx-n^
= (x^+'mx-{-n)(x" -^vi'x -7i)
= X* + (in.-\-m ')x^ + mm 'x" — n(m — m ')x -n^.
Equaticg coefficients, we have
mm ' = o
■J-
r
m—m' —
n
ni^+2mm'+m'^=p^
4:mm'*=4.^ ;
;. {m-mY=P'^ -"^q =
r3
r2
9 . =n-' = —s.
p^ — ip9
Exercise xlvii.
1. What is the condition that {a — x){b — x) — c^, may be a per-
fect square.
2. Find the value of n which will make 2x"-\-8x-{-n, a perfect
square..
3. Find a value of a; which will make x^+dx^ + Ux' + dx + dl,
a perfect square.
4. Extract the square root of
(a-b)^ - 2(a3 +63)(a _ b)^+2{a^+b*)
5. Find the values of m and n which will make
4a;* — 4a;3_{-5a;3 ^ffix+n, a perfect square.
6. What must be added to x^~V{ix^ -lGx^ + 16)-^x^ in
order to make it a complete square ?
7. The expression x'^-^x^ —IQx^ — 4:x + 4:8, is resolvable into
two factors of the form x^-{-mx-^6, and x^-^nx-^8; determine
the factors.
ex
8. Find the value of c which ^Yill make -ix^ — cx^ + 5x^ -f .y + 1,
a complete square.
COMPLETE SQUARES,
135
9. Oouiiii ihe square root of
10. If {a-b)x^ + {a + b)-x+{a^ - b^){a+b), is a complete
square, then a =36, or b=.'da.
11. Find the simplest quantity which, subtracted from
a^x^ +4tnbx-\-4:acx+5bc -{- b^c^ , will give for remainder an es.act
square.
12. a*— 4a;3—a;3 4- 16a; -12 is resolvable into quadratic factors
of the form x" +mx-rp, and x- +nx-^q : find them.
13. Find the values of m which will make x^+niax + a^
a factor oi x^ — ax^-\-a^x^ —a^x-{-a*.
14. Shew that if x^+ax^+hx'' +cx+d be a perfect square, .the
coefficients satisfy the relations
8c =a{4:b — a^), and
64r/= (46-ftS)2.
15. Investigate the relations between the coefficients in order
that ax^-{-bii^+cz'^+dx)j-]-eyz-\-fxz may be a complete square.
16. If a; 3 + ^2 -f to +c is exactly divisible by {x + dy, shew that
1(5 -d-)=-^=d{a-2d)
17. Determine the relations among a, b, c, d, when
ax^ — bx^-\- cx—d, is a complete cube.
18. The polynome ax^ + Shx^ + Sex + d is exactly divisible
by (a-x)^; shew that (arf- 6c)3 =4(ac-i3)(fcd_ cS).
19. Find the relation between p and q, when x^-'rpx^+q, is
exactly divisible by {x — a)^,
20. If x^+nax+a^ is a factor of a;4+ axS-J-aSaj^+a'aj+a"^,
shew that n^— 7i— 1 = 0.
21. Tix^-^-ax^ + bx^ +cx-\-d,he the product of two complete
squares, shew that
(46-a2)2 = 64(?, (46-fl2)rt = 8c, as/{3u^ ~m = Sb.
130 ' RELATION IN INVOLUTION*
22. Prove that a;* + px^ +qx^ -\-rx+s is a perfect square, if
,2
9 P''
p^s — r, and q = — -{. 2^s,
■ ■.23. If ax^ 4-36.C- 4-3cx + f/ contain ax'^-\-2bx-\-c as a factor, the
former will be a complete cube, and the latter a complete square.
24. If m"x^ +2^x+pq + q^ be a perfect square, fiud^ iu terms
of m, g, and x.
25. Find the relation between p and g' in order that
x^+px^+qx-\-r may contain (a;4-2)2 as a factor.
26. If x^ +px^ -{-qx+1- be algebraically divisible by
Sx--\-^px-{-q, shew that the quotient is a; + ^.
o
Relation in Involution.
Art. XXXVIII. If «a' = W/ = cc', prove that
1. (a + b'){h+c'){c + a') = (a' + b)ib' + c){c'+a)
a+b')xa' = aa' + b'a' = bb' + b'a'={b + a')xb'
b+c')xb' = bh'+c'h' = cc'-^c'b' = {c4-l')xc'
c-{-a')xc' = cc' + n'c' = aa' + a'c' = {a-{-c') xa'
a-\-b'){b-irc'){c-\-a')xa'h<c' =
a'-{-b){b'-\-c)lc' + a)xb'c'a'
a-\-b'){b + c')lc + a') = {a' + b){b'+c){c'-\-n).
a + b){a-\-bi){a'-c){a'-c') = {a'+b)[a' + b'){a-c){n- c').
a + b)xa' = aai + a'b = bb' + a'b = {b'-\-ai)xb
a+b')xa' = aa' + a'b' = bb'+a'b' = {h + a')xb'
a' — c) X a = aa' — ac = cc' —ac = (c' — a)xc
a' — c')xa = aa'-ac' = cc' — ac' = {G — a)xc'
a+b){a+h'){a'-c){a'-c')x{aaY =
b'-{-a'){b-i-a'){c'-a){c-a)xbb'.oc_'
But bb'.cc'={aa')^,
and (c' — a)(c — a) = (a — c){a — c')
:. {a+b){a + b'){af - c){a' -c') = {al + b)(a'+b'){a-c){a-cr).
RELATION IN INVOLUTION. 137
Exercise xlviii.
If rta' = 66' = cc' prove that
1. {a-b'){b-c){c'-a') = {b-a'){a-e){c'-b').
2. (b-c'){c-a){a'-b') = (c-b'){b-a){„'-c-).
3. (^c-a'){a-b){b'-c') = {a-c'){r.-b){h'-ai).
4. (^a-b'){b -c'){c-a') = {a-c%b- a!){c ^b').
(a-b){a-b^ ^ (a-c){a-c')
(6_c)(6-c') _ (^>-a) (/;-«')
^- (;/_,,),,,_«')' - (c'-6)>'-fc')'
8. Shew that the seven preceding relations may be derived
trom the single relation
(a-\-a')(bb!-cc') + ih-^b'){cc'-aa'} + {c+c'){aa'-bb')=^.(i.
138
SIMPLE EQUATIONS.
CHAPTEE Y.
Simple Equations of one Unknown Quantity.
Art. XXXIX. Preliminary Equations. AHhnnofh the
following exercise belongs in theory to tliis chaptei% in practice
the numerical examples should immediately follow Exercise I.,
and the literal examples Exercise III. Likd" those exercises, this
one is merely a specimen of what the teacher should give till his
pupils have thoroughly mastered this preliminary work. But
few numerical examples are given, it being left to the teacher to
supply these.
Exercise xlix.
What values must x have that the following equations may be
true ?
1. x-b = 0. a;-3i = 0. x-a = 0. x+3 = 0.
2. a;4-4i = 0. x-]-a = 0. a;+3 = 5. a;-4 = 6.
3. x — a = b. x + a = c. x-b=—c. 6— a;=3..
4. 8~a;=10. 5 + a;=ll. 9+.t = 4. 7-a;=-5.
5. 8+a;=-6. a-x = 3b. 2a = x + 3h. 8a = 5b~z.
6. 2a;- 6 = 8. 3a;+8 = 20. ax = a^. mx = bm.
7. Sx = c. ax = 5. ax = 0. (a + b)x = b-^a.
8. {a-b)x = b-a. (a + bx) = (a + b)^. {a—h)x = a'^-b9.
9. {a + b)x = b^-a^. (^a^ -ab + b^)x = a"^ -\-b^.
10. {n^-b^)x^a-b. (a^-b^)x = a + b. {a-'+b^)x=l.
11. (a+x — b) = {a-{-b). x — a-{-b = b—x + a.
12. 2a — x = x — 2b. ax-\-bx = c. ax — b = cx.
13. ax~b = bx — c. ax — ab = ac.
14. ax — a^ = bx — b^. ax — a^ = bx-b^.
SIMPLE EQUATIONS. 139
15. ax — a^ = h^ — bx\ ax+b-\-c = a+hx-\-cx.
16. a — bx — c = b — ax-\-cx; a + hx-{-cx''^ =ax — h-\-cx^,
17. hx - ex- -\-e = ex-b- rx~ ; 3a; = | ; 4a; = f .
18. 10;c=i — 1; rta;= — ; ax= — .
c b
in I (i ''' i «c^ ab^
19. aoa; = — + • — ^3 oca;= — _i, — ..
b a be
20. |a; = 5; |:e=:8; -5.^ = 2; -30;= -06.
21. 020; = 20: -80;= -2; •4a;=-6.
a; ax
22. •lBa;=l-8; — =. 6 ; t = c.
ax h X ax ..
a + ft a a — h a-t-b
*>4 a; = X =
" ' a- b b ' a-hb b~a'
a a b —a a — b
b—a a — 0 rt + o b-\-a
a-^b a — c 112 3
a-f-c rt+o a; 2 a; o
1 11 an hi 1.1
27. — = -7-5 — = -T' — = — ; — = -^ + -^
X ab ^^x b X c X 3 4
«^ -^ , 4 33 1 « , ft n
28. H7T + — = ^ IT' — + — =0.
yu oa; oa; o a; c
29. A = 6-JL; ^^ =7+-^-.
a;-7 a;-7- 3a;-4 4-3a;
30. (a;-4)-(a;4-5)+a; = 3; 2x- (a;- 5)- (4 -3a;) = 5.
31. 2(3-a;) + 3(a;-3) = 0; 2(3a;- 4) -3(3 -4a;) +9(2 -a;) = 10.
32. a(l-2a;)-(2a;-a) = l ; a;- 5(a-a;) = te- 5a.
38. wa;(3a-4)+3w<a;-3a+l=0.
34. a{bx- c)+h{cx — a)-'rc{ax-h) = 0.
85. a{ax — b)-\-h{cx — c)-\-c{cx—a)=;iQ,
140 SIMPLE EQUATIONS.
86. a{bx-a)-\-b{,:x-h) + c{ax-c)=-0.
37. a(x-2b)-\-h{x-2c)+c{x-2a) = a^-\-h''i+c*.
38. 3(3{3(3a;-2)-2}-2)-2 = l.
39. 9(7{5(8a;-2)-4}-6)-8 = l.
40. i{iaU(^+2) + 2}+2)+2} = l.
41. i{iaU(x + 2) + 4}+6) + 8}=l.
42. UUUh^-^)-i}-i)-l=^-
43. m{m^~H)-n}-n)-n=o.
44. f|{-A(f{f(!a;+4) + 8} + 12) + 20}+32 = 58.
45. |{|(f{M-^-+7)-3}+6)-l}=4.
46. r{5'(jLi{w(?/ia; — a) — 6} —c)—d\—e = 0.
47. (l + 6a;)2 + (2 + 8a;)3 = (l + 10x)2.
48. 9(2a;- 7)2 +(4a;- 27)2 = 13(4ic+15)(.'c+G).
49. (3-4a;)2+(4-4a;)2 = 2(5+4a;)3.
50. (9-4a;)(9-5a;)-f4(5-a;)(5-4a;) = 36(2-a;)'. '
Art. XL. In order that the product of two or more factors
may vanish, it is necessary, and it is sufficient, that one of the
factors should vanish. ThuB, in order that (x — a)(x — b) may =0,
either a;— a must = 0, or a; — 6 must =0, and it is sufficient that
one of them should do so.
Hence the single equation {x~a){x — b)=0 is really equivalent
to the two disjunctive equations, either x — a = 0 or x-b = 0, for
either of these will fulfil the condition of the given equation, and
that is all that is required.
Similarly, were it required to find what values of x would make
the product {x—a){x — b)(x — c) vanish, they would be given by
a; — a = 0, or a; — i = 0, or « — c = 0 .'. a; = a or 6 or c.
Hence the single equation
(x — a){x—b]{x—c) = 0
is equivalent to the three disjunctive equations
X — u = 0, ox x — b = 0, or x — c — 0.
simple equations. . 141
Examples.
1. Solve j;2-x- 20 = 0,
The expression = {x — 5){x-\-4), which will vanish if either of its
factors does, that is, if ic - 5 = 0, or ic+4 = 0,
.". x= 5, ov x= —4-
2. Solve a;4-j;3-a;2 4-^ = 0.
This gives a;3 (a; -l)-a;(.«-l) = <x-l)(a;2_l)
= x{x—l)(x + l){x—l), which vanishes fo»
x = 0, x= 1, x= —1.
3. Solve aj^+a-it;^ —ax- aS = 0.
This = x{x^ - a)-\-a "- {x"" - a)
= (x + a*)(x^ — a), which vanishes foj
x+a^ = 0, and a;* — rt = 0, or
x= —a", anda;2=a.
4. Solve x^{a~h) + a^-(h-x) + b^{x-a)=0.
The factors of the expression are (Ex. 2, page 79)
» —a, x—b, a — h; hence the expression vanishes if
x—a = Q, or a: — 6 = 0.
5. Solve 221a;2-5a:-6 = 0.
Here we have the factors 17a; -3 and 13a; +2 ;
.•. the equation is satisfied by^ 17a; — 3 = 0, or x = ^,
and 13a;+2 = 0, or a; = - /j.
6. Solve 2a;4+2a;3 + 6a; -18 = 0.
In this case we have 2(x4-9) + 2a;(a;3+3)
= 2{x^-\-3){x^ —S-{-x}, which vanishes for
a;3+3 = 0, ora;2+.c— 3 = 0.
7. Solve (.r-a)3+(a-6)3+(6-a;)3=0.
The expression is equal to 3(^x — a)(a — b){b - x),
ftnd therefore vanishes for x — a = 0, or a; = a ;
and for a; — i = 0, or x = 6.
142 SIMPLE EQUATIONS.
Exercise 1.
1. If an equation in x has the factors 2a; — 4 and 2a; — 6, find
the corresponding vakies of x.
2. If an equation gives the factors 2a;- 1 and 3a? — 1, %vhat are
the corresponding values of a; ?
3. If an equation gives the factors dx^ — 12 and 4a;- 5, find the
corresponding values of x. '
Find the values of a; for which the following expressions will
vanish ;
4. a;2-2a;+l; 4a;2-12a;+9.
6. 9a;-^-4; x- — {a+bY; x^-2ax + a».
6. a;8-0a;+20; 4a;2 - ]8a;+20.
7. x^+x-6: x2-a;-12; Oa;^ ...9a;-28.
8. 6a;3-12a;+6; 6a;2_13a;+6; 6a;2-20a;+6.
9. 6a;2- 5a;- 6; 6a;2-37a;+6; 6a;3 +a;- 12.
10. A certain equation of the fourth degree gives the factors
x^ —x — 2, and 4.<;3 — 2a;— 2, find all the values of x.
Find values of x in the following cases :
11. x^-2bx''-Sb^x=i).
12. x^-ax^+a^x~a^=^0.
13. a;3-2a: + l = 0; a;3-3a; + 2 = 0.
14. x^ — 2ax^+2a^x—a'^-0-
15. x^ + {b + c)x'^-bcx-b^c~bS^=0.
x-a x-b _ {a -by a;«-a^
x — b x-a {x—a){x — b) {x~a)(x — b)*
17. x^-bx^-a^x-a-b = 0.
18. 3.c3-f4rt6a;3-6«263a;-4a363=:0.
19. x^{a-b) + a3{b-x)+b^{x-a) = 0.
^^- (^a-b){a-c) "^ (6-cj(6-«)
/a;-2a\ 3 /2a;- o\
21. X -.— + <^\^n-i
= a;2-as
SIMPLE EQUATIONS. 143
ah , hx , ax 1
{b-a){x-a) ^ (x-a){a-b) ^ {a~i){b-x) a~b
24. Form the polyuome which will vanish for x equal 5, or
-6, or 7.
25. Form the polynome which will vanish for x = a, or 4a, 01
3a, or — 4fl.
26. Form the equation whose roots are 0, 1, -2, and 4.
27. Form the equation whose roots are l-}-\/2, 1— v/2, 1 - ^.'S,
and 1 + /3.
Art. XLI. In solvino; fractional equations, the principles
illustrated in the section on fractions may frequently be applies;
with advantage, as in the following cases.
When an equation involves several fractions, we may take two
or more of them together.
Examples.
1. Solve ^J^ + 'L^-ll^ = '^^±^.
14 ^ 6iB4-2 7
Here, instead of multiplying through by the L. 0. M, of the
denominators, we combine the first fraction with the last, getting
at once
7a;-3 7 1
g^2 ^ n ^ T ■'• '''x- 3 = 3a; 4-1, and 05 = 1.
2 2^+8| _ lSa;-2 ^ _ Z^ a;+16
~9 17a; - 32 "^¥"12 "36^'
In this case, taking together all the fractions having only
numerical denominators, we get
8a;-f34+12a;- 21a;4-a;+16 _ 13a; -2
36 17a; - 32 ' °'
25 13a;- 2
18 17a; -32'
.'. 425a;— 800 = 234a; -36, hence a; = 4.
144
SIMPLE EQUATIONS.
It is often advantageous to complete the divisions represented
by the fractions.
4a;- 17 _ 3| - 22a; 6 / x^
^- 9 —33" = *- ^i^- 54
Here, completing the divisions, we have
4x 17 1 2a; & x
9 ~ir~'9""^y^~"^''"'9"'
10a; a; 6 6
-9--2 = a;+- - - .-. -2 = --, ora; = 3.
ax-\-h ex -\-d
ain+h cn-\-d .
a -\ + r + = n-\-c
X - VI X — 11
{a7n + h)(x—')i) + {cn + d)(x — vi)=^0
{ani-\-h + c7i + d)x={(i+e)mn-i-bn+din.
6. Similarly may be solved
ax+h cx-\-d cx^-^-fx—g
+ + TT \ = a-\-c+e-
x — m x — n \x — m)[x — n)
am+b cn-{-d {e{)i}+n)+f}x—emn—g
x—m x — v {x — m){x — n)
{am-^h){x — n)^{cn-\-d){x — m)->r{e{m.-^n)-\-f]x — rmn—(i — 0.
{{a-^c)m-\-b + {c-{-e)n-\-d-\- f]x = {a-\-h+e)mn-\-Jin-\-dt)i-^(j.
6. il^^+l 8x+5 ^
3a; + l ^ x-l
43 13
••• 44 - 3^1 + 8 + — 1 = 52, or
; .-. 39x + 13 = 43a;-43, anda;=14.
a;-l 3a;+l
- 25 -Xa; 16a;+4^ ^ ^ 23
a;+l ^ 3a;+2 a; + l
Taking the last fraction with the first, and multiplying the re-
sulting equation by 15, we have
SIMPLiE EQUATIONS. 145
210x4-08 r-r . 5a,- -30
■ — 11. 75 + ■ ;
Sx+2 x+1
:. 80 - ^^ = 75 + 5 - ^^, or
'^L_ = ^ ; .-. 8a; = 27, and * = Sf.
3.f+2 a; + l
8. J- - 4- — = d.
., «^» _ 1 + ^-1 _ 1 + ^^ _ 1 = 0;
6+c «+c' 6-|-ii
fe+c "*" rt + C "^ 6+rt
which is satisfied by a; — (a + /y + c) = 0 ; .-. a; = (^/ + A + c.
+
10
x-\-(i x—b x — c
vilx — r) nix — f)
x—a x—b
which may be solved as-in Ex. 1.
3x+5 4^+9 _ 15a;+7 _ 12a;+17
a;+T ~ 2x+4 ~ 3.K+1 3a;-|-4 '
• 3 + — - - 2 - -1- = 5 + J"- - 4 - -J_ , or
" ^ x+1 2a;+4 ^ 3a;+l 3a;+4'
2 1 _ ^_ _ 1 _ .
:^ ~ 2a; + 4 " 3a; + 1 ~ 3a;+4'
3a;+7_ _ _ 3a; + 7
" 2x3+6a;+4 ~ 9a;2 + 15a;+4'
This can be divided by 3a; -I- 7, giving 3a;+7 = 0. or x= -|.
The r^siilt of the division is
1 1
— , or
2a;3 4-6.i-+4 9x2 + 15a; + 4
9a;2+15a;+4 = 2a;2 + 6a;-f 4, or Ix^ = —9a;, whicl' we can divide
by a;, /. x = 0 ; the result of the division is 7a; = —9, or x= —%■
1.
146 SUJIPLE EQUATIOirS.
Exercise li.
lQa; + 17 12a; + 2 _ 5x-4:
18 ~" 13a;- 16 ~ ~ir"
6x-\-lS ' 9a;+l5 ^ 2a;+ 15
^' 15 5a;- 25+^= ~5
7a;+l 35 a;+4
3 ■ — = — X — ' — 4- 31
4a;- 7 2- 14a! 3^ +a; _ 10 - 3fa; 19
^- 2a;-9 "^ 7 + "IT" "^ 2 ~ 21*
2x + a 3x — a
^- 3(^^ "^ 2(^2M^"^^'
a;-4 3a;— 13 _ 1
^' fo+5 "^ 18a; -6 ~ T*
3x+ 1 a;— 11_ a;— 9 a;— 5
^- 2a; -15 ~ 2a;-10" ' ^^ "^ aT^"^'
8 ^~^^ 4. ^~ ^ _9 _1_. Sa;-19 3a;- 11
a;- 7 "^ .^■-12 -""^^--v' a;- 13 + x+ 7"^-
a;-2 a;-l _ _5_ . a;+l J^+l. _ ^
^- 2a;+l "^ 3(a;-3) " 6 ' 4(a;+2) + 5a.-+ 13 ^ 2u'
5(2a;^ + 3) 5-7a; , / 3 ^ 1 4
^^__^ _ 3^^ 3a;^±^. 17 _15 32
^^- 9..:!. ^ 2 -^ 2a; ~4' a;-lG "^ a;-18 = ^-1?
12.
15.
16.
_ 3-2ia; _ 28-^ _ 10a; - 11 x
15 14(a;-l) ~ 3 30 ^ Y'
_L _ ^+2|a;g-a^3 _ J^ _ _5^
^'^ a;-2 G-5a; + a;2 ' 2*"~a;-
1, 30 + Oa; G0+8a; 48 , ,.
14. _L. — ' _ + 14.
a;-fl ^ X+-6 x + 1
5a-2+a;-3 _ 7a;3-3a;-9
5a;-4 ^ 7a; -lO"'
X x—9 x + 1 x — 8
a-- 2 "^ a; -7 a;-l "^ a;- 6'
17.
18.
19.
20.
21.
22.
23.
25.
26.
28.
SIMPLE EQUATIONS.
a;2-3a;-9 a;2-7a;-17 _ x^-6x-U
x-B'" "^ a;-9 ~ x—Q
4a;-f7 4a;+9 _ 4a;-f-6 4a; +10
Ax+l "^ 4x + 7 "^ 4x+i "^ 4a; + 8'
2a;- 3 2a;-4 _ 2.r-7 2a;-8
2a; -4 ~ 2a;- 5 ^ 2a; -8 ~ 2a;- 9*
7a;+6 2a; + 4| x _ 11a; Xj-^
28 ~ 23a;-6 "^ T ~ "21 ~ 42 '
x'^-5 a;3-ll _ a;2-7 a;2-9
^2^^ "^ a;2-l2 ~ a;3-8 "'' a;^- lO'
a; - Iff 2 -6a: _ 5a;-i(10-3a;)
2 ' ' ~ 13 ~ ^ ~ 39
l-2a; \+x 1
U7
8(a:3-a; + l) ^ 2{a;3 + l) ^ 6(a;+l) 9(a;2+l)
o. 2a;3+a;-30 x^+^x-A. x"-ll 2x^+7x-V6
2a;-7 ^ a;-l ~ x-4 ^ 2a;-3
x — a x — h {a — by 1{a — x)
x—b ' x-a {x — a)(x — b) ~ a-{-x
12a;+10a ^ 28a;+117rt ^ ^g
'3a;+« 2x+9a
07 l^|a;-5 13|a;-ll _ ISjx-l 13ia;-9
13i':e-6 ' 13|a;-12 ~ 13^a;-8 ' 13^a;-10
1 1.x 16a;
+
2(a;-l)2 "^ 2(a;-l) 2(.t3+1) " ^x-l){x^ + l)
29. i(|.^+4) - Zi^^ = ^ (-1 _ 1
o 2 \ a; /
3a;
81a;2-9 3 2a;2-l 57 -3a;
30. _ - ^^^ ^ = 3a- -
2 (3a;-l)(a;+3) ^ 2 a;+3
31 1 + ^^ + 1 "^^"fl^, _ a;2 + ^^ + ^ _
2(a;-l) ~ %+l) ~ a;3-2a;+l ~ '
148 SIMPLE RQUATIONS,
7a: -30 5x-7 2 -21a;
32.
83.
10^ ^x-B 21
42a;- 171 2a;-9 1
63 ^^'^ 63-14^ ~ Y^^ ^
18^^2 l + Hx 101 -04,
13 -2x +^^+ ~8~ =131- 'q —
4 -9a; 5 - 12x 24x2 - 5
^^- l-'6z ~ 7- 4x'^^" 7^ 25iB+T2a;2*
8a; +25 16a; +93 18x+86 6a;+26
qe I ' - I .
*^^- 2a;+ 5 ^ 2a;+ll ~ 2a;+ 9 ^ 2a;+ 7
1 1-1 1 ^
x+a + w a;— a + o x-+a— <> a; — a— o
Art. XLII. The results deduced in Section III., Chapter
IV., may often be applied with advantage.
Examples.
ax -i-b m
(page 123).
ex + d n
{ax-\-h)d — {cx-\-d)h md — nh
(ex +d)a— [ax + b)c ~ na— mc
md — nb
X = •
na — mc
ox^-\-hx-\-c a
mx^-{-nx+p VL
(rta;2+5a;+c)-r/.r;3 n
/- — iT . — o = "■ (page 122).
{mx^ +nx+p) — mx^ m ^^ ° '
hx + c a
:. — r = — &c.
nx-\- II III
3a;+J _ ?>x- 13
.r4-4 ~ X— 4
SIMPI.K EQUATIONS. 149
By (5) eacli of these fractions =
difference of numerators 20 0«+7
difference of denominators " 8 a;+i x+i'
or -^ = ---7, •• x = G,
vrx-\- a + h mx -\- a -\-e
4. . = ] r»
nx — c — (I vx — o— a
mx-\-(i -}- b nx — c — d
, r— = 7 — -, ; or By 4, page 122,
mx-\-(t -\- c nx—b- a j ■> i o »
or (» — >?/)»
vtx -\-a -{■ h nx — c — d
h—c b—c
— a-\-h-\-r-\-il, .'. x=kc.
^ l/{n+x) + i/(>i-x) ^
''• A^'Jrx)-^/{a-x} •
Here by (0), page 122, we have
r" ^-r^ — —, = :: ; or, Cancelling the 2 in left hand mom*
2p/{a — x) a—1 '^
ber, and squaring,
n+x (a+iy ■ -u /n^
= / ,^rj, whence, again by (6),
a — X {n — ij
2x _ (rt + l)--(«-l)3 _ 4a 2a^.
2a2
6
^/{x-a + b) - y{x \n- h) ^ n-b^
y^^x-a + h)-^\7{x-{-a-h) ~ a+b
l/{x — a+h) _ ^.
V%«+a-6) ~ b '
squaring and again applying (6),
1x ^'=+&i 1 _ ^"Vh^
150 SIMPLE EQUATIONS.
Exercise lii.
1+x
2. - ~
6.
2x^-lx+B ~ a;2-9x-|-2
az+b — c (b — c)^
ax~h + c {h + cY
7. If i/{x+n)+^/{x-y) ^ x_^ ^^^^^ ^^^^ x+y^
V'{x+y)-v'(x-y) y x-y
= 1.
8.
0.
2a:- 7 _ x+l_, 4a- -5_ _ lOx-32
57a; -43 _ ^x-1 . 23a;+5i _ 36a;-- 7
19a; 4- 13 ~ 18.r + 2^5 ' 115.r-29 ~ 180a; + 23'
210a;-73 21a;+7-3 . v»a; — a - i mx-a-c
3i0a;-86 3]a; + 8 ' nx—c-d nx-b — d
Zx+^jix-x'^) _ ^ ./(12.e + l)+y(12y) _
■ Sx-^/{4x~x^} ~ ■^■- {/(12;c+l)-|.^(12.tj " ^^
12.
13.
x^-{-ax- —hx+c x~+ax—h
x^ —ax^ + hx+c ~ x^—ax+b
^/{2a''-x-) + b]/(2a-x) _ /a+b
|/(2a2_a;3)_&^/(2a-a;) ~ ^a-b'
' V(x-+a^)-V{x^-a^)
15.
8a;3+12x3-8.'c+r) _ 4x^+6x-A
8x3 -1 2.1-3 +Ba;;+ 5 ~ 4a;2 - 6a; + 4 '
SIMPLE EQUATIONS. 151
16
17
18
28-K/a; _ 9 +3yx
28 -ya; - 9+2i/.f'
a^x^+a^bx^ —acx + d a^x-+abx—e
a^x^—a^bx^+acx+d ~ ci'x''^ -ahx+e
5T/(2a;-l) + 2t/(3a;-3) _
°- 4i/(2a;-l)_2i/(3aj-S) " '^^•
/2a;+y^(3-2x) _ 3
72aJ-i/(3^;^-) - T"
2£(3aH-3)+jf (7x + 8) _
" ■ 2^(3a;+3)-i^(7:c + S) " ^•
22
23
20
24.
33{13-2/(a:-5)}-3{13+2|/(a; -5)}.
(l/«+l){l/(».6- + l) - ^oix] = {^/n-l){i/{nx+-i) + y'nx]
l/{x-hc) + -y/b _ ^x + i/a
l/{x+c) -yb ~ ^x - i/a*
o" V^J^^ _ lAjf38. ^2x + 17 _ f2x+^l
l/ic+ 4 ~ ^x+ G' ■^2a;4- 9 ~ -^2^ + ir/
y»+2a _ -1/5+ 4a. 3a; -1 _ 1+j/dx
l/'x+ b ~ -j/aH-^i' |/3a; + 1 ~ 2
ya— |/(a - a;) ^/x + ^/6
26
27
31.
■\/a+ ^/{a—x} ' s/x — \/b b
ax + l + ,/{a^x-^-l) _
^^- «a;-t-l -N/(a3.t-3-l) -"'
on '«+a; _ H-1 . l-fx+a;2 _ 62 14a;
v/(2aa;+u;3) " 6-1 ' l^^+a;^ ~ 63 T~£
5a;4 + 10a;2 + l _ a^ -\-lQa-^ -\- 1
a;^4-10a;s+5^ ^ Sa^ + lOa^ + l*
Art. XLIlI. Various other artifices may be employed to
simplify the solution of equations.
152 simple equations.
Examples.
1. Solve 2-f v/(-J:Cc2-9a3-f 8)-2x = 0 : here there is but one
surd, aud it is convenient to make that su^d one side of the equa-
tion and transpose all the rational terms to the other ; this gives
>/(4cc2 - dx + 8) = 2x— 2 ; squaring both sides,
■la;2 — 9,t+8 = 1.^-2 -8x-+l, .-. x = 4.
2. \/{a + x)-{-V{a~x) = 2Vx. We might square this as it
stands, but the work will be simphfied if we first transpose, thus
\/(a+x) = 2 \'^x— \/{a-x) ; 'low squaring,
a-\-x = 4-X + (i- x — -i \/{ax—x^), or
ic = 2 v' {((X — x^ ) . Again squaring,
x^=A(ix — 4:X-, whence x = 0, or — -.
5
3. Clear of radicals
-^x + f/// + ^2 = U. Transposing,
^x-^ {/y = - f^z ; cube by formula [0] ,
ic + y + of^xy{^x+^y)= -z; and subsiitwtoig wr
^x-V-^'y ''-3 val". • —^2, t^is becomes
iK + ?/ — 3 -^^xyz — —z, ur
xi-y-\-z = o^''x!jz; :. out lug again,
{x+y + z)^ = 21xyz.
. a+x+Vj^ax+x^)
a + x ~
Dividing and transposing, we iiave
a+x a-' +2ax+x'
division in left-hand member,
fl = (6-l) .-. -^= v/{l-(/.-l)^},or
(a-i-x)^ a-\-x
^ + " = 1 v/il+(fc-l)n. or
— -f 1 = "fee.
SIMPJ^E EQUATIONS. 153
5. Solve v/(4.r3-fl9)+A/(4x3-19)= -/dT+S.
We have the identity
(4;t3 4.i9)_(4a;2-19) = 38 = 47-9.
Now dividing the membera of this identity by those of the given
equatioju, we have
V(4a;3 + 19) - v'(4x2 - 19) = a/47 - 3. Adding this to the giwB
equation, then
2A/(4a;3 + 19) = 2A/47, /. 4^2+19 = 47, and a. = ± ^7.
6. if(25+a;)+^(25-a;) = 2.
Cubing by formula [6] , (See Ex. 3), we have
25+x + 25-a; + 6if(253-a;2)=.8, or
T^(f525-x3)=-7, or (625-x2)=-343;
.-. a;3 -525-1-453 = 908, and a; = ±22|/2.
Exercise iiii.
2. y(3:e + l)+i/(4.r + 4) = i.
S. V(2.f-t-10) + i/(2a; - 2) ^6.
4. i/{i)ix)~i/{nx) = m — n.
G. l/-+/(x+3) = ^|^).
7. A/(aa;+a;3^^(1.-|-a:).
8. ^(17a;-26)= A.
, a
10. 6-i-a;-i/(63+a;2) = c2.
11. i/(8 + a-)--/a; = 2i/(l+a:).
12. ■/(2a;-27rO = 9/a-/(2.r).
154 SIMPLE EQUATIONS
13. ^(l-^cyi-f{H-x) = ^3.
14. f'i:>+.c) + f/{'d~x) = ^l.
15. ^(.c + l)_,.X(.t— l) = ^lh
IG. ^[a + x) + f(a-x) = -^b.
17. r(l + iA) + ir(l-l/^) = 2.
18. |/.^;-/{a-|/(a:c+a;2)} = i/«.
19. Clear of radicals ^a+^^/ — ^6-,
20. Solve aj+,/(a2^x2) =
7ja*
l/(a2+a--)
21 . Clear of radicals yx+ \/tj+\/x — |/m.
Solve the following equations :
22. v/(i-^x) + ,/{l-l-.c^-|/(l-a;)|• = l/'(l-»>.
23. -/(x-+ i/x) - Vi-'-V-^) = '*Vx +17i*
24. ^(l+x-+u;2)+/(l-.t;+.7;3)^/y,x.
25. ■i/(a3-.<;2)+.c/(^<3 _!) = ,( -2(1 „^.').
2o. —7", — " — = <^'
• i/{hx)+c- n
27. i/C-..;-+5)+\/(2a;3-5)= v^l5+ v'5.
28. -s/(3u;3 + 10)4-a/(3x3-10)= ^17+ /a.
29. v^(3a;2+'J)- i/(3x'3-9)= \/31 f 4.
30. A/(3«-o/>+a;3)-t- ^(2(i- 2/^-^x-) = V a+ v js.
81.. V(4«2_ai-i-2.c2)+;/(3«2_3/.2_^3)=,,4.^.
32. Clear of radicals, ]^(2x-) - ^{%i] - ^(2z).
83. |/(«+.t) + /(" - ^) - 2x -: -,/{ -t + i./(a:^ 4-^-) K
84. y(.. + 2..) + ^(-^-2«.)= -^^
85. J f<-^+^-i-v (-"?-) ==^(^^^^^^^^
86. ,/{(2«+x)--^-h/'-} f/U2<t-^)-+^-;- = 2a.
SIMPLE EQUATIONS. i£)&
Art. XLIV. Sometimes a factor can be discovered, and tbe
principle of Art. XL. applied.
Examples.
1. t+a^^^ji^ = x^Jr{ci-b)x^ + {a^ -ah)x-an.
x — a
Factoring we have
x — a
ox x^ —ax-{-a^ = {x — a){x—h) \
fl2
.'. {a-{-h — a)x = ah — a'^,?in<lx = a— —
Transpose '— and factor, then
a
•+'4 (o+hy) ^1 a\ (a-Vh)-] \
«6
a~+b - ^•
x+a a; — 6 a; — r ^'4-c
^' (^^^)(7^c) " {a^b){b-c) ~ {b - c)(^-^) " {a-o)[b'c){c-a)
Add term by term the identity (Th. iii., page 54).
x—a x—b x—c
(ir-b){c-a) ^. {a-b){b-c) ^ {b-c)(c-a)
2x _ h + c
*• {^b){c-a) ~ {a-b){b-c){c-a)
1 h+c
156 SIMPLE EQUATIONS.
Tlie left hand member vanishes for x = 0, and /. oy symmetry
for a = 0 and b=0; :. it is of the form mabx in which m is
nume7-icaL
Put x = a = b, and m is found to be 6,
.*. the equation reduces to
6ahx = ahc, :. and x = ^c.
Ix — a\ 3 x — 2ia+b ■
5. r) = oz. I '> let x — b = m, x — a = n, and .*.
\x — bj x — ^b + a'
m—n = a — h, then we have
m^ n—{m — n) 2n—m
«3 ~ m-\-{m — n) ~ 2m — n
2m* — 7i?» 3 = 2^4 — w^m, and
2,{m^—7i^) — nin(m^ — w*) = 0, which is divisible by 7n^ — n^,
.•. mf —n^ =0, or m-\-n = 0;
Bnt m+n = 2x-a—b = 0, :. x = l{a-\-h).
1 a;2-4a; + 2 1 x^-Ax+S _2_ a;g-4a;+3 5
6- y a;2_4a;ri + (5 ^2^ 4^~3 ~ 9 * x^-4x-6 ~ 18*
Let ?/ = a;-— 4a;, then this equation becomes
1 ?/+2 1 y+3 2 y+3 5 , ,. . .
— • q+-7r- ^ — TT" ^e = T3' or by division,
3 y — 1 6 2/ — 3 9 ^ — 6 18 •'
1 . J_ 1 1 2 2 _ 6
■3" "^ 2/"^ "^ T "^ 1/-3 ~ "9" ~ y-a ~ 18'°^
_^ : - = 0 ; this may be written
^—1 y-S y — (j
1111
y - I y — b y — o y — o
^ + ^ =0, .-. 5y-16+32/-3 = 0, or
y = 2i .-. a;2-4a;=2^, ora;2-4a; + 4 = 4+2^,
and a; - 2 = + 1. We might assume (as— 2)^ = y, when the eriven
equation would take the form
SIMPLE EQUATIONS. 157
3' y-o '^ a' y-7 9* 2/ -10 18'
And reducing as before, we should find
y = 6l = (x-2y, .'. 3^-2= +f, as before.
Exercise liv,
2. ?l±i^* = x^ + 2a{a-b)x+{2a-b)x^ -2a2b .
x + b
o a^ a^x _ 2c
' a^^ab+b^ ~ a^-b^ ^ ^^ ~ ^^'
4. ^- _ i- _ J- _ i- = 2ab\x^b)x'^.
a-\-b-\rX a b X
1 1
{x-b){x-c) "^ {a+c){a+b)
1 1
+
{a-\-c){x-c) "^ {a-\-b){x-b)
^ bz Bab an^ b^x 2a -b
9. — — — -7 + 7 TT^ = 3x — — •
a a-b "^ (a-6)3 "•*' a (a-b)*
x^-{-2ax x—a
' a;* -lla;2a2+«* ^ X'-Bax-a^'
2 \a;+a/ x+a
= (.c — rt ) (x — h) {x — c).
aa; bx ex 2 \bcx acx abxj
,, 1-ax l-bx l-.cx 12 2 2\
(a — b)^ a a^-b^ I
12. y^L-^ _ 1 + _ = __ + 1 +
abc 0 abc \
19,, x^J^{h->rcY-\-Bb{b-^f)x = b^.
a
. X,
b
158 SIMPLE EQUATIONS.
14. x—a-B-^(ahx) = b.
15. lla;4 +10x3 _40.r = 176.
in X ac c ax
Id. , ^— 4-
-„ « — & 2c.k2 «.— & \ — cx
a+b ' 1+cx a-\-b l+cx.
lo 4x*+4rt4— 33a;2a2 w^ , o « o s o "\
19- „. -..,.,00 +
a;3-lla; + 28 ^ x^-llx + lO a:2-14a; + 4U
20 8 , 8 ^ ^
x'^-Qx+5 "^ a;2-14x+45 .r^-lOx-^O
{a-b){c-a) ~ {a-b){b-c) "^ (fe_c)(c-^,)
a + c
~ {a-b){h-c){c-a)
22. (a;-a)34-(a-&)34-(6-a:)3=a;2-a2.
ix+ 2a\ 3 /a+ 2x, 3
23. X + « =2m.
\.r — aj \a — xj
24. {x+ay -(a +h)s + {b -x)^ ={x + a){x+b){a-\-b).
25. a;3-(x-6)3-(a;-a+6)3-rt3 + (.c-a)3+(r/-i)3+&3
= (a-'))c2
26. {x+a)^-{x+b)^-{x-by'-(-2ayi-{x-a)^+{a+by +
(a-by = {a^-b»)c.
x-\-a x — a (7*
28. (a; + a-t-5)4 .-{x + n)i-{x + b)*+x*-{a+by + a^ + b^
= 12ab{x-+{a + by}.
a — x
b—x c — x. 3.r
29- ^m^c "^ /)2— ca "^ c2"_«6 ~ a/;+^c + c^"
30. x^{b - a^)-\-a^{x-b^) ^b^{a-x-) +abj{nbx-- 1)
= («-a;2)(62_«4).
SIMPLE EQUATIONS. 159
81. (l[x+x^-)" =^~-..{l + x^+x^).
CttLf J.
•;>o I aj-l-a a — h
^ 1.
o.
x + b ^ \2,.c-\-b + c
34. x/(x3 -f27.j + l 80) --p/(a;3 +26.^ + 168)= J^±l^\.
35. {{,■ + a + y' (x^ +2ax + b-^)}s + {x+a- ^/{x^+^ax + b^)}^
= 14(./j+a)3. (See page 17, Ex. 1).
36. {x+u+ y{x^ -2ax-2b^)}- + {x + u-'/{x' — 2ax-2b^}}^
= .^-b-^+2a{a-b).
orr , x+a 1^ :c4-2a + 6
■ — ^ / ;<; — «— 2/* ■
38. (5a:-7)'-(2.c-4)3^27(a;3-l).
39.
_L '^-<^'-l 1 :t:3-6a;-4 2 a;2-6a;-7
3 * .c2-6x--4 "^ T ' X--6X-9 ~ T ' a;--6x-16
14 4
40.
15 x'^ — &X — 9
1_ :<;^-2a;-3 J_ a;^ - 2a; - 15 _2_ x^ -2x- 35
2.(;-8 "^ 9 *a;2-2.r-24 ~ 13 * a^-2x-~48
I
I
'_2
~ 585"'
41. {r-{-a-b+^ (.2^^43_J3)i3_
a;+a - 6- -/(.i^s ^ a2 _ ^2) ;. 3 -_: s(x + a - 6)3.
42 ^1 , 1
|.f+a)3 -h~ "^ (.,+6)2 _«3 -
1 1
a;3_(a + 6)2 + x^-{a-by
„ ,./U 45 7.C + 67 \ ^.^^
*3. 41 — — - 4- ^ -r 130 =-
\ X + 1 .c-r-i I
^^ 8a;-;- 57 9.v-^68i
39 ^ H 5-
t.60 SIMPLE EQUATIONS.
44. 51 1?£±15 _?£t^)+e63 =
. X—1 X—4: j
45. {x+a){x + da){x+ia){z+Ga) = x'^ + 6a^{x' +7a«-f 6aS).
4« 1 2 3 6 '
4b. + +
x+Ca X - da x+2a x + a
Exercise Iv.
1. a{b — z)'ih{c — x) = b{a — x)"{-cx.
2. (a-i-/;.c)(a— i) — (flx — 6) = «6(a;+l).
3. (a-6)(a;-c) + («-f-6Xa;+c) = 2(Z^a; + arf).
4. (a-')(a;-c)-(a + &)(a;+c) + 2a(6+c) = 0.
5. (a-6)(fi-c)(a-}-a;) + (a+6)(a + e)(a-a;) = 0.
6 ( .-6)(a-c4-x)+(a + 6)(a+c-j;) = 2a2.
(solve in {a;— c}).
7 {m+a(a + b — x) + {a — m){b -x) = a{m-rb).
8. m(a + L — a;) = n(a;— a— ^).
9. (7n+n)(m — 7i—x) + m{x—n) — n(x — m) = m^ — »*.
m — a; n — x v—x
10. + S- ^^ — =3.
m n p
a^b — x b^c — x c^a — x
11. -f- 7 + " =0.
a 0 c
a — x b — x c—x ^
12. -1 V + — T- = 0.
be ca ab
1 —ax 1 — bx 1— ex ^
18. - -r 1 1 r =0.
be ca ab
(Deduce the solution from that of No. 12;
a—hx h — ex c—ax
14. — , — H 1 r- = o.
be . ca ab
15. {a + b + e)x-—^ = , . + —J.*
^ ' u—b a-\-o a—o
SIMPLE EQUATIONS. ItJl
Sahc a^b^ ' {2a-hb)b^x _ {b + 3ac)x
^^- a+b "^ (a^^ ■*" «(rt + 6)2' ^ a
10 4 9 2 /^ , . 1 >
17. — + — = h -5- Solve m — •
X y X o \ xj
^^- X "^ 3 ~ 3a; + 12 ~ ii*
I. 1^ _ 2(5a;-12) _ 17 10
^^- 3 "^ oa; ~ 3a; ~ 20 "^ T*
10-a; 13-fx _ 7^+266 4a;+17
20.-3 "^ 7 ~ x+21 ~."2i
6 3 17
21. -T^ +
22.
23.
a;+3 ^ 2(a;^3) 2 2(a;+3)*
6a;+5 l + 8a; 1-a; x-%
8a;-. 15 '
15
3
1
1
a —
X
1 *-
a
1
1
a;
a
a2
X
2.5. (a;-l)(a;-2)-(a;-3)(.r-4) = 3;
(a;-3)(a;-4) = (a;-2)(a;-6).
26. 2(a;-4)(3a;+4) + l2a;-3)(3a;+2)-6(a;-2)(2a;-3)=0
27. {a-x){b-x)=x^ ; {a-x)(z-b)=x^ -c^.
28. {a-x){b-^x) = b^-x^.; {x-a){x-b)=x'' -ai.
29. {a^x){b+x) = {a-x){b-x);
{ax-\-b){bx+a) = {b-ax){a-bx).
30. (if-a;)(6-a;) + (a-c-a;)(a;--6+c) = 0.
IR2 SIMPLE EQUATIONS.
31. {a~x)(h — x) — (c — x){d — x) = (c + d)x - cd.
32. {x~a){x-b)-(x-c)(x-d) = {d-^a){d-h).
S3. {(a2 -h^)x-ah}{a—(a+b)x\ +2ah^x =
{{a+b)2x+ab}{b-{a-b)x}.
34 (ar-f l){x+2)(x-\-B) = (a;- 3)(a:+4)(a:+5).
35. (a;-fl)(a;+2)(a;+3) = (:«-l)(a;-2)(a;-3) + 3(a;+l)(4.r-?-l)
36. (a;+l)(a;+4)(x + 7) = (a;+2)(a;+5)2.
37. {x + 2)(x+5y'={x+Sy-{x-^-6).
38 (a;-l)(.r-4)(a:-6)-a;(rK-2)(x-9) = 13G
39. (a+x)(7; +a;)(o+a:) - {a~x){b-.r){c -x) = 2{x^' +abc).
40- (3^-^)(^-^)(^-^)-(^-^0(^-^)(^^-g) ^ (a;-rf)=.
a; — (i
4 1 .^•(x• - a) 2 - (ic - a + t) {x - a-^-r) (x - b - c) = (a 2 + he) (b + ').
42. {x-fi + b){x- h-}-c){x-c + d) - x-(x~ a + d) = bc{d - a).
43. (x - a + b){x — b + c){x- c + cl) - x{x - a + c){x- c+d)
= bc\d — a).
44. {x-2(,Yx-2h){x-2c)-{x-a-b){x~b-c){x-c-a)
= {a-k-b + c){a^-\-V~+e-)-%abc.
45. x^ —{x — a-\-h)(x—b-\-c){x—c+a)
46..L_i)(„_l)f._i)+i,."+M:£.
■ \ X I \ X I \ X I X^ X
47. (x + a){x + b)+{x+c){x+a) = {x + b){x + d)+(.T + d){x+c).
48. (aa;+/0(«-«-O-«{^-ic)(«a;+&)=a2(a;-c)(a;-i)_
a((7a; — c)(c— a;).
2a; -3 3a; -2 _ 5x2- 29a;- 4
*^' ~x-l "^ "ai^ ~ "a;2-12a;+32'
60 ^'^^ 3a;+2 _ a;^- 30.x- +2
8(i-fl) ~ 2(a;-l} ~ 6^21^6
SniPLE EQUATIONS. 163
3x-7 3(^--fl) _ lla;+3
7x-5 8^-7 _J-^±1 n
^^^- 3x--2 "*" 'dx-1 "^ 9a;3-9a;+2~ •
2a:+7 3x-Q 5(:k- 1) 3x-2 5a;- 8 2^+2
^* 'Sx-7^-Ix-o ' Qx-'Io ~-Zx-6^'J:c-25^-6x-l
^^ „^ 4^3 +2x. x — a a; -6
„ - , a ex c au)
66.
2(a;-l) «+8 _ 3(5^+16)
^"^^ ~a;-7 "^ a;-4 ~ 5a;-28 "
_^ ax ex a c
58. + = ^ _;
mx—p nx — q m n
ax+b cx-\-d a e
nix—p nx — q m n
„„ 6 — a; c-^x a(c-2x)
a+x a—x a- —x-'
a+b b+c a+c + 2b
x—a x — b x — c
ax+b bx ax (ax^ — 2b)b
ax~b ax + b ~ ax—b a^x^—b'^
60.
a
rt-j ux — b cx — d (bn + dm)x — {bq-\-dp)
mx—p nx — q {mx—p){nx — q) ~ m "*" n
^c, w. n p m n v
62. -1. + — ^— = -t. 4. -±L^.
x—a x — b x — c x — c x — a x—b
164 SIMPLE EQUATIONS.
fi8
1 1
ax— 2a ax — 2b ^ a « _
1
X
X
ax— 2b ~ ax-^2a ' 1 1
1 a +
a X
2a;2-3a;+5 2
7a;2-4a; + 2 ~ 7'
64
ax'^—bx-\-G a. ax^ —hx^ ■\-ax — <L
az-
-h
mx^—nx+p to' mx^ —nx^ + >}tx — q
mx-
-n
65.
i-a; 1 aj . 1 .
i+a; •" 4 ^+a; 4 '
%^-i 2 2 fx+l .
l-a; 3 3 "^ a;-|'
66.
21 71 21 71
X— 98 a;- 94 ~ a;+44 ~ a;-52'
67.
7 3 9 1
-
x-6 + a;-ll ~ aj-7 ^ x-12'
9 9 2 2
x-51 x-16 ~ x-Ql a;+81
5.4' 8 1
x-
68. T^ze + ^ = ^ + ^3io'
70.
1
aj-6
+
8
x-3
B
X
5
-2
+ i-
4
-5'
m — n
-
a —
X —
h
m
=
m -
-n
a-
X-
-b
x — a
a;-
-b
■n
a-\-h
a + c
b + d
69.
x — a X — m x — o x — n
x—b x—c x—{a + b+2c + d) x — {a-\-2b+c-\'fi
71. (x-a + b)^ -{x-a)^+{x-b)^ -x^ +a^ -(a-b)^ -b^
= {a-b)c^.
72. (x+a+by - (a+by - (x+by ~{x-\-ay +x'^ -\- a" -hbii
= 10ahx{2x-\-a + b){x+a+b).
(m-n){x-a) {n-p){x-b) {p-m)(x-c)
o+c c-f-a a'f6
SIMPLE EQUATIONS. 165
ax—1 bx — 1 cx~l Sx
74. 7Tnrrh\ + h2(^si7,\ "t"
a^{c+b) ^ h^{c+a) ^ c^{a-\-'h) ah + bc + ca
x—2a x—2b x—2c
75. ,— -f r H -, - =3.
b-\-c — a c+a — b a + b — c
X— 2a X — 2b x — 2c dx
b+c-a c-jra—b a-{-b—c a-^b-\-c
a — x b — x c — x 6
77. r,-^- + T-, +
a^ — be b^ — ' c c^ — ab a+b + c
x-\-2ab 2aO-x x-2ab x->r2ab
no '_ I ^ 1
' a +b -\-c b -{- c — a a — b-\-c a-{- b — c
a b a — c b+ c
x-{-b—(i x-\-a — c x-\-b x+a
m^(a-b) n^(h-c) , pHc-d)
80. — ^ + — ^ + -— +
x—m x—n z—p
q {pd+{n—p)c-{-{m — n)b— ma}
x — q ~
81.
(a;-2)(a;-5)(a;-6)(a;-9) + (a+2)(a-4)(a-5)(a-ll)
X
+
(,+l)(6 + 5)(64-8)(6-fl2) ^ (,_43(,_7)(,_ii)_,
X
(a2-l)(a-8)(a-10) + (&+2)(6+3)(6+10)(&4-ll)
X
Art, XLV. Employing the language of algebra, the princi-
ple illustvated in Art. XL. may be stated as follows :
Definition. — Any quantity which substituted for x makes the
expression f(x) vanish, is said to be a root of the equation f{x) = Q.
Thus, if a is a root of the equation /(a;) = 0, then /(a) = 0.
By Th. I., if x — a is a factor of the pohjTwme f{xY, then
/(a)" = 0, and a must be a root of the equation f{xY = 0 ; hence in
solving the equation we are merely finding a value, or values, of
X whicli will make the corresponding polynome vanish, ^u-^.
pose/(a;)" = (a;— a)i?>(a;)"~^ = 0, we are re(j[uired to iind a value, or
166 SIMPLE EQUATIONS.
values, of x which will make (x — a)(^{xY'~'^ vanish. The poly-
noma will certainly vanish if one of its factors vanishes, whether
the other does or not, and will not vanish unless at least one of
its factors vanishes. Hence {x— a)(p{x)"~^ will vanish if a;- a = 0,
quite irrespective of the value of (p(ic)"~^ Also, if (p{x}''~^ = 0, the
polynome will vanish, irrespective of the value of x — a. It fol-
lows, therefore, that if /{x)" can be resolved into two or more
factors, each of these factors equated to zero will give one or more
roots of the equation /(x)" = 0.
When there can be found two or mor?; values of x wliich satisfy
the conditions of given equations, they are sometimes distin-
guished thus : Xy, iCg, x^, &c., to be read " one value of a;," "a
second value of x," " a third value of x,'' &c. Thus, if
(x — a){x — b){x — c)=^0, ,
Examples.
1. Solve 2a;3-13a:2-f27a;-18 = 0.
Factoring,
(a; - 2) (a; -3) (2a;- 3) =0,
2. x"-(a + b)x+{a+G)b = (a-\-c)c,
... x^~{a + b)x+{a + c)(b-c)==0,
:. x^-{(a+c)-{-{b-c)}x + ia+c){b-c) = 0,
... {x-{a-\-c)}{x-(b-c)}=0,
.". Xi =(t+c, x^ = b—c.
3. x-{a-b)-{-a^{b-x) + b^{x-a) = 0.
.-. x^{a-b)-x{a^-b^)-\-ab{a-b) = 0,
.-. {x-a){x-b){a-b) = 0.
li a — b = Q, the given equation holds irrespective of the values
oix — a and x — b, and therefore of the values of x ; but ii a~b is
not zero, x^=a, x^ = b.
z =
SIMPLE EQUATIONS.
a;-l ~ b^-{x^l) ■'■ U-i/ ^-
aJi + l a ^ a+f)
-^ — - _ -— = 0 .'. X, = r»
Kj-l 6 ' «-«
+ -— = 0 .'.2:2 =
167
iCg + l , ^ _ n - ^^~^
(a-xy + jb-x)^ 34
{^^^+{a-x){b-x)+{b-x)^ - 49*
(a-x)^+2{a-x){b-o^ + {h-x)^ _ 2(49) -34 ^
{a-x)^-2{a-x){b-x) + {b-x)^ ~ 3(34)-2(49)
((a-a;)-(6-a;)j " '
(a-a;i) + (6-a;i)
a — b
-4 = 0, .-. a^i = ^(5&-3a);
(a-a,,) + ib-x^ + i = 0, ...X, = ^(5a-86).
a — 0 ■* ■"
(a;-«)(x-6) (y-5)(a;-r) _
"*• (c_a)>-Z>) ^ (a_6)(«-c) " ^•
Subtract term by term from the identity (See page 53),
{x-a){x-b) {x-b){x-c) {x-c){x-a) ^^
[c-a){c-b) "^ {a-b){a-c) "^ {b-c){b-a)
:. {x — c){x—a) = Q, :. x^—c,x^=a.
7. Find tbe ra/Kma?. roots of x* -12x-3+51a;2-90a:+56 = 0.
Factoring the left-band member by the method of Art. xxviii.,
(a;-2)(a;-4)(x2-6rc + 7)=0
.-. 0^1=2, 2:2=4, or a;2-6x-}-7 = 0.
Since x^ — Qx-\-l cannot be resolved into rational factors we
know that it will not give rational roots, .". a;^ = 2, a;, = 4 are the
only vftluefl that meet the condition of the problem.
168 SIMPLE EQUATIONS.
Any literal equation of the second, third, or fourth degree, and
many equations of the higher degree can be resolved tnto a series
of disjunctive equations. A full analysis for the first four degrees
will be given in Part II., meanwhile the following special forms
of the Theorem in Art. XLV., will enable the student to solve
nearly all the equations commonly proposed.
(A). In order that two expressions having a common factor
may be equal, it is necessary either that the common fiictor
should vanish, or else that the product of the remaining factors of
one of -the expressions should be equal to the product of the
remaining factors of the other expression, and it is sufficient if
one of these conditions be fulfilled. In symbols this is
li {x-a)f(x) = {x-a)(p{x), :. Xi=aoYf{x)=(p(x).
[B). If an equation reduces to the form (^nx+n)^ =c*
(mx-\-n)^ —c^=0,
()"a;j+w)— c = 0 and .'. x^
or {mx2-\-n) + c = 0 and .*. x^ ~
(C). If an equation reduces to the form
ni
— c — n
m
mx-hn] * a^
■px + q) &3
qa — nb —qa — nb
tlienx,= ^-^— . X, = -^:,:^- (See Exs. 4 and 5 above).
(Z>). If an equation appears under the form
(a — x){x — h) = c, (1)
then x-i = l{a + b + r), aJs = K^+^-»')»
in which r^ = (a — 6) 2 — 4c.
From the ideu tity {a — x)-i-{x-b)=a — b
vfe get (a-x)^ +2{a-x){x-b) + {x-by = (a-b)^ (2)
(2) -4(1) .-. (a-x)^ -2{a-x)(x-b) + (x-h)^
= (a-i)2-4c = r2 say
.-. {[a-x)-{x-b)}^-r^=0,
:. {{a—xy)-(xi-b)}+r = 0, anxd .: Xi=^{a + b + r);
or {{a-x^)-(xr,~h)}~r = 0, &nd .-. Xg = l{a+b-)).
SIMPLE EQUATIONS.
fill 1 ^
169
x — a x — a
• • -I ^^
1 aa;
.•. ic — « = 0, or aa;= 1,
_ 1
Applying (.4),
rt
9. {:c-^i-\-mx^h^c) = {x-^a^h){^x-?>a^1h-c)\
3a + c
^[x,~a-\-h) _ x-a-hb
'' ^x'^^+V ~ 3a+c '
.♦. {A) x^=a-h
Page 122. (5).
^'^- ic3-2a;~ h' " m{x+2)^ ^n{x^ -'Ix) mn-\"nh ^>
But (C) can be applied if vi and ?i are so determiued that
m(2;+2)2+7t(a;2 — 2ie) is a square.
This requires that ^m{m-Jrn) = (2m - n)^,
Assume ?m = 1, then ?i=8, and (1) becomes, on substitution and
seduction,
fa + 2)2 _ -J" 2
(3:c-2)3 - a + 86-'" ' '""y
2(14->-) 2(r-l)
3r-l ' '*'2~'l + 3/-'
(a; + 1)4 _ a_ (a;2+2a;-(-l)g _ _«
^^- (a;3+l)(a;-l)2 " T" •*• (a;--^+l)(a;3 -2a;+l) " 6*
For X- -j-l write a^z
"*• .c^(:r»-2a;) ~ 6 *** 2(z - 2) 6
.". a?! — ._, , X.J— -I
170 SIMPLE EQUATIONS.
This '»quation was solved in Ex. 10, hence z may be treated as
known.
i5ut -^- =z, :. ^^^^ = ^32'
[x-\-\\ P 2-1-2
UITil ~ '• — 9? ^ formed solved in (C).
12. {a-xY^{h-xY^c.
In the identity
Letw = o— a;, v = x — b, .•. n+r = a. - & andii^+v* =c,
.-. (a-i)4 = c + 4(a_^,)2(„_a;)(a:_6)-2(a-.r)2(a;-/;)2
Write 2 for (a—x)(x - h)
.'. z2_2(a-6)2z+(a-/;)4 = i{c+(a-6)4}=f2, say,
.-. {2-(a-6)2}3=t2
.*. by (Z>) 2, =(rt — 6)2— i; Z2=(a— &)' + «, ',*. gis known;
But (a—x){x -h)-z
.-. by(Z>) a;i=i(a + i+r);£C2 = l(«+6-r) (1).
in whicli r^ = [a — b)^- 4z,
= (a-Z/)2-4{(a.-i)2-«}=4«-3(a-fc)2
or («-i)2-4{(a-ft)2+£} = -4i-8(a-6)2
and^2 = i{c + (a-/))4}. (3)
Hence x is expressed in terms of a, b, and r,
r is expressed in terms of a, b, and t,
t is expressed in terms of a, b, and c,
4nd the expressions for r and t are cases of {B).
13. (a-jc)(6+a;)4 + («-a;)4(i+3;)=fl&(rt.3+/,3)
Let a— a; = ?? -2 and i+a; = w4-2 .". n = ^ (a + b) (1).
The equation reduces to
(n2 _22)r(„4g)3_^(TO_2)3} =ab{a^-\-b») '
:. {n^ -z^)(2n^-[-Gnz^) = ah(a^ +b^)
(2)
SIMPLE EQUATIONS. 171
«2 may now be found by {D), and from the resvilt z may be
found by (/>'), and from (1) x=-i^{a-h) + z ;
3z2 = 3(rt _ ly or i(iO«6 - ^<- - h^)
.'. x = 0, or a-b, or -^(«-i)+i\/(30a&-3«2 _ 3^3).
14. {Via-{-x)+ y(a-x)mV{a-\-x)+ ^{a-x}^^cx.
Divide the terms of tbe identity
\/{a+x)'^ - y(«-K)* = 2x
by the corresi^onding terms of the equation,
•■• \ (a-u:/ - c-r •• a-"- a; " \c-l/ '
(C4-I)4_(c__l)4
' ••• *• - «-(,,4:i)~4 + (c_l)4-
15. ir(.^-x)2 + ir{(*-^)(^-^)} + l^('^--^)' = ^(«^+"^+^')
Divide the terms of the identity
f/{a~x)^--^{b-xy-=a-b
by the corresponding terms of the equation.
.-. ^(.-.)-r('-)= ^(,,^|-+-,.)-
Cube, using the form (u—v)^ = u^ -v^—3itv{u-v).
(»-.)-(6-.)- 8#'U«-K'-)[ • ^(^+,,)
.-. -^{(a_^)i6-a.-)} =
.-. (a-x){b—x) =
a form solved m (D).
a6
a363
(rt«+flZ;-i-62)3
Assume -i/(" - x) = 2 y (as - 6)
a — b
,'. x- b = glTpi
1 72 SIMPLE EQUATIONS.
The proposed equation now becomes
^/(x-b){z+lr'
2-1
(^_6)(2-l)4
l/«
= c.
a form solved in Ex. 11.
17. {x-2){x-5){x-Q)(:x-d) + (u+2){y-4:){ij-5){y-n) +
{z+l){z + 5){z-^S){z+i2)=x{z~4.){x-l){x-ll) +
(2/+l)(y-l)(z/-8)O/-lO) + (z+2j(z-|-3)(z+lO)(0 + ll).
Let x' = x^-llx, y' = y'^ -% and a'^^^+lSz,
.-. (a;'4-18)(u;' + 30j+(//'-22)0/'+20)+(z'+12)(2'+40) =
a;'(a;'+2B) + (y'-10)(/y' + S)+(2' + 22)(^'+30)
.-. a;'3+48a;'+510+?/'2-27/'— 4'i0+z'2 4.52<;'+480 =
ic'2 + 28a;' +2/'2-22/'- 80+2'2+522'+660,
.«. 20x' = 0, .-. a;2-lla; = 0, /. a;i=0,a;3 = ll.
Exercise Ivi.
What can you deduce from the following statements ?
1. A'B = 0. 2. A-B-C==0. 3. (a-i):t; = 0. 4. 12a;]/ = 0.
5. What is the difference between the equation
{x-5y){x-iy-\-3) = 0
and the simultaneous equations
x — 5y = 0 and a; — 4^+3 = 0.
What values of x will satisfy the following equations ?
6. x{x-a) = 0. 7. ax(x+h) = 0. 8. (a;-a)(&a;-c) = 0.
9. ax^-=Sr,x. 10. a;2 = (a+6)a;. 11. a;(a;2 -a3) = 0.
12. a^x^^b^x. 13. x-2+(a-a;)2=a2,
14. ;i;3 4.(«_.^)2 = (a_2.r)2. 15. (a-x)3 + (^ -6)3 =a2a./,2
16. (rt— ic)(3:-&) + a& = 0.
17. {a-xy - (a - x){x - h) + {x - hf = «= + o6 + 6^
18. x^-{a-'b)x^nb = Q.
19. x3-(rt+6+c)a:2 . ah->rhc+ca)x—abc ^0.
SIMPLE EQUATIONS. 178
If X must be positive, wiiat value or values of z will satisfy the
iollowing equations ?
20. (.'e-5)(;f+4) = 0. 21. xS-H 29a; -30 = 0.
22. x--17x-84: = 0. 23. 3a;3 +10x+3 = 0.
i4. :<;4_i3^3_f_36 = 0. 25. x^ -^x^ - 5x+6 = 0.
■m
Solve the following equations :
26. (a-a;)2+(x-6)2 = («_A)3.
27. (a-a;)3-(a-a;)(x-6)4-(a;-/))2=(a-5)2,
28. «2(«_^j3=i2(6_a;)2. 29. a^{b~xy = b^{a^xy.
30. (a;-a)3 + (a-&)3 + (i-x)3=:0. 31. (a;-l)- =«(.x-3 -1).
09 (i — x x — a „£. a-j-6 — a; a—c-{-x
x — li c-\-x a — c — x ~ a-\-c—x
84. (a;-a+fe)(a:-a+'') = (a-6)3_a;2.
85. (a;-a)2-62 4-(rt4.6_a;)(6+c-a;) = 0.
36. (a+fc+c)a;3-(2^-l-5 + t>+« = 0.
Q_ a + 6 — x a-\-h — G
01. ^ .
C X
38. (a - x) 3 + (a - &)3 = (a+J - 2:.)*.
39. a;(a+&-!B)+(a+6 + c)c=0.
40. (n—p)x^+(p—7n)x+m — n = 0.
.-. ax^ —bx-\-c c .0 ax^ — hx+e __ a — h-^e
v}x^ —nx+p p mx^ —nx-\-p m—n-\-p
43. 4a;-^+a2-i3_2(a + ft)a; = (rt-x){6+a;)-(rt+x)(fc-x).
44. (2a-t_a;)3+9(a-&)3 = (a+i-2a;)3.
45. (2rt+2c-x)2 = (26+£c)(3r(- J-f3c-2a;).
46. (3a-5&H-a;)(5a-3&-a;) = (7a-6-3a;)3.
47. {%a-b+x)(9a-\-h-x) = (5a+3i - 3ic)3.
48. a{a-b)-h{a-c)x-\-c{b-c)x^ =Q.
174 SIMPLE EQUATIONS.
-r{a — c}^x.
50. {■'• + l){x + S){x-i)(x-l) + {x-l)(x-3){x-lr4:){x + l) = 9G.
51. (j--l)i.': + 3){x-^)[x+'J, + [^^:j{x-3){x+o)'^x-^Q}± IS
= 0.
52. ^-1- — =: 8i. 53. ^ + _ ^ _J]_ _^
■'■ ic « — o a-\-b
KA '^ ^ ^ .„ C' + x b+x ^,
54. X— — = — - — 55. -, + =2},.
X
1 — tyc. , , -r-
o a 0-\-x a + x
m
KG ^~^ x—b 13 a-x b + x in. n
Ob, . + = — -. 57. - — — = —
x~ 0 a—x () b-^x a — x n
« a; m x^+nx+a'^
5b. — + — = — 59. -, , = c.
X a n X' —ax +a^
{a-xY^+ix-by- 5 a~x ^ x-b m
64.
65.
66.
67.
{a-x)(x — b) 2 ■ a; — 6 a— iC jj
(a; + a)2-(a;-i)2 " 2«6 '
(a-a:)3-(a;-Z*)2 4.ab
{a~x){x-b) {a^-b'')
(a-x)^ + {a-x)(x--b) + (x-b)^ _ 49
{a~x)^-{a-x){x-b)i-(x-b)2 ~ lO'
2a2 + r/(a-a;) + (a + a^)2 r+l
2a^+a{a+x) + {a-xr' " c^l* (^sofo^^ = 5)-
68. (5-x)^+{2-xy = 17.
69. a;44.(«_,j.)4=c. a;4_{.(_j._4)4 ^ 82.
70. {a-xy + {x-b)^ = (a-b)4'. 71. (rt-a:)5^-(a;-5)5=r,
72. af^+(a-r)»=rtS; a;' -L(C-a-)'^ = 1056.
73. (a-»)'(x-i) = + (<f-a;)-(x-6)-'=«-//2(„_/,).
SIJIPLE EQUATIONS, .175
74. {a-x){h+x)'^-\-(a-x)^{b+x)^ + {a-z)-{b+xy^-^
{a-x)^{h+x) = {a + b)c.
(a-x)^Jx-b)^ _ 41
_ (a-xV+(x-hY _ 211
' . (a-a;)4-t-(a;-i)4 " 97 ('^~'')'
77.
78.
79.
80.
81.
82.
85.
86.
(a-x)^ + ix-b)^ _ a4 + ft4
(a-a;)2+(x-/;)3 ~ a^ + b^'
(a—x)^(x-^)i _ rt4 + J4
(a— a;)54-(aJ— &)^ a^ — 6
(a -a;) 3 (Z»-a;)3
<7 — re a;— ^
a h
(a;- 6)--^ + (a-x)^
~ b2 a^
(^y-.r)4 + (a:-t)4 rt4+54
(a+6-2a;)2 ~ (^f+fe)^
g.j {a-x)^ + {x-b)^ a^-b*
(a+T-2a;)2 = (a+l^a'
84 (^i:^'+(x-i)-5
(a-a-)-(a;-Z)) («-a;)(x— 6)
(a-x)^+(x-b)^
}^ i — - ■ L = c(a—x)(x—b).
{a-x)^+{x~b)^ ^ '^ '
(a-a:)4 + (a;-6)4 " (a-;c)(a;-6)'
88. ^l+a;2)3 = (a;3_3)2.
176 SIMPLE EQUATIONS.
89 ^* + ^ ^ ± 90 (a; + l)-(a;=^ + 1) a_
ix-l)^x a (a;3+a;+l)2
a
f7
93 ('^'+^)' ^ iL 94 (^+1)' _ ±
a;(a;+l)3 6 ' ' x{x'+l) ~ b '
96. ^(•'^+^)^ ^ 96 a;- + .r + 1 icH-^-1
(a;-l)4 -ft* ' ■ j^x+1)-' ■ (x-ip" - T
97 ^IZj^m _ JL 98 ^J^ii+ll _ «
99 (^ + 1)(^' + 1) ^ 100 (^+l)fa''^-l) _ _^
(a;-l)(a;3-l) * 6* ' (x-l)(a;'' + l) " 6'
101. (^+11^ = JL 10^ (x+iy ^ a_
x^ + l &■■ "■ x'^ + 1 h'
103. 2(«.-.7-)4-9(a-a;)3(3;-/;) + 14(a-a;)2(x-6)3--'
9(a-a;)(a;-6)3+2(a;-/>)4 = 0.
104. 4(a-a;)4-17(a-a;)-(a;-6)2+4(x-/;)4 = 0.
Find the rational roots in the following equations :
105. x4-12a;3+49.r2-78x+40 = 0. [Let 2 = a;2 - Oa;] .
106. x^-Qx^+lx^+Qx-Q.
107. 2;4_i0x3+35a;2-50a;+24 = 0.
108. 32a;4-48a;3_i0a;2+21a;+5 = 0.
109. a;3 - 6x2 +5a;+ 12 = 0.
5 4 9 4
— -j_ — _ 0
X x — a x — ^2a x— 3a ' x—4a
14 5 4 14 5 4
a;+20 ^ a;+5 ar-4 x-s>5 ^ x-40 'x-2{',
110.
111.
j-|o 2a; + 5a a;+8a
iC a; — a
x — a a;+5« 2* — 5a
+
+
9 — 3a x-4.a x—C
ilMPLE EQUATIONS. 177
ii„ x+i x+2 ic4-4 'x + 3 z-1 x-3
lis. — _1_ 1— _|_ _J_ a- ' — ^_ <
x + 2 X x — 1 x~2 x—'d x — 5
... 1 31 20 8 20 31
114. — _ _j_ 4- -^ _L _ — — J-
X x-l ^ a;-2 ^ x-3 ^ x-4: x-5 ^
' =0.
X— 6
^^g ^(a^+2x) + i/{a2-2x) _
■^/{a^-i-2x)-^{a^-2x) ~
m^ V{m^x-\-2) + y/{m^x-2)
a2 • 7/(m~'x'+2)--/(w3a;-2)
117. a/(.t2 -a2) + ^(ajs _i2) + ^(a;2 -c-S) =.r.
118. {\/{a-x)-\-\/{h-^)]{y\a-x)-W{lj-Ji)\^fi*
119 ■^(« — a;) — 1^(0; — 6) _ a+h-'lx
f{a-x)-\-f/{:x-b) " ~^u^-b
120. V(a+a;)4-V(«-a;)=y(2a).
[Write u luJ.' |^\a -x>, and v iov i^(.c — 6)],
178
SIMULTANEOUS EQUATIONS.
CHAPTER VI.
Simultaneous Equations.
Art. XLVI, There are three general methocls of resolving
simultaneous linear equations, 1° by substitution, 2° by compar-
ison, 3° by elimination. The last is often subdivided into the
method by cross-multipliers, and the method by arbitrary multd-
pliers.
In api^lying the elimination -method the work should be done
with detached coefficients, each equation should be numbered,
and a register of the operations performed should be kept.
Ex. Resolve
u+i+x+y-\-z = 15.
u+2o + 4:X+8y + iez = 57.
u-\-Sv+dx+ 27?/+81z= 179.
u+4:v + IGx-f G-1//+2563 = 45S .
M -f 5 <.■ + 2 5x-+ 1 25.// -f G252 = 97 5 .
Register
(2)-(l).
(3) -(2).
(4) -(3).
(5) -(4).
:7)-(0).
(B)-(7).
(9) -(8).
(11) -(10).
(12)-.(11).
(14)-(13).
(15)-- 24.
ii(13)-G0(W)}.
A[(10)-{12(17) + 50(16)}].
(6)--{3(18)+7(17) + 15(16)}.
(l)_|(19) + (18) + (17)-i-(16)!
u
V
X
y
z
1
1
1
1
1 =
= 15 (1)
i
•2
4
8
16
57 (2)
1
3
9
27
81
179 (3)
«.
•i
16
64
256
453 (4)
1
5
25
125
625
975 (5)
1
3
7
15
42 (6)
1
.5
-19
65
122 (7)
1
7
37
175
274 (8)
1
•J
61
369
522 (9)
2
12
50
80 (10)
2
18
110
152 (11)
2
24
194
248 (12)
(5
60
72 (13)
6
84
24
I
96 (14)
24 (15)
1 (16)
1
2 (17)
1
8 (18)
J
1
4 (19)
6 (20)
SIMULTANEOUS KQUATIOXS. 170
An eXcamination of the Register will sliow how easy it ■would
i;ave been to shorten 'the process, thus (lO)'is (7) — (6) which is
(:)) + (l)-2(2); similaily (11) is (4)-i-(2)-2(3) ; .-. (18) is (4)-^-
H(2)-3(3)-(l), &c.
A ojeneral systematic arranfrement of the ehmination-metliod
will be given in Part II. For two or three simultaneous equa-
tions it may be stated as follows.
r/2.'K-fA._,//-f (-2 =0.
Arrange the coefficients thus —
^1 ^1 ^1 ^i
«2 '^2 '^2 "s*
Form their products diagonally from left to riglit downwards,
thus — a^feg ^'i^2 '*i'^2'
Form their products diagonally from right to left downwards,
thus — ^if<2 ^J'2 "^i^a-
Subtract the latter products in order from the former, thus —
a^b^ — b^a^, b^c^ — c^b^, c^a^—OyC^.
Divide the 2° and 3'^ remainders by the 1° remainder, the first
quotient will be the value of w, the second quotient will be the
value of y. >
[Writing i?^, B.^, 7/, for the three 'remainders' respectively,
the general result is [iiix + v7j)li^ = niE^ +nR^] .
Ex. 1. Solve lla;+57/-68 = 0
6a;-7?/+31 = 0
11 5 -08 11
6 -7 31 6
-77
30
155
47G
-408
341
-1U7)
-321
-749
3
!
X
7
v.
1^0 SIMULTANEOUS EQUATIOKfc
Ex.2. 12
X
25
y
= 1.
22
X
^80
y
= 17.
12 -25
- 1 12
22 30 -
-17 22
360
425
- 22
-550
-30
-204
910)
455
182
1
1
2
6
II
H
1
i_
X
y
.'. x = 2
and y =
-.5.
2° Let the equations be
a^x+h^if + CjZ+d^ =<>
Arrange the coefficients thus
«1
6i
"i
-<^I
-«i
-*I
«2
^2
^'2
-t/..
-S
~6,
«3
h
H
-dz
-^3
-\
«I
K
<^i
-d,
-«1
-*,
«2
\
^2
-d.
-«2
-^2
Selecting the first three cohimns form the diagonal product**
from left to right downwards, thus :
SIMULTANEOUS EQUATIONS. 181
«i ^1 Ci giving a^b^c^
\
\
ao ^2 Cj a2*3''l
\ \
«3 ^3 '■$ ^S^l'^fl
\ \
<*1 ^1 ''l
^2 • ^'2 ^2
Form the diagonal products from right to left downwards, thus:
«i ^i c, giying c^h^al
«3 ^3 ''2
/ /
Cg&ga,
a, ^3 c.
Cjjftja,
a, b-i Cj
/
«2 ''2 '^2
From the sum of the former products take the sum of the latter
products obtaining a remainder, which call R^.
Similarly form a 2° remainder, Eg from the 2°, 3° and 4° columns
a 3° " i?3 " 3°, 4° and 5°
a 4°» « i?4 " 4°, 5° and 6" "
Then a;=7?3 4-Z?i, y^R^-i-Ri, z=R^—R^,
and generally *
(wa;-{-wt/+;5z)Ei =mR^-\-nR^-^pR^»
Ex. 8. 3a:+2y- 42+20 = 0
5x-ly-Qz- 1 = 0
7a;+5y + 5i!-24 = a.
:82
SIMULTANEOCS EQUATIOIiS.
a
2 -4 -20 -
• >
— ^
5
-7 -G 1 -
5
7
7
5 5 24 -
7
-5
3
2 -4 -2!) -
3
-2 '
5 •
-7 -G 1 -
5
7
1
-105
-288 28 -500
(3x -7x5= -^105
-100
700 432 14
5 x5x -4=-- -100
- 84
- 20 500 -504
7x2x -6=- 84, &c)
-289
392 9G0 -990
19G
GOO - 15 240
(-4x-7x7 = 19G
-90
10 480 980
-6x5x3 = -90
50
G72 -840 15
5x2x6 = 50, &c.)
156
1282 -875 1235
-445)
-890 1335-2225
X y t
Exercise Ivii.
Solve the followiug systems of equati ns :
1. 2x+Btj = 4:l
Sx+2y^S9
3. 11a; + 12?/ = 100
9a;+87/ = 80.
5. nx+ly = 7
5a; + 3?/ = - 36.
7. 5a:+%+2 = 0
3.^;+2?/ + l=0
9. 10a;+77/ 4-4 = 0
.Gx + 5// + 2 = 0.
11. ^x+hj = ^'
3a;-4y = 4.
2. 5a; + 7?/=17
lx-5y= 9.
4. 18x-?j5y+ld = 0
15a; +28?/ -275 = 0.
G. 3a;+lG/i-5 = 0
28// = 5a;+19.
8. 21a; + 82/+66 = 0
23?/ -28a; + 13 = 0.
10. 23a;+15?/-4i = 0
32.r+21?/-6 = 0.
12. lx-ly = l.
SIJIUI^TANEOUS EQUATIONS. 183
13. ly = lx-l. 14. §x+§y = n.
15. l-5x-2y = l. 16. lx=10y + -l.
2-5x-dy = G. Ux=lGy + -l.
17. 5x-iy-{-l = 0. 18. ■lGx--Oi>j = l.
l-7x-2-2y+l-d = 0. •ldx--lly = l.
19. 3-5.c + 2^v/ = 13+Ua;-3-57/.
2i*-+ -8^ = 221+ -Tu; - 3|!/.
20. 1 + JL = A.
ic y 6
Jl J_ _ J_
a; 2/ 6
22. i:? = ?:! _ 1.
« y
•8 3-6
— + — = 5.
X y
24. |, A.,,. .
IT + y - '^^•
20. ix-\{:y+\.) = \.
k{x+l)+l{^y-l)^'d.
1 o
28.
3x-+l 5// + 4
1 2
4a;- 3 ~ 7//- 6
80. l^^^±i == 8.
45-1/
21.
^ + '^ = 3.
X y
15 4
— — — = 4.
X y
23.
•3
17a; - — = 3.
y
•4
IGa; - — = 2.
2/
25.
';= + ■' = 6.
lOa; 9
- - -j = 31
5 7
27.
.^+2^/ ~ 2a;+2/
7 ■ 5
3a;-2 ~ G-y'
2'J.
x+^y _ g
7a;-13
3//- 5
31.
3a;+l 4
4-2y - 3*
{B+|/=l.
= 4.
J 84 SIMULTANEOUS EQUATIONS.
32. 1--^ = 1. 33 ^^±^1±} _ 2
5-3?/ 2 2a;-?/ + l ~
7-22/ _ _2 8.r-y+l
5 -3a; ~ 3* «-!/ + 3 ^ ^'
3^ a;+3y+13 x+1 y+2 2(a;-y)
*" • •4X+-52/-2-5 = ^^' ^^- ~3 r = ~5^
•8a;+ajH--6 _ J_ ^-3 y-3
6a;H-%-23 ~ T' 4 "" 3 = ^^ ~ '■
36 ^^-y+^ a;-2y+3
3 4^~
3a;- 47/+ 3 4a;-2?/-9
i — + — y— = ^•
87. 20(.c+l) = 150/ + l) = 12(a;+7/).
38. (a;-2) : (y+l) : (a;+.v-3) :: 3 : 4 : 5.
39. (a;-5) : (y/+9) : (a;+^ + 4) :: 1 : 2 : 3.
^^- ^ = ^- ^1- (^-4)(.'/ + 7) = (^-3)(i/+4).
X+i 2/r"
^) = 5W (.+6)(,-2) = (.+2)fa-l).
42. (a;-l)(5y-3) = 3(3x+l). 43. (a;+l)(2// + l) = 5a; + 92/ + l.
(a;-l)(42/ + 3) = 3(7a;-l). (a;+2)(3//+l) = 9a; + 13y + 2.
44. (3a;-2)(5i/ + l) = (5ar-l)(2/+2).
(3a;-l)(s^+5) = (a;+5)(7y-l).
45. a; + 7/ = 37. 46. 2a;+2z/ = 7.
?/+z = 25. 7a; +192 = 29.
z+a; = 22. i/+8«=17.
47. l-3a;-l-9</ = l. 48. 5x-+3i/+2z = 217.
17//-Mz = 2. 5a;-3s^ = 39.
2-9z-2-lx = 3. 3?/ -22= 20.
49. \x-\xj = ^. 50, l^a;+l|?/ = 10.
^a;-:|z=l. 2§x+2|0 = 2O.
^«-jy = 2. 3i-s/+3iz = 30.
SIMULTANEOUS EQUATIONS. i^"
//+z-a;=l3. x+2y + iz=li.
z+x-y= 7. a;+3/y+y2 = ^B.
53. ^.a.^^^^= 3. 54. 7x+6?/ + 7i = 100.
x-y
= 0.
2^-+4?/-f- F3=13. :c-2(/+ z =
3a;+%4- 272 = ;U. ?ix+ y ~2z = G,
Sr,. 3j5+2!/ + 82-110, 60. .c + y+z = d.
5x-\- ?/-42 = 0. a; + 2y + 3c-14.
2x-3y+ z = 0. a:+3.v + Gz = '20.
57. aj+2// + 82 = 32. 58. x + y + 2z = %4t.
2^ + 3y-(-z = -42. «+2.v+z =33.
3x+ ?/ + 22=10. 2a; + ^i2^82.
59. 3.C+3//+ z = 17. 60. c+'ly-z= 4-6..
3a;+ y4-Cz=15. y + -2z~x=l(yi.
x+Sy-\-Bz = lS. z + 1x-y= 5-7.
61. x4-2i/--73 = 21. 62. x+^^l^z + S.
3a; + -27/- z = 24. </+z = 2f/y-U.
•9;c + 7^- 2^ = 27. z +a; = 3|.,--82,
63. ix+il/ + |3 = 30^. 64. %x + ?>\y^A\z=\m
ix + iv + 12 = 27. 3^x + 4iy/ + .^^z = ? 75.
1:^ + ^2/ + ]2 = 18. 2§.c + 3|^+l?2=iuV.
65. -+i = 2. 66. -^_p^- = ^
z+3 _ , gg + ^ ^. 9
:^ - ^' 2/ + 1 " '
67. -_±^ = 10. 68. ^^ = 2.
a + z
186
SIMULTANEOUS EQUATIONS.
69. i - A ^ 3. y 70. — + :l + :l = i
v y
2 '6
X z
y g
= 4. _ + — + — = 4.
X y z
^ 12 10
V. 0. '- + = ^r
71. ^- . 4- 72.
J/2 _ 1
j/+z ~ "6"
za; 1
2+a; ~ 7 '
73. (.r+2)(2?/-i-l) = (2a;+7)//.
(a: - 2)f32 -^ I) = (x+3)(3z - 1).
(y + l)(2+c =(2/+3)( z + 1).
74. (2a;-l)(2/+i;=2(a;+l)(v/-l).
(a,+4)(z+l) =(a:+2)(0 + 2).
(2/-2){z+3) .-=(y-l)(2 + l).
75. (rc-ui)(%-3) = (7.t-+l)(2//-3).
(4a;-l)(z+l) = (a;+l)(2z-l).
(i/+3)(2+2) =(3^-^)(3z-l).
76. 21a;+31?/+42z=n5.
6(2aj+i/) = 3(3a;+2) = 2(^+2)
77. 15(a;-22/) = 5(2a:-3;?) = 3(?/+2).
21a;4-31^ + dlz = lSrj.
78. 6a;(i/+2) = 4?/(z + x) =^ 32(x ^ vj.
1 1 1
79.
6
4
y
6
+ — ■
z
— +
X
8
5
ft/
'- +
X
12
10
y
z
xy
iy — 3x
=
20.
xz
2x - Si
.-=
15.
yz
4»/ - oz
=
la.
a; 2/ ^
= 9.
^
3aj +7/4-2 = 20.
3«+a; + 4?/ = 30.
3?f. + 6.c-fz = 40.
6u-f8;/-f3^ = 50
60.
a-+;'-f S// = 33.
5» f ?/-f-2 = ii,
i^z+A-f 2 = 11.
3?. ^ a'+^ = J.l.
ST:\ItTLTANEOUS EQTTATIOXS. 187
81. a-{-x + i, + z = lU. 82. ■u + .r-\-jj-^z = ^4:.
It +2j;-{-2ij + 2z=-2,67. ti + 2x+ 3i/ - L'z :- 0.
n + 2x+Bu + 3z—SrA). Su-x-5ij-^z = J.
7( + 2x-^3i/+iz = 110. 2u+3x-4:y - 5z = 0.
38. u-{-x + y+z = GO. 84. :i-{-x-\' ,/-\-z = l.
u-\-2x+3ij+-iz = 100. 2u-\-4x ^ 8// + IQz = G.
u+Sx+G>j-\-10z=150. 32i + 9x+27.(/ + 8l2 = lo.
?t + 4.«;+10// + 202=210. 4?H-16x-f 04// + 25Gz = 35.
85. ^x+y-U = l. 85. ht-^x-\-iy-:^.^Vi.
lx-^P-lu = l. ^u-Sx+l-,j-l^^l7.
f.'/- 12-^(6 = 0. pi^}^x-y + i?=n.
Art. XLVII. Tlie principle of symmetry is often of use in
the solution of symmetrical equations. For from one relation
which may he found to exist between two or more of the letters
involved, other relations may he derived by symmetry ; also,
when the vahie of one of the unknown quantities has been deter-
"lained, the values of tlie others can be at once written down, &o,
1 • (x4- y) {x+z)=a.
{x+y){y+z) = /^-
(x+z){y+z)=c.
Multiply the equations together and extract the square root.
•■• {x+y){y+z){z+x) = i/{ahc).
Divide this equation by the third.
.'. x + y = }— i— — i-, and therefore, by symmetry.
.*. J/+Z =
.". z-\-x =
Hence we get
X =
c
1
a
V[ahc)
nh — bc-\-ca
2Viabc)
wlienee y and z may be derived by symmetry.
188 SIMULTANEOUS EQUATIONS.
2. x.+v-^-z-O (1).
vx + by + cz = 0 (2).
br.r+caij-^ahz-{-{a-b){b-c){c-a) = 0: (3).
cx(l)-(2) gives {c-a)x+{c-b)y = 0.
.'. y = ^ L, and similarly,
b — c
_ (a-b)x
— r •
b — c
Substitute in (3) these values of y and z, and reduce,
,-. x{a — b){c —a) = (a — b)(b — c){c — a),
.'. or x = {h — c), .'. y = c--a, z = a — h.
8. a{yz—zx—xy) = b{zx — xy — yz) = c{xy — yz — zx) =xyz.
Divide tlie first and the last equations by axyz ;
.-. — — — _ — _ — , and hence, by symmetry,
axyz
J- _ i- _ -1 _ i-
b ~ y Z X
1 _ JL _ 1 _ 1,
c z X y
,♦. _)_ — = — —, and by symmetry,
be X
L i_ - _ A
c a y
!_ J_ _ _ A.
a b z ■
i. ax -Y by -\- cz = \ (1 )
«2.f+A3^+r32 = l (2).
a^x + b^y-\-c^z = l (3).
<;x(l)-(2) gives a{r-a)x+b{c -h)y = c-\ (4).
cx(2)-(3) " a^-{c-a)x + b^{c-b)y = c-l (5).
4x(4)-(5) " ah{c.-a)x-a^{c-U)x = h{c-l)-{c~l),
ox a[a — b){a — c)x=-{G-l){b—l),
... X = (1-^Hi-c).
a[a — b)[a — c)
wlienae y and z may be derived by symmetry.
SIMULTA?^SO'JS EQUATIONS. 189
6. Eliminate a;, y, z, u (wLich are supposed all diflferent) trom
the foUowiug equations :
x=^hy-\-cz-\-du.
y-cz + du+ax.
e = du-\-az+by.
u = az-\-hy+cz.
Subtracting the second equation from the first»
.•. x — y = hij—ax, or
(1 + a)x = (1 + h)y = (by symmetry) (1 + '^)z = (l+dju.
These relations may be &lso obtained by adding ax to both
members of the first equation, by, to both members of the second
squation, &c.
Now divide the first equation by these equals,
1 b__ c_ _d_
" \ + a ~ i + 6 "•" r+c "*■ \-ird
And since = 1 — we have
1+a 1+a
^ _ a b c a
Exercise Iviii.
b-c-bc'
1 . (jriven ax -\- bi/ = c and that x = ,— t~i*
'' b a -ba'
a'x+h 'y = c' derive the value of jr
a(dm— en)
2. Given bx = ay and that x = — ] t — »
"^ be — ad
dx + ""? = cy-\-n c derive the value of y.
8. Given ax-{-bi/-^cz -d. and that a; =
a(d-b)(d-c)
a^x-{-h^ij-\-c^zz=d- "-T pr/ \. write down
<3iSx-\-b^y -\- e^z =d^ the values of y and z
190
SIMULTANKOUS EQUATIONS.
4, There i? a set. of equations in x, y, z, u, and w, -witli corres-
coucling coefficients {a to x, &c.), a, b, c, d, and e ; one of the
equations is
x = by + rz-\-du-\-ew, write down the others.
Solve the following equations :
K ^ , y V z ^ X 2
m n n p m p
6. x-\-a)/ + J>z = 7n, y + az ^hx = n, z + ax-\-hij =p.
7. x + rtv = /, y + bz = )n, z + cu = ii, u-\-dw=p, w + ex = r.
8. Eliminate x, y, z, (supposed to be all different) from the
following equations :
z=by-\-cz, y = cz + ax, z = ax-\-liy.
9. Eliminate x, y, z, from
X y z
= a, — . — = i), = c.
y-]-z ' z-\-x * x+y
10. Having given
x — hii-\-rz-{-dv.-\-ruj,
y = cz + dn-\-en--\-((x,
Z =dit + (ii-\-ax + by.
u = ew ■]-ax-]'by + cz,
w = ax-^by-^cz-\-dii,
abode
Shew that ,-7— + ip- —7 + tT' + i~r~) + rT~ = ^^
1+a 1 + 0 1+c l+(t 1+^
Art. XLVIII. Eesolution of Particular Systems of Linear
Equations.
Ex. 1. • x+y-\-z = a (1)
y+z-^u = b (2)
2-fH + . r = c (3)
ii-{-x+y = d ^ (4)
(l) + (2) + (3) + (4) B(u + x-\-y-^^)=r> + h+r+d (5']
8(1) 3(a;+7/+z) = 3« (C)
|t{(5')--(C')} tt =i(-2rt4-AV + ''-)
SIMULTANEOUS EQUATIONS. 191
The values of x, y and z may now be written down by sym-
metry.
Tiie following is a variation of the above method, applicable to
a much more general system.
A-Ssume the auxiliary equation
u-\-x-\-y-{-z = s, (5)
.-. (1) becomes s — a-.a, (6)
(2) " s-x=^b, (7)
(3) " s-ij = c, (8)
(4) '• . s-z = d, (9)
(5) + (6) + ^7) + (8) + (9) 4.s = s + a + h^-c+d.
:. s = \{a ->!■■}> -\-c-^d),
s is now a known quantity, and may be treated as such,
in (6) giving u = s — a
(7) " x = s-b
(8) " y = s~c
" (9) " z = s-d.
Ex. 2. yz = a{y-\-z), (1)
z.c = b{z+:^, (2;
xy^c{x-\-y), (3)
111
(l)^ayz, ^ -\ = — ,
y . z a
/o^ / 111
Z X o
111
{Sy-i-cxy, 1 = — .
X y G
This may now be solved like Ex. 1, using the reciprocals of a
b, c, X, y and z instead of these quantities themselves.
Ex.3. ai«+ii(a;+7/+2) = r, (1)
a„x-\-b2{y+z + u) = c^ (2)
'^3.'/ + ^3(2+ i + 'i^) =^^3 (3)
a4Z + b^{u-^x+y) = c^ (4)
Assume the auxiliary equation
u-tx+y+z = s. (5)
V\;iJ SIMULTANKOUS EQUATIONS.
\i; becomes b^^s—ib^— a ^)it = c^
by ' r,
«i — «! Z>i - a, \ /
Similarly from (2) , ^ s -x = j — ^ (7)
2 "2 2 ~ '2
^ ' ^B-'^3 ^ h-^3 ■ ^ ^
From (10) we can at once get the value of s, which may there-
fore be treated as a known, quantity.
bjS — Cy
in (6) giving u=, _
"\ "i
'and the value of a;, y, and z may be obtained from (7), (8) and
(9), or they n'va^y be written down by symmetry.
Ex. 4. ax-\-b(}j-^z) = c ' (1)
ay+-h{z + n) = d (2)
az-\-hl(ii-\-x) = e (8)
au-irb{xJf-y)^f (4)
Assume tt+.T + i/ + z = s (5)
(l) + (2) + (3)-f(4) (a + 26)s =c + ^ + r+/ (6)
Hence s is a known quantity and may be treated as such.
From (1) and (5) hs—bu-\-{a — h)x = c,
:. bi( ~(a - b)x=bs—c, (7)
Similarly from (2) and (5) bx -{a — h)y = bs—d, (8)
" (8) " " by-{a-b]z^bs-e, (9)
*• (4) " " lz~{a-b)n^bs-f, (10)
b{7) -{-{a- b){8) i" a - (a - b) -y = abs -bz- {a - b)d,[l 1 )
6(9) +(^i - b){\i)) b-^y - (a - h)"- a = ab>i - hi,^ {a-Oy, (12)
SIML'LTANEnuS EQUATIONS.
1'15
-«(/>2./ + ("-6)3/-}-6{63(c-fZ) + (a-6)2(.-/)} (13)
The values of x, y, aud z may now be written down by sym-
metry.
Ex. 5. a^ + a^.v-\-ai/+z = 0,
b^ + b^x + by-\-z = 0,
The polynomc t^ + .<;;- -{-yt+z vanishes for i = a, t = h, t = e,
.'. by Th. II., p. 4G, for all values of t.
t^ +.rJ2 +yt+z --= {t-a){t- h)(t - r\
= t^-{a + h-\-c)t-+{ah + hc + ca)t-ahc.
,-. Th. III., p. 53, x= -{a-\-h-\-c),
y = ah-\-bc-^ca,
2= —ahc.
Ex. 6. a;+?/-|-2 + ?/ = ],
ax + /^^ -\-fZ-\- du = 0,
Employing the motliod of arbitrary multipliers,
(4) + /(3) + »»(2)+?i(l) a^x+h^y+ c^
+ ma + ?.'fi + VIC
-\-n I 4- 71 I -{■ n
To determine x assume
63-f /Z/2-l-H?7> + n = 0,
0 + d^
+ ld'^
+md
+ n
u = n
(2)
(3)
(^)
(5)
iC =
(7)
{^)
(9)
But the system (6), (7), (8) has been solved in Ex. 5, from
which it is seen that
1= —{h-\-c-\-d), m = hc + cd + db, n= —bed,
and a^+a~l-]-ain + n = {a -b)(a-c){a- d) i
'i;H SIMULTANEOUS EQUATIONS.
•. using these values in (9)
-hcd
( a — h){^a— c){<:i — d)
The values of ?/, z and u may now be written down by sym-
metry.
Ex. 7. ^^ + _L ^ _^ = 1. (1)
VI — a m — 0 m — c
- +^J'+^^^l, (2)
n — o n — h n~c
-f- — ^V + — = 1- C'^)
p — a p—b ]j — c
Assume 1 - ^_- _ . L' _ _^_ _ t^+Bt^~-\-Ct+D
t — a t-h i-c ' (t-a){t~b){t-c)
But in virtue of equations (1), (2) and (3), the first member of
(4) vanishes for t = m, t = n, and t=p, and .-, t^ + Bt- + Ct + D
♦janishes for the same vahies of t, and .•. Th. II. p. 46,
ti-\.Bt^ + Ct + D = {t-m){t-n)(t-p),
.'. (4) becomes 1 —
X y z
t — a t — h t — c
_ (t-m){t~H){t-p)
'" {t-a){t-b){t-c) '
To obtain the value of x multii^ly both sides of this equation
hj{t-a),
t-a-x- y^^~^) _ z{t-a) ^ {t-m){t-n)(t-p)
t-b t-c ~ (t-b}{t~c)
Now t may have any value in this equation ; let « = a.
(a — ni)(a — n){a — p)
~ {a—b){a — c)
The substitution (xyz\abc) will give the values of y and t,
Ex. 8. ^^t^ = ?^ = ^±i . (1)
p q r
lx + 7Hi/4-nz = s^ (2)
simulxa!
By Art. XXXVII.,
x+a y + b z-i-c
??5
v
(2)
q r
s^-{-la + mb-\-nc
Ix-\-))iy-{-fiz-{-ia-{-miy-\-nc
lp-\'-viq-^nr
= II, say
lj)-\-niq-\-nr
,'. x=pR-a, y = qR — b, z = rR—c.
Ex.9.
yz+zi«+xy = (a-fi +Cjxyz
(1)
yz +zx ZX+ xy xy + yz
a "^ b ~ G
(2)
{l)-xy»
11,1
— + + - a+b + c.
X y z
(3)
{2)^xyz.
11111
— + — — + — — +
X y ^ y z _ z
y
X
(4)
a b c
Page 122 and
(3) " 4- 2 + 2
- X y 2 _
a-\-b-\-c
2
c^)
(4) and (5)
11 11
.-. h — = 2rt, — + — =
X y y z
20,
1 1
— + — = 2c.
Z X
(6)
(3) -(6)
111
— a — b + c, —a-\-b—c, —
X y z
— a
+ h+c.
Ex. 10.
aJ + c , y+b
a+b a-\-G
W
"-'' + ^'-f - 2.
«— c a—b
(2)
(1)
x-\-c y + b
a-\-b a + c
x—a—b+c a+c — b—y
a+b ~ a+G
(3)
1C6 SIMULTANEOUS EQUATIONS.
Similarly from (2) 5Z/^1±£ ^ a-h + c-y_ ^^^
a—c u—b
(3) and (4) ... x-a-h-\rc = '^-(a-h-\-c-y)
a-\-G
But unless ^- — — ^ ~^., this cannot be the case except for
a + c a — b
a-b+c-y = 0,
in which case z—a — b-{-c = 0 also,
giving x = a + b — c a.Tady = a — b+c, (5)
Ti; a + b a~c
If ---— =: r- .-. a3-£3 = ,,2_,.2 (6)
a.+-c a — b ^ '
63-c2=0, or (6 + c)(5-6-)=..0,
& = c, or b— —c.
But if &= +c or — e, (1) and (2) are one and the same equation ;
hence if (1) and (2) are independent, (6) cannot be true, thus
leaving only the alternative (5).
Ex.11, "lax^ib + c-aYii-^z), (1)
2by = {c + a-b){z^x), ■ (2)
(a;+?/ + z)-'+a;2 + ?/-'+23=4(a2+62 + f2) (3)
(1) and page 122 (5)^_ = Ip- = ^+l±f (4)
b + c — a 2a o+c+a
(0) u u y _ x+z _ x + !j-\-z .gv
c + a-b 2b ~ c + a+b ' ^ '
^4), (5)and " ...^+?/+^ =. ^ = ^ = __J__
« + /' + (^ 6+c — « c+a-6 rt-|-6-c*
«3 (a;+?/+z)2+x2+?/2_(_22
Reduction and (3) = 4(^24.^,2^^2) = ^*
SIMULTANEOUS EQUATIONS.
1&7
1 1 1 ...
Ex. 12. . ax =by = cz= ~ + — ^- ^ \^)
.a h c _ ff+6+c
{l)^xyz :. - = ^ = :^ • • - ^H^+---» ^^
a 1 /I 1 1\ a;(/ + ?/z + z.c '
(2)x(3) • 4\- = ^i^ •
:. a^x^ = a+b-\-G.
Ex.13.
a ~ b '
c
(n
xyz = m^
(?)
(1)
z X y
a+ b ~ b+c ~ c + a' ~
m
suppose
(3)
then
xyz m^
{a-\-b){b + c){c + a) rs
.-. r^ = (a + b){b + c){c+a)
Hence the value of r is known and from (8)
rx — m{b-\-c).
Ex. U. y+z = 2axijz (1)
2+u; = 26.<;^z (2)
a; + ^ = 2rjc2/« ,: (3)
y-\-z z+x x-\-y x+y + z
•*• ^y^^ ~2^ = ~2r "^ ~2r "^ ^+y+o
a; 7/ z
b + c — a c-{-a~-b a-{-b — G
xyz
:. x^yH^ = (^f)^c-a){c + a-l)){a+b-c)
1
.-. x^yH^ = ^^r^ a) (c -I- a- b){a + 6 - c)/
(-t>
t9B 8IMULT4NE03S EQUATIONS.
Hence the value of x^y~z^ is known, call it —^ and substituto
in (4) •
1 X
r ~ b -{- G —a
.-. rx = b + c—a,
in which r^ = [b-{-c — a)(c-^a — h)[a + b — c).
Ex.15. y^+z^-x{2j + z) = a (1)
z^-+x^-y{z + x) = b _ . (2)
a;3+2/3-2(.x-+^) = t (3)
(l) + (2)+(3) 2{x-+y^+z^-xy-yz-zx) = atb-\-c (4)
(1) may be written , x'-^ +y- +2- —x{x+y-]-z) = a (6)
(2) " " x^ + y"+z-^-y{x + y-\-z) = b (6)
(3) " " x^+y"+z^-z{x+y+z):^c - (7)
.•. x + y+z ■=
a — b b — c c — a
y — x z — y x — z
\y -x)- ^{z -yy + {x-z)^
a'i^h^j^c^-ah-bc-ca
x^+y'^-{-z^ -xy -yz — zx
2(a3+&3_f.c3_3aJc)
Write r2 for 2(a3 + &3+c3 _3^,;,c^^
(9)
(9) ....+,+. = __L__ (10)
Eeturning to (8) {x-^y^zY = ^^"^ + ^^t+r+t "'^"''^ <^)
(4) 2(x-^ +r"+2"-^i/-2/2-2-0 = ^^^±^ (11)
BlMUL'rANEOUS EQUATIONS. 1'^
i[(8) + (n)} x^'-^^r-^z-^ = ^l±}l±^ (12)
(5) ami (10) x-+7/-^z"^- — ^_ = a
a + b-\-G
(12) ^a^j^h^-j^c"_a{a + h-\-c)
-h^'+c^ -a{h+c).
(5), (6), (7) are symmetrical -with respect to (.-ryzjfiroc); (10) sliowg
this substitution does not affect r, and consequently the values
of 2/ and z may be written down at once from that of «»
Exercise '.
lix.
1.
az+h]/ = c,
mx+ny = d.
2.
(iz~\-hy = c,
mx — ny = d.
3.
ax + by = c,
mx-\-ny = c.
4.
-2- + ■' = 1.
a b
x+y = G.
5.
a '^ b
1,
6.
a ' b - ^-^
X V
b ^ a
1,
X y
b a
7.
ax+bc = by-{-
x + y =c.
ac.
8.
a h
b a
— + -=«.
X y
9.
{n-^c)x-{a-
(« + %-(a-
■b)x =
■- 2ab,
= 2ac.
10.
x—c a
y-c ~ b'
x — y = a — b.
ll"
X a
- /■'
y ''
x+m c
1
12.
x+y a+h-{-c,
7/-|-l ' a — b+c'
y—1 a — b^c
y + n d x+l a+b — c
200 SIMULTANEOUS EQUATIONS.
y-a+h c ' a+b ^ a+c '
y-^b c + a x — b y-c
" " = 7 — • 4- — 2
x-\-c h-\-a a — c a — b '
vi-a '^ m-b ~ ' (b-^c)x-\-{a-\--)//~\-{u-{-b)z
X y ^ =0,
+ - ^ = 1
« - (7 n-b ' hex-\- acy + ahz = 1 .
17, x-{-y+z = l, x-a y-b z-c
15.
-»
r
19.
;s-a y-b z-c
P q ~ r '
hc-{-my-\~vz = l.
21.
x-\-y-\-z = a-\-l>JrC,
hx\cy + nz = a^-\-b'^+c^.
cx-\-ay-\-hz = a^+b^ -f c2.
ax + hy + cz = m, ^^- 'p q
Jl_ s V_ , « , l{£-a)^m{l^-b)^-n{z -c)
I- a ■•" Z-i + TZTc " ■^- =1.
20. rf(a;-^)=%-/j) = r(2-c),
ax\by-\-cz~ in'^.
22. a;+?/+2 = 0,
ox + 6y + c« = rt/)+Sr f r/7,
(^; - c).« + (c - «)^ -f (a - b)z
= 0.
23. a;+^-}-2; = /n, 21. ax+by + cz = r,
X : y : z = a : b : c. inx= <ny, qy=pz
25. xy + yz^zx = 0, ayz-\-hzx^cxy = 0,
hcyz+a.cxz-\-ahxy ^{a -b){b — c) {c — a)xyz = 0.
26. (a + 6)x4-(?'+c)?/ + (c + a)z = a6 + 6c+ca,
{h-irc)x-\-{a + c)y-\-(a + b)z = a^+b^+e'».
27. *ox + ny+jiz + qu = r,
X y z u
a b c d
SIMULTANEOUS EQUATIONS. JOl
a b c o ^b-c
1 1 1
— + — + — =■- a-\-h-\-c.
X y e
29. {<i-b){c-\-c)-a:j-{-hz-={c-a)[ij -{■})) -C2-faaj = 0,
x-\-ij +z = '\'< +^ + c).
30. a:'-+h// = l, 31. lf/+mx = n,
b// + cz = l, ' vx + lz = m,
cz + ax —1, viz-\- ny = L
82 x + y = a, 33. ^+2-r = '-^^
, /)i
2/+2 = o, H-.C -y = — f
x-^-z-c. z-\-y~z = — ,
n
84. 1 = 2a. 85. ^ — ,
y Z y z X a
1 ^ o. 1112
« X z X y b
a; 2/^' ^ y z ~ c
86. (a + ?>).r4-(a -5)^ = 2ic,
(6 + c)7/+ (& — c)a; = 2ac,
(c4-n)z4-(c — «)?/ = 2a6.
87. a: + f - - = a. ^8. ^ + -^ = 6-.,,
/j c b-\-c c-\-a
Z ^ _ -L ^ JL ?-
T ~ T " ' T+7 "'' ~^b
y + — - — = ^y -TTT + TTTIT = ^-^*
XV y z ,
a
b ' c+a a + 6
'202 SIMTTLTANEOUS EQUATIONS.
39. a:+Z/ -2 = ", 40. jt-f-v-x = a,
y -^-z — v — h, v+ X — y ~ b,
z+v — x=c, x+>/-z = c,
v-\-x-y=d. y+x-u = d,
z + u — r = e.
Exercise Ix,
Eesolve
1. {a+b)x+(a-b)y = 2{a''-\rb^) 2. x + y = a,
{a-b)x + {a + b)y = 2{a^-b2) x- - y^ =b.
3. 2x — 3y = m, 4. {a — b)x-l-(a-\-b)y = a + h,
2x- — 3y^ = n^ -{-xy. x y \
a + b a — b a-{-b
K V 7\ . a^-h + 1 {a-irb-c)x-{a-b^c)y
5. [a ~- b)x+y = — — rr ' "• i n \
a — b-^1 X a-\-b—c
Z-\-ia-\-b)y= T~ ' — = T~7~'
' V ' /:/ a — b y a — 0+0
x-\-ii a x — a a — b
7. — ^ =- j . 8. = — T-
x+c y-\-^ ^ a^~b^
a + b ~ a-\;-0 y a^ + b^
x-y+1 _ .Q ^+y_±l _ "+}
x — y—1 z—y + 1 a — 1
x+y+1 ^ ^ ' x+y + 1 _ 1 + h
x+y-1 ~ ' x — y-1 1 — b'
x+y—1 a+b a—b ' *
x+y+1 , ^ , ?/ o
x—y—i a 6
13. {a + c)x+{a-c)y = 2ah, 14. rt^+r/.r 4-7/ = 0,
(a+6)?/ — (a-fe)jc = 2ac. b'^ + bx + y = 0.
* SIMULTANEOUS EQUATIONS. IZHB
15. y + Z — :f = rt. 10. 7:r-{-ll;/ + z = a,
j: + i/—z = g. 7z-{-11x + ij — c.
X 1/ z '^' ' • .
— < — _ IL ^2hc (c-a){>/-rb)-ci,+ax = 0,
X
y
c a b x+y^z = 2[a + h^c).
— + — — — = Aca.
z X tj
•0 , y - %->n ^ y ^
19. T—- + -r—. =«+^ 20. — -- -4. ^ ^^0,
0 + c c —a o-\-c c — a a— b *
y ^__ _ / ^ y ^
c-\-a a — b ' h — c c — a a'^b~ *
. Z X X If ^
21. ^ + -J^ + -^ = 1, 22, ^ . „.
a a — 1 a — 'A x+y
T + 6-1 ■*■ &-2 - ■"' i/+s - *»
c ^ c-1 ^ c-2 ~ ^' z+x ~ ""
23. ± + f + i--+^ + A^
rt 0 c oca
a; ,y 2 1 1 1
04 ^ •'/ ^
"" ' rt b c
u
d'
25. ax = bt/ = cz:r=ifi
mx + ny-]-pz-\-qu = r
7/3 — 23 _^„..jj.
26. ?/+z = att,
r
27. x+!/ = m,
x + z =4)U,
y-^z = n.
x+y = cn,
z+^i = a,
1-x a
1 — u ~ b
u — x=:b.
•201
SIMULTAXEOUS EQUATIONS. •
28. Ux+9i/-\-z-u = a, 21J. x + ay + a^z + a^u-^a* = 0,
Uy + Vz+u-x = h, x-{-bi/-i-b"z-^L^u+ij'^ = 0,
11z+9u+x-y = c, a;+«/ + c22+6-3w + 6-4 = 0,
nu-{-9x+y-z^d. x+dy + d-zi-d^H + u^ = Q.
30. .x + y = a, 31. x + Iy = a,
v+z~b, i^^niz = b,
ii-\-i- = d, u^pi: = ,i^
v-\-x = e. v+qx = e.
82., xf//+3 = a, 33." x- !i-\-z = a,
y + Z + H=:b, y^z-\-H=b,
u + V'\-x = d, u—v-{-x=d,
v-\-x-\-y = e. v—x + y = e.
34. x+y+z~u = a, 35. , x+y+z-u — v = a,
y+z + u-r = b, 'f^' y^z + u-v-x = b,
z + u + v — x = c, z-\-u + v-x-y = e,
u + v+x — y = d, u + i- + x-y~ z = d
v+x + y-z = e. v + xj-y-z-u = e.
86. 2x-y-z+2io v = 3a, 37. v~ 2:c + SH-2y + z = a,
2y-z- H + 2v -x= ob, x—-2y^Sv -'■22 + 11= I,,
2z ~ u — c -\-2x- y = 3c, y-2z + 3x-2u+v = c,
2u-v- x + 2y — z = 3d. z- 2u + oy — 2v+x = d.
2v-x-y+2z - u «= 3e. u—2c+'dz - 2x+y = e.
Exercise Ixi.
Eesolve the following systems of equations :
J l+x + x^
" i+u + y'
1 + y+x^ _ x-+x-{-\ ,„,.c-l
i+*-+^^ ' y'-^y+^ \y-l
(\+x)a+y) _ l±a
(l-x)(l-y) " l-a 4.
x+1
ix-l\
i/+l -
X-+X+1
i/'-i-y+i
:* b2
x + y
1+xy ~
■SIMULTANEOUS EQUATIONS. 205
(l+a;)(l-//) _ 1+6 x-y _ b^ - g-
a;+?/ a ,, x-\- y 2a
1+a;// ~ 6+6-' 1—^ ~" l — a^
x—y b — c -^ ^—y 25
1— icy ~ a l + a;y ~ 1 - 6'
1 — a;// ~ a^ — a^' ^ + y l + -<^// "'-
a; - ?/ _ 2.''/3 1 - a;// x - // 2/^
l+'V/ ~ 62_^3" ;j;_^ "f" 1_ ^y ^ ,t
9. :yU+-^') _ 10 Z/ + z = 2rta;;v2,
a;(l + ?/2j ~ » a;+z = 2ia;?/^,
i/(l-a;3) x + y = 2cxyz.
= 0,
11
41 -r')
?/+5 — .1- z-{-x-y x+y-z 12. ax-hy = cz,
a ~ b ~ c ' I 1 i
= — 4- — 4- .
xyz = m^. ' X ' y z '
13. y- +z'^ -x{y + z] = a,
x^+z^-y(x + z)^b,
x^-ty^-z{x-\-y) = c.
14 2ax = {b-\-G — u)(y-^z),
^hy = {c + a — b){x-\-z),
{x+y-\-zy +x^+y3-^z^ = i{a2 +b^ +c^).
x^ +xy-\-y^ x^ -\- 11^ xii
f
15.
x—1 a — 1
1/-1 - 6-1'
a;3-l a3_ 1
j,3_i - 53_i-
17.
x^ \-x^y^+y^ = a,
x2+xy+y^'=b.
19.
xy + ^ - «(a;2 + j/2)
16.
x'^ — xy-\-y'^ a
IQ
'-• c^ + fr =
3 _
a
">
a;^y - r//2 =
x — y
' cl
20. x^=a{x'^-iry^)-b.nj.
206 SIMULTANEOUS EQUATIONS.
21. 4.c[x-^ + l) = {a + b){x-ii)2,
4.c{y^-l)^{a-b){x~y)\
22. x^-ti^ = -^ {x'^xy + y^'){x-^y),
23- r^ = a, 24. ~^^ = a,
ll-ry xy '
ic+2/2 - ^- xy~ = *•
25. % + 2) = a., 26. (x-+7/)(a;+0) = a,
«(a;+ //) = (% (z^x){z^y)=c.
"in. x{x+y+z) = a-yz, 28. a;^ - (3/- «)2 ^«^
^ V{x+y-rz) = b-,-:x, y2_^z-x)-=b,
z{x-}-y+z) = 6- - x^/. z'^~{x- y) 2 = c.
29. a;2+^2^,,2 1 1 2a
30. ^ + — = — .
X- y 22
a- + ?/==/^z, 1 1 26
^3 0,2
-»
ai^* ?/^ z3
x-y = cz. l^ 1^1
X y ~ c'
32. xy = ^--i,
x+y = I'z, . (a; - y)(3 + 1 ) - 2a,.
x-y = cz, {x^-y-'jiz+iy-i^iht.
EXAMINATION PAPERS. 207
CHAPTEE VII.
EXAITINATTON PaPERS : EdUC\TION DEPARTMENT AND UxiVSRSITT
OF Toronto.
L
1. State the rules for the addition and subtraction of Algebraic
riiiautities. Express in the simplest form.
(b+c — a)x+ {c-\-a. — b)y-{-{a-\-b - c)z
{c+a-h)x + {a + b-c)y-\-{b+c-a)z
(a-}-b — c)z-{-{b +c — a)y + {c+a — b)z
2. State and prove the Index Laws. Assuming these to be
general, interpret a;"™.
Find the products in the following cases :
(1) (x^ + 6x^y + 12xy^ +8y^){x= -6x'^y+12xy^ - 8yS).
(2) {a + b-\-c){b-\-c-a){c+a-b)(a + b-c).
3. Prove the rule of signs in Division.
Divide : [Apply Horner's Method to (1)]
(1) a;6_22x4+60a;3-55a;3 + 12a;+4 by x^-^Gx+l.
(2) a;4+9-l-81a;-4 by a;2-3-f9a;-2. (3) a;"' - 1 by a;* - 1,
4. Find the square roots of
4 ]
(1) 4a;4™-— a;^"* + — x"""
• o J
b^ c^ a^ c b a
5. Distinguish between an algebraic equation and an identity.
Solve
(1) 1^/(1 -2.1)4-1^(1 + 2a;) = 3.
808
EXAMINATION PAPERS.
(2) '-^2 _^ .r^ ^ ^.+,
x+1 ' x-2 ~ x-3
6. A person bought a certain number of oxen for $320. If he
had been able to i^urchase four more for the same sum, each
would have cost him $4 less. Find the number |of oxen. Ex-
plain the negative result.
7. (1) If ^ = ^ shew that «_!±M±!^' _ ^i^J^
b d c2 + 2c(/+3d2 - ^(c-3f/)*
(2) Find the value of a;6 -200a;^-fl98a;4+200a;3 -197;r2
-397a; when x=199.
S. Three towns, A, B, C, are at the angles of a triangle. T'rom
A to C, tlirough B, the distance is 82 miles ; from B to A, through
<; is 97 miles ; and from C to B, through A, is 89 miles. Find
the direct distances through the towns.
11.
1 . Prove x^ -=- a;" = x"'~".
Simplify {a+h + c)^ -S(a-{-h-\-c)'^c + ^a-\-h-]-c)c^ _fS.
2. Prove the rule for finding the L. C. M. of two quantities.
F«a theL. C. M. of
a^-\-h^ + c^-dabc, and (a + b)^ +2{a + b)c+c^.
^ a c ac
8. Prove -^ x —r = —;•
b d bd
Pimphly [^^^ + -^— ^,) . (^-^_^-^-, _ ^-^^.
4. Eoduce to their lowest terms — :r„7-; — ;;; — ir, and
(I- '-{-a. — 2
a(a^.2h)-^b(b + 2c) + c(c + 2a)
a2_^2_(.2_26^
5. (1.) li a^-pa^+ga- r = 0, then x^-pxl+qx~r is exactly
divisible by a; — o.
(2.) Prove that {n + b + c){bc-{-ca + ab)-{b+c){c+a){a^b)is
divisible by abc. Is there any other divisor ?
6. Ux =
EXAMINATION PAPERS. 209
la + 6\ 2^ , a - -b- /« + b\ "+"
7. Solve the equations —
^ ^^ l-"2x ~ 7^^2i ~ -^ ~ 7^1fe+lx--^*
(3 ) •^•+^ ^+1 _ 4x-l-9 12a; +17
a; + 4 ~ x+2 ~ 2uH^7 ~ Gx'-PTu'
8. A pei'sou going at tlie rate of ^j miles an hour, and desiring
fco reach home by a certain time, finds, when he has still r miles
to go, that, if he were continuing to travel at the same rate, he
would bo g Inufs too late. How much must he increase his
speed to reach home in time ?
9. Of the three digits comprising a number, the second is
double of the third ; the sum of the first and third is 9, and the
sum of the three digits is 17. Find the number,
10. A owes B $a due in months hence, and also .$6 due n
mouths hence. Fiud the equation which determines the time at
which both sums could be paid at once, reckoning interest at 5
per cent, per runum.
III.
1. If 3;= 10, ?/=lL 2=12, find the value of
\ x'^ — (i/-[-z)- X — , — ; and subtract
( V^ ' ' j x+)j + z'
(ij — 2)a2 + (z - x)ab + [x — j'/)b'^ from
{u-x)u^ — {y—z)ah~{z — x)b^.
2. Divide a + (a + 6).f + (a + i + c)a;2 ^^a+b+c)x^ + ib^c)z*
+ rx^by l+x-{-.i^ -\-x^ ; and find the square root of
9 - 2-l.iH-58.c2 - llC./;3 + 120^1 - UOr^ + lOf 'a;«.
c. r. ^ ■-, •^•c+5 a; + 5 2x-\-')
3. Solve (1; ^T + ^7~( = .> - TT"
eio
EXAMINATION PAPERS.
i. A boy bought a number of oranges at the rate of 45 cents a
dozen ; if he had received 20 oranges more for the same money
the whole would have cost him only 40 cents a dozen. How
many did he buy ?
6. A farmer took to market two loads of wheat, amounting to-
gether to 75 bushels ; he sold them at difierent prices per bushel,
but received on the whole the same amount for each load ; had he
Bold the whole quantity at the lower price he would Lave received
$78.75 ; bnt had he sold it at the higher price he would have re-
ceived $90. Find the number of bushels in each load.
6. Show how to find the square root of a -h y^b.
Find the square root of 1 + |/(1 — w^)
^^+^ 4x— 1 7a; -1-1
7. bolve ^ — "^ + X = ^ ; and find the value of i
^x — / X — S X — o
when ax^—^Qx + Sl =0, has equal roots.
a+b _ \/{ac)+ \/{bd)
*^a*a_i - -/(rtc)- V{bd)'
9. Show that a3(6 — c)+i3(c — «)+c3(a — i) is exactly divisible
by a-{-b + c ; and resolve the expression into its factors.
IV.
1. Multiply a^-f 5=- c2 + 2a6 by a^ -b'^ +c^ + 2ac, and divide
the product by a^ — 6^ - c- -f 26c.
2. Simplify
18a362 _ ^3ab{x-y) li(c-d) S(c^ -d^)\
■u
] ~1(^4 " \ 2T^3~ -^ a{x^ - j/^^l r
EXAMINATION I'ATl^RS. 211
3. Find theL.C.M. of 4x•--9//^ 4x2-10x'y + G»/e. and 6.r'-
13a;?/+6?/3, and the G.C.M. of l+x^+x{-x^ and 2x + 2x' +
Sx'^ + Sx^
4. Obtain the square root of i — fj/A^, and find the value of c
when 4:X^ — 12x^u+cx^l/^ — 12x7/^+4^* is a perfect square.
5. Distinguish between an equation and an identity. Give an
example of each. What value of m makes (a;— 3)2 —{^x — V){x—'6)
= m an identity ? Can any value of m make it an equation ?
6. Eeduce to its simplest form
l/(2 + :e)-T/(l+x) ^ l+y^ll -1 -f- (l+a-)\
1/(1+0,-) -v'x "^ l + i/'{l + l-(l+x-j[
7. Solve the equations
(2) 73^-5rr=:.;-5^)(.T + 3^),
2 ' S _ 7
X— 5p x + '^!/ ~ 33
8. A person performed a journey of 22. V miles, partly by car-
riage, at 10 miles an hour, and partly by train, at 36 miles an
hour, and the remainder by walking, at 4 miles an hour. He
did the whole in 1 hour 50 minutes. Had he walked the first
portion, and performed the last by carriage, it would have iaken
him 2 hours 30^ minutes. Find the respective distances by car-
riage, train and walking.
U^^, ^^^v
°
9. Solvo
•
a: + 3
x+1
4a--f9
12a;+17
x+4
ic + 2 '
" 2x-+7
6x+16
10. What value of y will make 2x*-^Sxy +Qy^ eracily divisible
by a:- 3?
If a and ?^ are lae roots of the equation x^-i-x + l = 0, show
that fl3_?,3^0.
212 BXAMINATION PAPERS.
V.
1. Multiply
Ax^'-^.x+\\ by 2ar + |.
Prove that
{h'' — '>j)^ — (x — ^y)^ is exactly divisible by x+y.
2; Express in vrords the meaning of the formula
{x + a){z + b)=x" + {a-\-h)xi ab.
Retaining the order of the terms, how will the right-hand
member of this expression be affected by changing, in the left-
hmd member (1) the sign of b only, (2) the sign of a only, (3)
the signs of both a and b ?
C. S:m\My {a + by-\-{a-h)^-2{a^-b^)» ; and show that
{a+b + c){b + c - a]{a-\-c - b){a-{-b - c) = iaH^
when a^+b^ = c».
a c (id
4. Prove that -, — =---■ = -r—
u a he
Simplify
a^-\-h^ \ I ah^ \ ^ 4«(a+6)
2a6 ^ I \a^ + b^j ' a^-ab+b»
5. I went fi-om Toronto to Niagara, 35 miles, in the steamer
" City of Toronto " and returned in the " Eothsay," making the
r:>und trip in 5 hours and 15 minutes; on another occasion I
went in the " Eothsay " (whose speed on this occasion was 1 mile
an hour less than usual), from Toronto to Lewiston, 42 miles, and
returned in the '• City of Toronto," making the round trip in G
hours and 30 minutes ; find the usual rates per hour which these
steamers make.
6. Solve
^ ' X y a X y a
(2) a;3 + 5a; = 5^/(^2+5x+28)-4.
7. Find three consecutive numbers whoso product is 48 times
the middle number.
EXAMINATION PAPERS.
£18
8. If m and n are the roots of ax^ +bx + c = Q, then
ax^ -}-hx + c = a{x — vi){x -n).
Show that if ax^ + bx-i-c = 0 has ecjual roots, one of ihem is
given by the equation
{2a^-2ah)x^ab-b"=0.
m n , x"^ y^
9. If — = — and -r- + -r-^ =1, prove mat
??)3 7;-- III- -\- 71^
VI.
1. SimpHfy
2. Divide a^—h^ — c^—Sahc by a — b — c, and show, without
expansion, that
(1 +^+a;2)3 _ (1 .-x-{-x^)^-Gx(x^ +a;2 + 1) -8a;3 =0,
3. Eesolve into factors x'^—^x-y^-i-i/^, and
7^2 - G;/3 - x;/ + 19x + 33// - 36 ; and prove that
b^{c-\-a)+c^{a-\-!j) — a^{b-\-c)+abc is exactly divisible by
fc + c — a.
4. Apply Horner's method of division to find the value of
5a:»+497a;* + 200a;3 + 19Ga;^ -218a; -2000 when a:=-99, and
the va ae oi Gx^ ^Sx"^ -llx^ -Qx^+lOx-2 when 2a;- = -3^; + !.
6. Find what
^\ -r-c)-]- — 1^ -J becomes when x = -.
V{a+x)- V{a-x) 1+6*
6. If a and b be any positive numbers, prove that
la a b
214 EXAMTNATION PAPERS.
7. Solve the equations —
(]) ,.* + 7/- = 5,
.1
« +2/
K'
(2) a;+27/+3z=i4,
2a; + 3?/ + 2 = ll.
3a;+?/+2z = ll.
(3) (x+l)(x + 3)(.T+4)(«+6)=.iG
8, There are three consecutive numbers such that the sum of
their cubes is equal to 16|- times the product of the two higher
numbers : find the numbers,
9, (1) Form an equation three of whose roots are 0, \/{ — 2),
and 1 — ^/2.
(2) If one of the roots of the equation a;-4-i''C-f-g = 0, is a
mean proportional between p and q, prove that
p^=q{l+py.
10, Two trains start at the same instant, the one from Dto A,
the other from Aio B; they meet in 1^ hours ; and the train for
A reaches its destination 52^- minutes before the other ti-ain
reaches B : compare the rates of the trains.
VJI,
1. Give some application of tlie '• rule of signs " in Algebraic
alultiplication and Division.
2. Find the numerical value of the quantity
hc(c — a){a — h) — cii[((, — b)(b — c) + ab{b - c)(c — a),
when a a 10, b = -01, c = 0; and prove that if
X = , then will (a-+o) . — ;
a+b a+b—c+x
EXAMINATION PAPERS. 215
8, Investigate a method of finding by inspection the remainder
after dividing any rational and integral function of x by x+a.
Show that the quantity
a^b^-ab^x-{a^+2b^)x^+ax^-{-2x'^
is divisible by each of the quantities x-\-a, x-\-b, a—2x, b-x.
4. Investigate the rule for finding the H.C.F. ef two algebraio
quantities, showing under what limitations factors may be intro-
duced or suppressed at any step.
Find the H.C.F. of
(1) 6.c4-7x-3_l3«3 + I9a;-G anda;3 + 2x3-l. '
(2) {x-\-y){ax^-bir)-xy{a-b){x-\-y), and
{x-y)[ax^-by^-)+x>j{a-b){x-y).
6. Prove, by general reasoning, that the value of a fraction is
not altered by multiplying or dividing both the numerator and
denominator by tlie same quantity.
13 7 X \
Simplify (1)
12(2a;-3) 12(2.f+3) 4:X^-\-%
^^^ \{x-\-a){x~b) + {x-a){x-^b)) '
1 1
+
Xx+a){x + b) {x — a){x — h)
6. Solve, with respect to x, the equations
a;-18 2a;-24 lla;-34 _ 1_
-> '^T' ■'" ^Tl^^ "^ 22 " 44"
5.^2 +x- 3 7a;2-3a;-9 _ a;-3
^^1 5a;_4 ~ 7.C-10 ~ 35ar3-78a:+40'
(3) X- = ax-^by, and y^ = bx-\- ay.
VIII.
1. Definfi the terms "power," "root,'* " index," and "coeffi-
cient ; explain also the reasoning by Avhich it is shown that
<t ~ (6 — c) = a — 6 + c.
219 EXAMINATION PAPERS.
2. Multiply (a;3+x?/ -1-2/2)2 ^y (x—y)'*.
■ Find the values of a and b which will make
x^+ax-\-b divisible 'byx-\-p, and also by xi-q.
a. Divide x^ +7/^+2x^,0^ by (.c-(-//)^ and
4. Investigate a rule for the extraction of the square root of any
algebraic quautit}', and deduce the rule for the extraction of the
square root of a number.
If to any square number be added the square of half the num-
ber immediately preceding it, the sum will be a eomplete square :
viz., the square of half the number immediately following it.
6. Find the square root of
(1) a2a;6 + 2a6x4 + (62-f2a,)„2-j-c3a;--+2/jc.
(2) ix^ - ix' -h ix^ + lx^ - lx'^ + ^\J.
6. If x^+ax-\-b and x^+a'x-b have a common measure, it
will be a;+ — k — , and the condition that they may have a com-
mon measure is Ab — a^—a'^.
Find the H. C. F. of x^ +p^ x^ -\-p* and x"^ +2px^-\-p^x^ -p^.
Find the L. C. M. of 2^{x^ + x-20), d^{x^ -x-BO), and
mx^-lOx + 2,4).
7. Find values of a and b which will render the fraction
3x^-{'ka + b)x-\-a-{-2b^
5x- -{Sa-fb)x-a + 4:h3
the same, for all values of x.
8. Solve the equation 12 - i/{x+])[x+(}) - i/{x-'i)[x-{-5} = 0.
and account for the circumstance, that the values of a;, determined
from it, apparently do not satisfy the equation,
BXAMINATION PAPHRS. 217
IX.
1. Prove thata(2?i-}-l)(a2+7i-M + l)-n(2</-f l)(7i2+«-a-i-l;
= (a — n)^.
2. If a, b, and s are positive quantities, and if a>fe and c>a — 6<
prove that
e — {a — b)=c — a + b.
Assuming this equation to hold good when a, b and c are unre-
stricted, prove that the expression-( -a), occurring in an algeb-
raic operation, is equivalent to +a.
3. li x^-\-ax^+h and x^+px+q have a common measure of
the form of x^ + ni,c-\-n, then a^bq-{b — q)^
4. Find the H. C. F. of
a--b--abxy + abx~^y~'^, and a-x^ -b^y-^ +a'bx^y-b^xij''i',
5. A and J5 are two numbers, each of two digits. The left-
hand digit of A exceeds that of B hy x; the excess of A above B
is y ; but the sum of the digits of B exceeds the sum of the digits
of Ahy z. Prove that y-\-z = 2x; and give an example of two
such numbers as A and B.
a b c
6. If -r- = — = -r, prove that each of these ratios
^a a+b-^-c
= „ r-, and also = ,— — :-,♦
7. Solve the equations
x + a x — a b->rx h — x
x — a x-\-a ~ h — x b+x
(2) a{x'-+y^)-h{x^-y^) = ^a
(fl3-63)(a;3-7/3) =4rti.
8- A farmer buys a sheep for $P and sells h of them at a gain
of 5 per cent. ; at what price ought he to sell the remainder to
gain 10 per cent, on the whole ?
9. The sum of three numbers is 70 ; and if the second is divided
by the first, the quotient is 2, and the remainder 1 ; but if the
third is divided by the second, the quotient is 3, and the remain-
der is 3 ; whg^t are tlie numbers.
(1)
218 EXAMINATION PAPEB3.
X.
1. Divide ax^ + 1cxijz-\-hij^ +ax- {y-\-z) + hij- {z-\-x) + 2cxy{x + y)
by x+y-\-z.
2. Prove that if x*^-{-p.v--^qx+a^ be divisible by x^ —1, it
is also divisible by x^ —a^.
3. Explain the reason for introducing or suppressing factors in
the process of finding the H.C.F. of two algebraical quantities.
Why is the name " Greatest Common Measure " objectionable ?
Find the H.C.F. oi x^-x^-x^ -x-2 and Qx^ -lx^+dx-2.
4. A traveller leaves A for B at the same time that another
leaves B for A ; the former walks at the rate of 3 miles an hour
till he has performed half the distance ; he then rests for an hour ;
after which he resumes his journey, walking now at the rate of 4
miles an hour ; the second traveller goes at the rate of 4 miles an
iiour till he has got over one-third of the distance between B and
A. ; he then rests for 40 minutes ; after which he resumes his
]aurney, walking now at the rate of 3 miles an hour. The tra-
vellers reach A and B respectively at the same time. Find the
distance between A and B.
6. Show by examining the square of a-\-b how the square root
of an algebraical quantity may be found.
Find the square roots of
(1) 25a;4_30rtx3-f49a3a;2-24a3x+16rt4, and
(2) ^+Kl _ (^ + JL)^2 + i.
2/2 a:- \y xj 2
6. Show that a = \/aJ^, when m and n are integers, and m is
divisible by.7i; and state the principle on which you would main-
tain the truth of the equation for all values of m and n. «
7. Solve the equations
^ ^ 5a: -4 ~ ' 7a;-- 10
(2) (y.c- 1)2 -f (-U- - 2)- = (ox - 3)3,
EXAMINATION PAPERS.
219
B. Two regular polygons are so related that the number of
their sides is as 2 to 3, and the magnitude of their angles as 3 to
4 ; find the figures.
XI.
1. State in words the several operations to be performed in
order to obtain the result expressed by the following algebraical
expression :
V
'ma^ +nb^
Also find its value when a = b = 4:.
2. Two men, A and B, dig a trench in 3f- days. If A were to
do more work by one-third than he does, and B more work by
one-half than he does, they would dig the trench in 2|f days. In
what time would each dig it alone, at his present rate of work ?
3. Perform the multiplications in
(1)
/ 2a;^-f 3/ j / 2x^-2/ I ( 4x^-\. 6xV+9z/' ) ( ^^^ " ^^V + %* 1
(2) ax^+\xy^pj^^){ix^-lxi/+iy^).
4. Divide
(1) a;4-f9+81a;-* by x^-3 + 9x-^.
(2) x^-{a-{-h+p)x^-{-(^ap-\-bp — c+q)x^-{aq-{-bq-C]>)x — qc hj
x^ —px+q.
5. Show that x~"'+^-x'^''~^ is always divisible by x±l, m and w
being any positive integers.
6. Define a fraction ; and from yonr definition prove a rule for
adding together two fractions with different denominators.
Add together the fractions,
a^ —be b^ — ca c^—ab
(a + b){a+c) (b + c){b+a) {c. + a){c + b)
220 EXAMINATION PAPERS.
7. Solve the following equations :
,^. 3;2^2x+2 x^ + 8x-{-20 x^+4tx + 6 x^ + Gx+12
x+1 ^ x+i a;+2 ^ a;-f3
X 100 , ^ o. ?/
(2) (x^+y^)-^ = ^^, (X3--/2) ^ 21
XII.
1. When OT and n are whole numbers, and m greater than n,
a™ 1
show that — - a'"-" and that -^ is correctly symbolized by a~" .
2. Multiply (a '-6)(a+^)(«-+^^)(«''+^'') • • • to (n + 1 ) factors.
3. Divide 1 — x by 1 — 2a;, to 5 terms, and write down the
<'7-+l)th term, and the remainder after (r+l) terms.
4. If the number three be divided into any two parts, show
that the difference of the squares is three times the difference of
the numbers.
5. Find the L. C. M. of l-8x+llx^+'^x^-24^xA, and
^ 1 -2a;- 18;<;2 +38.1-3 -24x4.
6. What relation must there be between the coefficients m, n,
p and q, in order that
(x3 4-wx+7i)2 -f pa;2 + gx
may be an exact square for all values of a; ?
7. Solve the following equations :
(1) (T+^ys + (l-x)2 " "•
ox — h^ ^/{ax) — b
(2) v{ax]-\-h = ^r~ -'•
C3^ — "- = 1, — ^- = 2, and — = 3.
8. Given x4-?/+2 = «a; = %, find (x+?/+z) -z.
9. Find a number expressed in the decimal notation by two
dio-its, whose sum is 10 ; and such, that if 1 be taken form its
double, the remainder will be expressed by the same digits in a
reversed order.
EXAMINATION PAPERS. 221
XIII.
1. Find the value, when a = 2 J, b-.3^, c- 4+ of
2. Show that the vahie of the expression, in the preceding
question, is not altered by changing a into a-^x, h into b-\-x, and
c inte c-\-x.
3. Multiply (1 -fa ia;)(l 4- rt 2^) (1+^3^) ••• (l+«„a;) to 3 terms.
4. A speculator borrows a sum of money at the yearly interest
of 7 per cent. ; part of the amount he Invests at 8^ per cent., and
the remainder at 9 ; and, at the end of the year, he finds that he
has made a profit of $75 ; but, had the former part been invested
at 9 per cent., and the latter at Qh, his profit at the end of the
year would have been only $65. Find the whole sum borrowed.
5. Given ax+hy = c, a'x+h'y = c', determine the value of
nix-\-ny, and find the conditions under which the value becomes
indeterminate.
(I I da 0„ 1
6. If-!- = ^= = -^^,
then will a, +^2 +^3+ . . . + a„-=-^
7. Eliminate x and y from the equations
S , -7. 5
X -^r y = a
« = x-k-^x^'y
a\-a^a^
«! — aj
1 a
5 ■^
P = ?/+3xSA
8. \iax- ■\-'bx-^-c = ^ -A d .<i»^ + 6ia;-fc, =0, then will
(«5j —a^b)(bci —b^c) = (aCj — a^c)^ .
9. Find that number of two figures to which if the number
formed by changing the places of the digits be added, the sum is
121 ; and if the same two numbers bo subtracted, the remainder
is 9.
222 EXAMINATION PAPERS.
XIV.
1. SimpKfy
a(i + c)3 + 6(r+rt)2 + c(a+&)2-{(a-?;)(a-c)(64-c) +
{b-c){b-a){c+a) + {c-a)(c-h){a-j-b)}.
2. State the law of Indices, and prove it for positive integral
indices ; and assuming it to be general, interpret the expressions
z~^, X , where m and n are positive integers.
3. Having given the equations,
prove that a^ [yz -y'z ') + ''^ [z-c- -z'x')+c'^{xy-x 'y ') = 0.
4. A traveller P sets out to walk from A to B, proceeding at
the rate ot 3 miles an hour ; and, 32 minutes afterwards, another
traveller Q sets out to walk from B to A, proceeding at a uniform
rate. They meet half way betwixt A and B. P then quickens
his pace by 1 raile an hour ; and Q slackens his 1 mile an hour.
Q reaches A at the same time that P reaches B. Find, the dis-
tance between A and B.
6. How are equations classified ?
Solve the equations —
(1) mnx+a'm.n = n^x-\-aiu^.
(2) x^-x^+y*-y^ = 84:,
x-+x^y^-\-y'^=49.
6. What two numbers are those whose difference, sum and
product are to each other as the three numbers 2, 3, 5 ?
XV.
1. What is the meaning of the symbols a, a-, a^ . , ?
Show a priori that a°=l ; how do you know that ab = ba ?
How is it proved that the multiplication of hke signs gives a
positive, and that of unlike signs, a negative result.
EXAMINATION PAPERS. 223
2. Find the value of
(^b~c)^+2{c-ay + {a-h)^-S{b-c){c-a){a~b)
when a = l, b= —^,
3. Simplify the following expression :
{ac-b^){ce-d2)-\.{ae-c2){bd-c^)-{ad-bc){be-ca)
4. P and Q are travelling along the same road in the same
direction. At noon P, who goes at the rate of m miles an hour,
is at a point A ', while Q who goes at the rate of n miles in the
hour, is at a point B, two miles ha advance of A. When are they
together ?
Has the answer a meaning, when m— n is negative ? Has it a
meaning when jn = )i7 If so, state what interpretation it must
receive in these cases.
5. Show how to find the Least Common Multiply of two or
more algebraic quantities,
(1) x2—ax~2a^, x^+ax^ andax^ — x^.
(2) x^ —x^y—a^x+a^y and x^+ax^ —xy^ —ay^.
In what algebraic operations is the Lowest Common Multiple
of two or more quantities required ?
6. State and prove the principle upon which the rules of Addi-
tion and Subtraction of fractions are founded.
Simplify the following expressions :
(a _!_& _ c)2 - ^3 (^b+c -a)^-d^ (c+a-by - d^
(1) (^^6)2_(c+d)2 + {b + c)''-W-fd)2 + (^+u)-^b+d)-'
J.2 ^y2 -z^ + 2xy a^-\-a^h a{a—b) 2ab
(2) ^2_\j2-z- + 2yz a^b - i>3 ~ {a+h)b ~ a^-b^'
7. liax-by-{-c{x-y) = {a-b){a + b-c},
by - cz+a{y-z) = {b- c){b + c-u),
cz--ax+b{z-x)=^{c-a){c+a — b)
then will a^{b-c)+b^{c-a) + c2{a-h) = 0.
8. P is a number, of two digits, x being the left hand digit, and
y the right. By inverting the digits, the number Q is obtained.
Prove that 11 {x + y){P-Q) = i) {x-y) {P+Q).
224 EXAJIINATION PAPERS.
XVI. '
1. Show that
{(ax + by)2 + (ay-bx)^}{{ax-{-hy)^-'{a!/+bxy} =
(a4-64)(:<;4_,^4) ; and that
2{a-b){a-c)+2{b-c){b-a)-i2{c-b)ic-a)
is the sum of three squares. '
2. It s = a + b-{-c-\-&c. to n terms, then
s — a s — b s — c,j, "
+ + + &C, r= n — 1.
s s s*
3. Show that a — b, b — c, and c~a cannot be all three positive'
or all three negative.
4. Extract the square root of
Ax^ + dx^ - V2x^ +16x- +9 -2x{Gx<^-8x* +9x^-12).
5. Given ab - ^{a+b){p + q)-{-2jq = 0,
cd-i{c+d){p-{-q)+2)q = 0,
find the value of p—q, and show that if either a or 6 is equal to r
or d, then p is equal to q, unless a+b = c + d.
or *
6. Find the value of — , having given
y
ic^" — ay^" x^ — Mx—yY
7. Prove that {a — b){b—c){c — a) is a common measure of the
quantities
c4(a-6) + a4(/v-f)+i4(c-a).
8. Find the conditions that a^x+h\y = Cy, cf^x+h^y = c^, aocl
aoX-\-b^y = c^ may be satisfied by the same values of x and y.
9. Two persons, A and B, start at the same instant from two
stations (c) miles apart, and proceed in the same du-ection along
the line joining tlie stations with velocities (a) and {b) miles per
tiour. Find the distance {x) from the stations where A over
bakes B, and interpret the result when a z b.
EXAMINATION PAPEKS.
225
XVII.
1. Ex^Dress in symbols the result of subtracting from unity the
quotient obtained by dividing the sum of a and b by their product.
2. Multiply together x + '^/a + b, x— Vn-\-b, x+^/a-h and
X- Va-h\ and divide 24a3-..22«2^, + 2fl2c_5a^,2^27a6c-34ac2
4-6^3 _22i2c-fl6&c2 + 8c3 by '6a-2b-\-4c. .
3. If x-\-a be the H. C. F. of x^+px+.q and x^+p'x+i',
their L. C. M. will be {x-\-a){x-\-p- a){x-\-p' - a).
Show that the difference between
X x X a b c
+ 1 + z — lana- — - +
x—a x — b x — c x — a x—b x — c
is the same whatever values be given to x.
4. Prove, if the four fractions ""
bx-\-cy-\- dz cx-\-dy + az dx+ay '+ bz ax-{-by ■\-cz
h^c + d — a e-\-d-\-a— b' rf-^rt-f-Z> — c' a-\-b+c ~d
are equal to one another, their common value will be equal to
x + y+2
— ^ — as long as a + h-\-e-\-d does not vanish.
5. What do you mean by solving an equation. Show that 3 is
a root of the equation
T/(a;3-3x+4) = -^^^
6. Eliminate x between the equations
a;3 4--^ + 3 Lc 4 ]=m, ani
a; * \ X I
a;3— -r — 3 a; = n.
x^ \ X I
1111
7. If — + -— _ — = — -—r , a, 6, c are not all different.
a b c a-\- 0 — c
8. A cask, A, contains m gallons of wine and. n gallons of water ;
an another cask, B, contains p gallons of wine and q gallons of
water, how many gallons must be drawn from each cask so as to
produce by theii- mixture h gallons of wine andc gallons of water ?
226 EXAMINATION PAPERS,
XVIII.
1. Multiply together the f.^rtors
1-x, 1+x, 1+x"-, l+a-*, arc! 1+x^.
and show that if n is any uneven number, the s im of the nth
powers of any two numbers is always divisible by the sum of the
numbers.
2. Find the numierical value of the expression
c \'n + \/c
b \/a— \/c
where a, h, c are connected by the equation a'b — c)^— c(h+c) '^ — 0.
3. A has a younger brother, B. The difference between their
ages is ^ of the sum of their ages. By adding twice B's age to
5 times A's, we obtain the age of the father; and by subtracting
twice B's age from 5 times A's, we obtain the age of the mother.
Show that the age of the mother is -^^ that of the father.
4. Find the II.C.F. of
x^-(2a+b)x^+a(2a+?>)x-a2(a + b), and
X* - {2b+a)x^ +b{2b+a)x-b2(b+a).
114
5. If — 4- = — . shew that
b c a
6. Show fully how the rule for finding the square root of a
«flven number is obtained. If w + 1 figures of the square root of
a number have been obtained, prove that the remaining n may be
obtained by division.
Extract the square root of
7. Find the value of the expression
t ^ when X = , y = —
l+xy «— « a
EXAinNATION PAPERS.
227
8. Solve the equations :
(1) ),{.v-'la) -l{x + Ba) + l{x-6a)=0.
(2) V(2x2+l)+v'(2.;3+3) = 2(l-a:).
9. Divide 21 into two parts, so tliat ten times one of them may
exceed nine times the other bv 1.
XIX.
1. Multiply together
X' + \^ax-a2 - |.«+|«-i.
Divide this product by
lx^+^ax-2a^-^x-{-2a-^;
and extract the square root of the quotient.
2. If x+y+z= 1 1 = 0, shew that
•^ X y z
(x'^ +y^ -\-z^)-^ [x^ +y^ -hz^) - xyz.
8. Find the H. C. D. of 20a;*+.r- -1 a.nd'ii>x^-i-lox»-3x-3;
also of (x+y)' -ic^ -y' and {x^-y^}'.
4. Given that ab - (a-^b)(x+y)-^4:xy = 0,
cd — {c+cl){x—y)->r4:xy = 0,
find the value of (j"-i/)^.
6. Having given
x^=y^-\-z- -2ayz
y^=z^-{-x^-2lzx
z^=x^+y^-2cxy,
x"- y"" _ 2^
Show that i-a^ ~ 1-b'^ ~ 1-^c^'
6- ir^+^{2x-\-x'^) - ^^•
7. Determine a; in terms of a and b in order that x^-\-2ax^ +
3523.3 _ 4^ 3a._j. 454 may be a perfect square.
8. A company of 90 persons consists of men, women, and
children ; the men are 4 in number more than the women, and
the children exceed the number of men and women by 10. How
many men, women, and children are there in the company.
228 EXAMINATION PAPEB3.
XX.
1. Divide (l-{-m)x^ — (in-{->i)xy{x — y) — (n—l)y^ hy
x^ -xy-\ry^.
•I. If x'^+px'^^+qx+r is exactly divisible hy x'^-\-mx-\-n^theti
nq — n^ = rm.
3. Prove that if m be a common measure of j? and q, it will algo
measure the difference of any multiples of ju and q.
•Find the G.G.M, oix^-px^ + {q-l)x--\-px-q and
x^ — qx^-{-[p — VjX^+qx —p.
4. Prove the rule for multipKcation of fractions.
simphfy -IjiS^yz^ X T-(?-^ ^ z^^ziix-zy): ,
(2/+2)'--^2 {^+^Y-y- (a:+^j2-23
, a a a^ 2a^~b^ — ab^
5,, What is the distinction between au identity and an equation /
li x — a=:.y + h, -piOYQ x — u = y-\-a.
Solve the equation
16a;- 13 40^-43 32^-30 20a; -24
4a;-3 ' 8x-9 8x'-7 ' 4a;-5
6. What are simultaneous equations ? Explain why there must
be given as many independent equations as there are unknown
quantities involved. If there is a greater number of equations
than unknown quantities, what is tlie inference ?
Eliminate a; and ?/ from the equations ux-tliy = c, a'x-\-b'y=.c^.
a"x + b"y = c".
7. Solve the equations —
(1) ^{n + x)+^{n-x) = m.
(2) 3a;4-Z/+z=13, 3//-f z-|--j; = 15. 32 + x+z/=17. '
8. A person has two kinds of foreign money ; it takes a pieces
of the first kind to make one £., and b pieces of the second kind :
he is offered one £ for c pieces, how many pieces of each kind
must he take ?
EXAIMINATION PAPEES. 229
9. A person starte to walk to a railway station four and a-half
miles off, intending to arrive at a certain time ; but after jvalking
& mile and a half he is detained twenty minutes, in consequence
of which he is obliged to walk a mile and a half an hour faster in
order to reach the station at, the appointed time. Find at what
pace he started.
10. (a) If — = — then will — ' — = — .
(b) Find by Horner's method of division the value of
a;«+290a;*+279a;3-2892a;2- 586a;- 312 when x= -289.
(c) Show without actual multiplication that
{a+b-\-c)^-(a + b+c){a^-ab+b"-bc-{-C'-ac)-3abc =
S{a-\-b)[b-\-c){c-ra).
Note. — In. Ex. 6, p. 87, after proving that a+h+c is a factor,
we may proceed as follows to discover the remaining quadratic
factor :
The quadratic factor must be of the form
m{a^ + b^+c^) + n{ab+bc+ca),
in which m and n are independent, being either zero, or a positive
or negative number. To determine them put <:- = 0, then the
given expression gives
{aS + b^-{-3ab{a+b)}^{a + b) = a2-^b^-+2ab,
but also = m{a^-\-b^)-\-nab. :. ?n = l andw = 2.
,*. a^ + h^ +e^ +S{a+p)(h + c){c + a)} ^ {a + h+c)»
a^+b^-i-c^ + '2{ab + bci-m) = {a-i-b + c)''(.
230 EXAMINATION PAPERS.
. XXI.
1. ?ind the value of a:3- I— 1 r] ^' + \~u ~)*+"p
when a = ^, b=^, x = 2. SimpHfy
2. Fiud, by symmetry, the sum of {a-{-b-\-G)^ —{a + b — c)^ -
(a-b-\-c)^-{b-a+cy, and of (a^-^r/^.t-f Sa^ajS - 2«x3 + 3a;4)2
and {a'>= +4:a3x+2a^j;- + 2ax^+Sx^)-.
3. Exphiin and illustrate the signs > , <
Prove: x^+y^'>2xi/, (x+y+z)- >3(.c//+//z+zaj), and
s J i 4 J ^
4. Determine the value of a; 4-2/ — ^ + 3a;' //"z , when a; +y — z —
0, &c. ; of a7+7f/x-^ + 8a;--3a3-(a;* + 7«a;3-8a;3-3a'-'), when
x= —1.
p Tnp
5. Show that (a"") «' = a7 .
SimpUfy I (-—J *| ' X (-^) ^ X t/(256), and divide
a; —Qax -f 5a x+la^x —2a by a; —2a a;+a .
0. If u = ^ ix-^—\ and v = jl y + ~) pi'ove that
7. Gold is 19j times as heavy as water, and silver 10^ times.
A mixed mass weighs 4,160 ounses, and displaces 250 ounces of
water. What proportion of gold and silver does the mass con-
tain ?
8. Shaw that l+px+qx^-i-rx^^is a perfect cube if p-=dq,
and (]' =3pr.
9. Solve the equations :
ix-2 lx+2
m ^.+-2 + ^'.-2 = *-
(3) — + --p^ =20 - , x+Q = ^y.
*
«XA.l\nNATTO^ PAPBR3. 231
10. A person buys two bales of clotb, eacb contaiaing 80 yards,
for $240. By selling the first at a gain of as maeh per cent, as
the second cost bim, and tbe second at a loss of as mucli per
cent., be makes a profit of $16 on tbe whole. Find tbe cost
price per yard of each bale.
SECOND CLASS TEACHERS, 1880.
XXII.
1. Find tbe value of x'^+x^ -IQGx'^ -lCn).c^ +81x + 81 when
x= —'•7 ; and the value of x^ — dpx- + {3p'^ +g)x — pq when
x=:a+p. (Arrange the latter result according to powers of a).
2. What is tbe condition that x-\-b shall be a factor of
ax^+hx-\-c?
Find the factors of
(a). (a^^-ah) + 2{l>^-ah)^3(a^~h2)-{.4:(a-h)^ ; and
(b). (ax+b){bx + c){cx-\-cb) — {ax ■}- c)(bx -{- a)[cx -j- b) .
3. What must be the relation among a, b, c, chat a.v-+bx-\-c
may be a perfect square ?
(a). Extract the square root of
(a-b)i-4:{a^+b^){a~b)2+4:(a4^+b*)-r8aH-\
(b). If 5 be subtracted fi.'om tbe sum of tbe squares of any four
consecutive numbers, the remainder wiU be a perfect .square.
(Prove this.)
a c c h In
4. If T~ = ~r = "I' a^f'l IT = — = — ■
b a t « in p
(a+c + e){h + L-^n) ah-\-cl-\-en
Pi-o^etbat |6 + J,-J:/-)(A;+iM+;;;) = bk+d^^'
ab{x^-y^)+xy{a^-b^)
5. (a). Reduce ^^^.^^^2^^_ry(^a2+b^) *° ^*^ ^°^®^* *^^°^^-
(6). II xi/-\-yz + zx = l prove that
X y z 4,xyz
+ 1-^ +
l_a;2 -r i_y2 -r i_^i - ^i^^2>^i^i_yi)^i_z^)
232 EXAMINATION PAPKB8.
6. Prove that
2{x-+2+4/(x2-4)} .
(h) {b+c-a)a'' + {c + a-b)b^ + {a-^h~.c)c a
a A -1- -^ ■■< a
{a + h-4-c){a'' + //' + c')-2(a' + 6' +c').,
7. Solve the equations —
(a), (b - c){x - a)^ + {c - a)(x-b)3 +{a~b){x ~c)^ = 0.
(b). x + ij = 4:Xi/; )j-\-z = 1yz; z+x-'dzx.
(r). x+y+z = 0.
ax-\-bii-{-cz = 0.
bcx + cay-{-abz-\-(a — b)(b—c){c — a)=^(j.
x — 1- »; — 3
(^) ,1^ + . + 1 + 2 = 0.
FIRST-CLASS TEACHERS, 1876.
XXIII.
1. Investigate Hornei-'s method of division.
Divide x^ ~dx^ -Six' +'2rjx<' +dx^ -8x^ -\-19x^ -{-Sx-^IO by
3.** — 21a;3-f 9a;— 6, showing the " final remainder."
Find the value of 2x^-^803x^ -3d^x^ + 1605x^- -1204x+i22,
when x= —402.
2. Ii/{x), a rational and integral function of x is divided by
X' +px + q, the remamder is a'—B '
where a, (3 are the roots of x^ +px+q = 0.
Examine the case where p- =4:q.
3. Show without actual expansion that
a^{b-c) + b^{c-a) + c^{a-b)
(a3 _ /;2 )3_|_(i2 _ c2)3 + (c2_a2)3
(a-i)3+(6-c)3 + (c-a)3
SXAMINATION PAPERS. 233
4. rind the value of x aud y that will render the fraction
o 9 , , n 5^;^ ^TT the same ior all values ot z.
^« + (2/ ~ '^)'^ + ^^'(i/ "■ ^^)
5. Show how to find the suro of n terms of a series in Geo-
metric progression.
(1) Show that the sum of n terms of the series
l + r + (l4-2r)(H-r) + (l+3r)(i+r)2 4- . • ., isn (l+r)".
11 1
(2) Sum to infinity the series o.^.p + 4^./-.o + p Q.in+ • • • •
6. Explain the notation of functions : prove that if
f (,„) = i + „,a;+ ^^^.7 •^' +<^c-» then/ (w) x/ (?i) =/(w.-}-70.
Show that in the expansion of (1 -[-«)" the sum of the squares
1-2-3 .... %i
ot the co-efificients = ._, ^^ .-, ,-^«
(l-2*3 • • • • ny
7. Solve the equations —
,,N x - « x— i x — c
^ ' h~-^c "•" l^c + a-f-A "" ^•
(2) a;4-10.6-3 + 35x2 -50.^ + 24 = 0.
1 1
(^) 21x2 - 13x+2 + 28x8 - 16x4-2 "■^^•^^"'^■''■^^'
8. Give a brief account of mathematieal induction, and show
that a square of a multinomial is equal to the square of each term
together with twice the product of each term into the sum of all
that follow it.
Find the sum of the products of the first n natural numbers
taken two and two together ?
9. If — - = ?/ + «, ^= 2 + X, — - = X 4- y. pi'ove
(1)
1 1 1+a 1+i 1+c
a 1) G \ — ah\ — 6c 1 — ca
^^ a{l-bc)~b{l-ca)~c{l-ab)
VT^^ Vl-ca . V\-ab ^1—bc Vl-ca VV^
234 EXAMINATION PAPERS.
10. AB is divided in C, so that AB, BC = AC^ ; from CA is
cut off a part CD equal to CB ; from DC is cut off a part DE
equal to DA ; from £1) is cut off a part equal to UC, and so on
cd inf. Sbow that the points of section continually approach a
point C such that AC' = BG.
14. Eliminate x, y, z and n from the equations
a^x+hiy + c^z+d^u = 0.
a2X-\-h2y + c^z-\-d.2U—0.
a ^x+ h ^y -{-c ^z + d ^n = 0.
a^x-^-b^y+c^z + d^K = 0.
12. A rail^vay train travels from Toronto to Colling-wood. At
Newmarket it stops 7 minutes for water, and two minutes after
leaving the latter place it meets a special express that left Colhng-
wood when the former was 28 miles on the other side of New-
market ; the express travels at double the rate of the other, and
runs the distance from Collingwood to Newmarket in ,1^ hour;
and if on reaching Toronto it returned at once to Colhngwood,
it would arrive there three minutes after the first train ; find the
distance between Toronto, Newmarket and CoUingwood.
FIEST CLASS TEACPTEES, 1877.
XXIV.
x-{y--z)+y''{z—x)-j-z^{x-y)
x^y^ -\-x^y^+x'^z- -j-xH^-\-y'^rJ+y''z^ + 2x^y^z^
ax+m-{-l nx-\-n ax + m ax-hn+1
2. Solve (1.) ^^^^—i + ^;^qr7i32 = ^ + ^ - 2 + ax^n:^! '
(2.) iri:^7^+rr:^y^=2.
3. A, B, and C start from the same place ; B, after a quarter
of an hour, doubles his rate, and C, after walking 10 minutes,
diminishes his rate one-sixth ; at the end of half an hour, ^ is a
quarter of a mile before B, and half a mile before C, and it is
EXAMINATION PAPEKS. 235
observed that the total distance walked by the three, had tLey
continued to walk uniformly from the first, is 6J miles. Find
the original rate of each.
4. {1) investigate the relations that must exist between the
constants in order that Ax'^-\-Iiij- +Cz- ■^-aijz + bxz + cxy shall be
a perfect square.
(2) Find the conditions that the values of x and y derived from
the equations ax+hy= — -4- — = c^ maybe rational.
X y
5. 11 x~+px+g and x^-\-mx+n have a common factor, then
will {n — q)--\-ii{m—p)- =in{m—p){it-q).
6. Prove («"*)" = «"*", whether m and n be positive or negative,
integral or fractional.
Show that (x--'"+ar")»» =a;"» ^ " x (af-" + -*;"-'")«»
7. (l.)Ifx = 4-then V^^5?! = {-^X^
1
of these fractions = — (a"4-i"+c"+t^).
8. If a; be very small, show that —
, — =2 - 4..!-, very nearly.
2+ox-(l+4.«)
^2(^2 _ 12) 7l2(,j2_12)(-^2_23)
9. Prove that l — n^ + "Ts " 02 12 — 2^ — P — + -. = 0
10. If a debt $a at compound interest be discharged in n years by
. a
annual payment of $ — , show tliat (l+rj"(l — m/-) = 1, where r
is the interest on $1 for a year.
236 EXAMINATION PAJfUAa,
11. Solve— (1.) 8a;2-2a:2/-55.
x--5xu + 8y^ = 7.
5 5
3 a p + ,j 1
i, 5
(8) a^b'^x" -4:a b'xm ={a-by-x
V
FIEST CLASS TEACHERS, 1878.
XXV.
2 ..3
1. SimpHfj W'^-±f_V-4-)'- (V^-V^r-7-"---,
■ a; a + a' \ « x ■ a{a-x)
x^-ii/-zr- y^-Jz-x]^- zl-J^-][Y
^""^ {x-\-z)^-y^- ^ '{x+y)--z' + (y + ^P-^c^
X b b^ b h'i
2. Divide — — 1 — — —-7,+ — +— , bvic— a;
a a a" X x^ "^
shew that (-9a2)* = i{ ^(6rO-t-A/(-6rt)}.
m n r x^ v^ z-
8. If — = — = — and-5r+7-^+^7 = l, prove that
X y z a^ b^ ' c' ^
4. Fiud the relations between the roots and co. efficients of the
equation ax^+bx-\-c = 0.
If m and n are the roots of the equation ox^ -bbx-\-':--0, show
that the roots of the equation acx'--i-{2ao— b'-^)x+ac = 0 are
ni n
— and — •
■II m
6. Solve the equations :
(1) a-24-2l/a2_2a;=2x+8.
,^, x^ y^ X y
(2) — -— = 10|, — - — = 4.
y X *' y X ^
(3) X2 = i/S x+y+z=12, x^-^y"'+Z'=[n.
EXAMINATION PAPERS. 287
6. Two men start at the same time to meet each other from
towns which are 28 miles apart ; one takes five minutes longer
than the other to walk a mile, and they meet in four hours. Find
each man's rate per hour. »
7. If P, Q, R be respectively the pth, ^th, rth terms of a G.P.,
shew that
12 3
Sum to infinity the series — + 773 4-":^+ ^^'
JC tL- Jb
8. Find the amount of $f ufc compound interest for n years, r
being the interest on $1 for one year.
Supposing %2^ to be withdrawn at the end of each year, what
will be the amount at the end of n years ?
9. Determine the number of combinations of n things taken r
together.
The number of combinations of n things taken two togethej;
exceeds by 6 the number of combinations of n — 1 things taken
two together : find n.
30. (1) Find the limit of (l-f^)"^ when x increases without
''imic.
(2) Find the (7*+l)th term in the expansion of (3 — 6a;)
x^ - 3a; - 8
11. Determine the limits between which lies o .3 lo^ii for all
possible values of x.
FIEST CLASS TEACHEES, 1879.
XXVI.
7. Prove that ^{{0 -hy ■{■(h-c)-' + {c- ay] =l{a-b){h-c)
{c-a){{a-bY->r{b-cy + {c-ay).
2. Extract the square root of ^<6—2a-j/(r/& — a8), and find the
simplest real forms of the expression
,/(3 + iA/-l) + v/{3-V 1).
238 EXAMINATION PAPF.K».
8. Solve the equations :
(1). 2a;4-f^3_ii^s_l_^4-2^-0.
(2). u;2+.y2+2-^=«3
yz-\-zx -^x^ = h'^
z-f- y— z—c.
(3). y (»3 + 5x+4.) + -^/(x- -f-Sx- - 4) = x+i.
4. Prove that the number of positive integral solutions of the
c
equation ax+hi/ = c cannot exceed — r + 1.
In how many ways may £11 15s. be paid in half-guineas and
half-crowns ?
5. If xy = (il>{a + b), and x'^ —n-y + i/^ =:a^-\-b^, shew that
!^ j[\ I ^ V
\ a o I \ o a j
6. Given the sum of an arithmetical series, the first-term: and
the common difference, shew how to find the number of terms.
Explain the negative result. Ex. How many terms of the series
6, 10, 14, &c., amount to 96 ?
7. Find the relation between p and q, when x^+px+x = ('* has
two equal roots, and determine the values of ni which will mane
u^ +max-{-a^ a factor of »■* —ax^ +a"x^ — a^x-\-a^.
8. In the scale of relation in which the radix is r. «hew that
the sum of the digits divided by r — 1 gives' the same remainder
as the number itself divided by ?•— 1.
9. Assuming the Binomial Theorem for a positive integral
index, prove it in the case of the index being a positive fraction.
«
Shew that the sum of the squares of the co-efficients in the ex-
pansion of (l-fx)" is [2n-j-d w )2^ n being a positive integer.
10. Sum the following series : —
' (1.) l + 3x-\-5x^ +lx^ +&C, to n terms.
(2.) o — o+Q — i'Q+ &c. to n terms, anil to infinity.
11. Shew that
EXAMINATION PAPESS. 239
be, — nc, — ah
b^^c^, a^ + 2ac,-^r^ -2nb is divisible hy
c2, 0^, (a+i)3
FIRST CLASS TEACHERS, 1380-~Grade C.
XXVII.
1. Kin ax^+2bxy+ci/^, hi+lv be substituted for a; and W7?(+n^;
for y, the result takes the form Au^ + IBnv + Gv'^. Find the value
of (£2 — ^Q'^.^(j)2 _ac) in terms of k, I, m, n.
2. Eesolve a{b—c)^-\-b(c — a)^+c{ri-b)^ into factors.
Prove that = ^
«i;i/; xi/z
i{ It = x{-By^ — Cz^) , v=-y(Cz^—Ax^), w = z{Ax^ — By^).
5. Extract the square root of
(a-i)2(6-c)2 4-{&-c)2(c-rt)2-f (c-«)8(fl_/,)3,
and tiy, cube root of
4. Eliminate a;, y, e from
' -^ ' X y z
k{x"-i-y- +z^)+2[Ix-\-my+7iz) + /i = 0.
^' Shnphfy ''|^1;|;^'^^ {l/(4 + 3i)+i/(4-3i)}2,
/-1+.?V3\^ -14-JV3, ^
and (^ 2 j + 2 +^'
in which j= \/{ — l).
6. Given the first term, the common difference and the number
of terms of an arithmetical progression, find (i.) the sum of the
terms, (ii.) the sum of the squares of the terms.
240 «XA5fD:s,TI0N PAPEES.
7. Solve tbe equations
ah
(11.) aa;+%=— + — =1.
—1 -1 —1
(iii.) oc{y-\-z )=a, y(z + x ) = h, z(.r + y ) = ^.
8. What value (other than 1) must be given to q that one of
the roots oi x^ —2x+q = 0, may be the square of the other.
If a, b, c are the roots oi x^ — fx^ -\-qx — r, express
'iah + 26c+ 2ca -a^-h^ -c"^
in terms of/-", q and r.
9. A vessel makes two runs on a measured mile, one with the
tide in m minutes and one a.gainst the tide in n minu^es. Find
the sjjeed of the vessel through the water, and the rate the tide
was running at, assuming both to be uniform.
10. Five points. A, B, C, 0 and P lie on a straight line. The
distances of ^, B, and C, measured from the point 0, are a, h,
and c ; their distances measured from the point P are x, y, z.
Prove that whatever be the positions of the points 0 and F,
«* {b - c) +2/ -' {c-a)-^i^{a-b)-^{b-c)l^c-u) {a -b) = 0.
APPENDIX.
Section I. — Elementaey Theorems on Polynoaies.
(See page 39, et seq.)
Theorem I- If the polynome/ (ic)" be divided by x — ti, the
remainder will be /(«)".
D'Alembert's Proof. J\xY is the dividend, a; -a is the divisor :
let /i (a;) "~^ be the quotient, which is necessarily a polynome o*
degree n—1, and let R be the remainder. Then, since the pro-
duct of the quotient and the divisor added to the remainder re-
produces the dividend,
But R does not contain x, hence it will remain the same, not
merely in form but in actual value, whatever value be given to x..
Take the case x = a, then («— «)/i (a?)"~^ vanishes for its factor x—a
does so. heucB R=:f{ay\ Thus the remainder is the value of the
dividend when x has the value which makes the divisor vanish.
It has been objected to the above proof " Division can be per-
formed only when there is an actual divisor, therefore in assum-
ing R to be the remainder ol f{xY -i-{x — a) it is assumed that a- is
not equal to a, and although R will remain unchanged for all
values of x that fulfil this assumption, it cannot thence be inferred
that it will do so if the contradictory assumption be made. In
such case the only legitimate conclusion is that there being no
divisor there is neither quotient nor remainder. Therefore,
although /(a)" may be the remainder in the case in which x is
not equal to a, yet the above argument does not prove it." This
objection confuses arithmetical or numerical division- with alge-
braic or formal division, division by a definite quantity with divi-
sion by an undetermined or variable quantity. The following
proof does not involve the assumption x-a, and consequently is
not open to the foregoing objection.
242 APPENDIX.
Lagrange s Proof. Lemma, af* - a" is divis\ble by .* -i, if «
be a positive integer.
By actual division ——zr~'^ -_^---- - ,
.-. «" -a» is divisible by x.—a if a;"-^-«»-i is so divisible,
hence u;'-^ - a"^^ " " x-a " ^''-^-^"-^ ....
Thus we can reduce the exponent unit by unit until at last we
arrive at, x^ — a"^ is divisible by x-a ii x—a is so divisible. But
x~a is certainly divisible by Itself, .-. x^-a^ is divisible by ic - <*,
.-. x^-a^ is also divisible by a; -a, .". so also is x^—a'*' and thus
we may go on to any positive Integral exponent whatsoever.
Theorem. Writing /(re)" m polynomial form arraugpa in
ascending powers of x,
f{xy=A^+A^x+A^x^+A^x^-h +A,a^,
.-. /(a;)"-/(«r=^i(a;-«) + /l2r.-/;3-a2)_^.^g(^3_a3)+ ....
+.l„(a;"-a").
But every term of this polynomial is divisible by a; — a, and the
highest power of x in the quotient is a;"~^ got from the term
A^{j:" —a"), so the quotient may be represented by/i(a;)"~S
fix)" f(a)
x — a -' 1 ^ ' 'x — a
Theorem II. If the polynome /(^)'' vanish on substituting
for x each of the n different values a-^, a^, a^, . . . . «.„,
i\\Qnf{xY =A{x-a^){x — a^) {x-a„),
in which A is independent of x and consequently is the coefficient
of.7f in/(.t;)".
Since /(ai)=0, ■• fixY ={:x-a^)f (a;)"-^ In this substitute
a, ior X, :. ^ineef{a^Y =0, it becomtrf 0 = («., -rtj)/i(a2)"~^- Of
this product the factor a^ —a^ does not vanish since by hypothe-
sis a^ is not equal to a^, therefore the other factor fiia^Y"^ must
vanish that the product may vanish, and consequently/j(a;)"~^ is
APPENDIX. 243
divisible Ly x—a.,. Let the quotient I)e denoted by /2(.'»)''~^, ..
f{xY=(x — a^){:x~a,^f^{xY~". Substitute tig for ./• and proceed
as before, and it will be proved that x—a^ is a factor of /(jc)" .
Continuing to n factors we get a quotient independent of a;, since
each division reduces the exponent of a; by unity, .-. finally
fixY = Ai;x — a^){x-a^^ (./j - a" ).
Cor. \ij\xy- and <p(x)"' both vanish for the same r> different
values <Ax,j\xY is algebraically divisible by (^{xY^.
»
•Let «j, «2> ^3' «m 1^6 t^6 "'■ different values of x for
which the poiynome.s vanish,
.'.J{xf ={x-a^)[x-a^) {x-a^F{xY-^-
and ^(./')"' = .i(»; — «i («—<<„ ) (.c — a„)
.-. 1\xf -=-(Z)(.r)"' = F(a;)"-'"-4-J ,
w.'iich is an integral function of a siuce A does not contaiu x.
Theorem III. If the polynome/(a;)" vanish for more than
n different values of a; it will vanish identically, the coefficient of
every term being zero.
Let (7 J, Oa,*a^ a„, «„+i be v+1 different values of a;
for which /(a;)" vanishes,
.-. f{xY =A(x-a^)(x-a2)(x-a^) (x-a„)
Substitute (1,1^1 iov X, and since /(a„+i)" =0,
.-. 0= ^l(«„+i - rt J )(«„+! - «._,)(rt„4.i - u^) .... (a„,+i - n„ ,
But none of the factors On+i — a^, a„+i — a^, &,e.. vanishes,
,'. A must be zero, or
f(xY =0{x-''ii)(x — a^)(x-a^) (x — a„)
and the factor, zero, will be a factor in the coefficients of every
term.
Theorem IV. If the polynomes f{xY , (p(.r)"' (n not less than
m) are equal for more than n different values of x, they are equal
for all values, and the coefficients of equal powers of x in each
are equal to one another.
244
APPENDIX.
f(xY'^-A.,-i-A,x+A.,x^ + A^x^+ .... +A„x''
(p{x)^ =B^-^B^z + B2x'' + B^x^ + . . . . +5^ic^
.'.f{xr-<i>(x)"^ = A,-B, + {Ai-B,)x + {A^-B^)x^-{-
{A^-B^x^^+ .... +(J„-i?>-
+ J,«+ia:"^+i-f.^^+2a;"'+2 -]-A„x\
aud this is a polynome of degree n at most. i>nt/(a;)" =(p(a;)"' foi
more than n different values of x, that is /(a;)" — ^(a;)'" vanishes
for these vahies, .-. hy Theorem III. /(a;)" - (p{x)"' vanishes identi
cally, aud*the coefficients yl„ — 5o, ^ J— 5j, A^—B^,
A„- B^, A„i^'i, Am+i, ^„ are all equal to zero,
.*. Aq ~ Bq, a j =B^, A^ = lJ2,.--Aj„= B^, Afn^i =0, Afn4-2 — ^ ■■■
Note to Art. XVII. To find, where such exist, the factors of
ax~ + bx;/ + Cxz+ey^ +fnjz+hz^.
Multiply hy 4a
4:a^x^ -^iaJ>x>j + 4:'^icxz+4:riey^ -^4:ar/yz-}-4:nhz^ .
Select the terms containing x and complete the square, thus
Aa^x^ +4:aJ)xy-i-4:acxz + b^ij^ +2bcxz + C^Z'
■^(b2-4:ae)tj2-2{bc-2arj)yz-{c^-4.fth)2^ =
(2ax+b]/ + Cz)^ - {(62 -4:ae)y^ + 2(bc-2ag)i/z + {c^-4:ah)z^}
If the part within the double bracket is a square say {my + nz)^
the given expression can be written
{2ax+bii + Cz)^ - (mf/+nz)'
which can be factored by [4J . Factor and divide the result by
4a. If the part within the double bracket is not a square, the
given expression cannot be factored. If b and c are both even,
multiply by a instead of by 4a and the square can be completed
without introducing fractions. If e is less thau a it will be easier
to multiply by 4e instead of by 4a and select the terms containing
y. A similar remark applies to h.
This method can evidently be extended to quadratic multino-
mials of any number of terms.
appendix. 245
Examples.
1. Eesolve x- ■{-x)/ + 2xz-2i/-+7ijz-iiz'^ iuto factors.
Multiply by 4
4x^ + 4:X>j+8>jz-S!/^-{-^S>jz-12z^ -
Complete the square selectiug i-nns m x,
4a;2 + 4<:.y+8x-2+.'/2+4?/s+4z3-9^2+24j/z-l(;z3«
(2x- + 7/ + 2z)"-(32/-42)2 =
{(2x+v/ + 22) + {3?/-4^)}{(2a;+7/+2z)-(3^-4r)}=-
(2x-l-4?/-2z)(2x--2^+6z)=.4(^4-2^-2)(a;-^+3.^)
. • . the f iictors are (u; + 2// — s) (a; — // + 3z).
2. 6rt2 -7rt/;+2^tc-2062 + 64Z;c-486-2.
Multiplygby 4 x 6 = 24
14ia2-168a6 + 48^c-480/^2+1586&c-1152c5=s
(12rt-7i+2'-)^- 529^2 _^1504/>c-1156c2 =
(12„— 7i+2f)2-(236-34f)2=
(12rt+16?^-32r)(12a-306 + 36r)=B
24(3«+4&-8c)(2«.-5/^ + 6c),
.'. the factors, are 3fl+4& — 8c and 2a— 5/> + Gc.
3= a;2_|.i2.r;/+2x2+26^2_8^2_923 =
(a;2 + l2x//+2x-2 + 3Gy2 4.12//Z +2-') _ 10^2 _ 20^2-102'* =
(x+6y + z)--{0/+2)v/10}2:=
{:r+(6+ v/10)^+(yiO+l)z} <
^a: + (6-A/10)y-(i/10-lM
4. 3a2+10r^6-14«p+12«rf-8/>2_8W+8';2_8(;f/.
Multiply by 3, not 4x3, since the coefficients of the other terms
in a, are all even,
9rt2 -i- 30ai - 42«c+36a6i - 246^* - 24^*d + 24c 2 - 24efi.
246
APPENDIX.
Select tho terras containing a and com^Jotc the square
706c - MIkI - 25c ' + GOcd - 3Gd^ -
(3a + 5h-7c-{-Gd}^ - {Ib-iic + Gd)^ =
{3a + 12b-12c + lM){'da~2b~2c) =
ii(a-^U-4:C + 4d){3a-2b-2c),
,•- the factors are a + ib —4c-{-id and da — 2b — 'Ac.
Work lixercise XXIX by this method.
m p m _
_ P
q
m
. b~n
nt m
{a -f h)Ti = an -
m
-ITn
m p mp
(a'n )'q = am
Section II. — Indices and SuPvDS.
The general Index-laws are
in p m i_ p
an , a g = a n V (1)
(3)
(^)
{-)
Tho law connecting the Index and the Sura byiubjls is
m
oJ = ^"/(a™) (G)
[The indices 1-, i, i, &c., are generally used to denote ' either
square-root,' ' any of the cube-roots,' ' any one of the fourth-
roots,' &c.
The surd symbols -j/, -^, V, &c., are by some writers re-
stricted to indicate the arithmetical or absolute roots, sometimes
called the positive roots. Thus
^4 = 2, but4*=±:2, .-. 4^=±y4
Also, V{{-2f-}= i/4 = 2.
lf27 = 3, b'lt 27^= 3 or 3/ ^ j /. S ' =(]/')-3/^':
J.
i ,X
4/16 = 2,but IG =+2or ±2j, .-. 16" = {r)t/16
APPENDIX. 24V
With this restriction the geueral connecting formula -wonltl be
a7 = (111) ;'(«"•)
In the following esercises this restriction need not be observer,]
EXEKCISE.
1. What is the arithmetical value of each of the following :
8G*, 27^", IG*, Q2\ 4^ 8', 27*, G4^, ?.2', 64*, 81*, (3f)^
i I h I -5 -2 -75
(5:Vr) , (IfV) , (-20)-, (-027)% 49 , 32 , 81
2. Interpret ar'^, «», a^' , («2) ^, aS ", r^, («"')-% a^ a"*.
3. What is the arithmetical value of
36~', 27~*, (•1G)~^ (-OOIC)^, (1)~^ {4^)~^', {^%)~\ {5^\)~^
1 -L
4. Prove {iry = (a")" ; [cC^f = (a" )"^ ; a"^ = (a-^)"* ;
and express these theorems in words.
6. Simplify a J, c^ x^ ,ni .vi'^n .n~^^\ (7i)*-(2^)*-(Bl)*
a c a e e 5 1 \h
1' T ^' "T"' "^' (2fr .(G^r-^(i)
a c a ex
6. Eemove the brackets from
(ae)^ (Z;)"^, (cJ)~^ ('^V', («~^A (.r^)~^
{an--)\ {al}y, («.^c-i)~*, (a--'^/')~^ (a;'^/"')""^
7. Eemove the brackets and simplify
1 .13- 1 1 2 1 1 i / 1 I \ 2 / 1 1^2 / 1 1 \«
(.« ) (x* ) [X ) ', X X X ;
r A' ; a? ;r :
{x''~' -^x-^~' ] {x''--''-^x-- ^}.
248 APPENDIX.
8. Sim^\i{j-x{x~^{-x)-^}^, x{{-x)~^{-x)-'-}~^,
{ — x) {x~^x }
9. Determine the commensurable and the sua.-cl factors of
12% 24^ 18"^, (-81)*, 12^ 64^ {^^J, {Gif^.
(The surd factor must be the incommensurable root- of an
integer.)
10. Simplify 8*-fl8*-50^72V(T¥;5/-(To5)'"^i
{(6 + 2")(6-2^)}S (2V3V+(^'-3S';
(2 +3^)(r+9^-6^); (7 -3^)-(7" + 3-)-.
[{{a-^x)[x + h)y'- {{a-x){x-h)}^] 3 ; '
{a'+(a3 a;3)^[* . {J -{a'-^ -x^)^}^
Express as surds,
i.1. a , x , jj , c , h
n + ^ _^ + § .25 -« + ,^
12. a; , ^/ , a , b
a m—H n~^
18. (ax-bf,{x^-4:x+l) 4 , (p -7:^) *•
Express with indices,
14. if a-, Vc^ "/a;"', if/r-", Vf^/.-K), 7/1-*
15. -^(a^+iB)^ ^(^34.^3)2, {ir(«3+63)p, .-y|^^_;).^>,
;/(a-ia;)"-\ V (a" -Z>" )'"-'"
16. (rt ) , (6 ) , (c ) , (x- ) , (a^x) , {a ^x ) ,
1 — i 14
(a? v/ ') .
Simplify the following, expressing the results by borb v^^-Vk-
tions,
APPEND1.V. 240
1 X ■ t 2 1
17. ii-(i' ",a^.a ', a .a , a.a , a '-^(1,0,' ■^"'n^ , a ^a
a h c . a b c
•^ "^ *^ a; y oc- c[ab)
X i 3 3 -^n Sn
It' + a " a —a * a~ ^ — (t^ a'-+l + a~'^
1Q - - , ., I
4 —h h —h _"~ -" « + l + «~^
a —a a — a a '^ -\-a'^
1 1 * 3-2- ■ ,■' \ I ¥
20. Divide x-y by a;" — ;v^; ic -\-a x" + a by "-+-« .c -f'* ;
ic+.'/H-^— 3.C // 2 by* +?/ +;s
i .\ i
2a6+2/;(;+2c;a-a!»-fe2 -q- by <(" 4-^''+f
ExERCISE.
1. Express the following quautities i. as quadratic smds, ii
as cubic surds, iii. as quartic surds.
' J. __ "
a, 3a., 2a2, a-x, a;", y . a~"\ -, mx p, •!, '01, I'l.c'-.
y
2. Eeduce to entire sui'ds,
xVx,a^a, l>^^h\ 3if3, 4if2, V^, ilT^, if%, ^\
n +
xJ^y I /j- + ?/\
.'//.
(x-y) ''■^{x^+2xy + y^} ^ (x-x ')f{x^ + iy,
250 APPENDIX.
3^ Eeduce to their simplest form
1/12, 1/8, ^,^50, -^16, 4if -250, Vh ^h i/A. 5ir(-320),
,/{a3(l-^^)[, ^r,,2(,,._i)4|, ^(,,i), »^,,.-.i^ »,^»+»^
y«^«-^^ I'ya'™-, V('f2x + «3), ^(a34.2.,4^^,,5^2)^
l/{(x-l)(^2_i)|, -^{(a2 + 2ffa;+;c^)(«3+;c3)},
,/(8.c3 -16^+8), ^{{x^-^+x-^){x'-'lx^-^l)f,
N \~^+2+.<;-i / N \27j;2 +18u; -r 3 N ( ^^ j
4. Compare the lollowing quantities by reduciug tlicm to the
same surd index :
2 : ^/3 ; 2 : -jf 9 ; 1/2 : -^3 ; ,/10 : -,^30 ; 2 ^/2 : ^^22 ;
a2 : Va^ : V-^ : iy// ; ^x : ^// ; '"^£' . y.c^; i/a : 'yb : ::/c ;
5. Reduce to simple surds with lowest integral surd index
Vif'a), f^{i/b), ^{^c), ^{i/x% t/ifx'), 4/(r-^")»
' f'it/'^''), 1^(1/27), A/(ir81), irif/81), yi.n/a),
■ f{aya), VWx), iTC^'-yf), iflov/S), |/{3-^'3),
V(3#/3), e'(^C/-^), V{a^'{h:yc)], xV{x-Wx-'),
y^{y-'fr''),z^/{z-''-fz-^-),y{x:y{yiyz)],x-^f{x"^Vx^)
6. In the following quantities, combine the terms involving the
same radical ;
3|/2+5v/2-7a/2; -/8-'/2; ^nQ+Qf^;
^16+ \/ 2; cq/aj-i/ic; a'2/x—bl/x;
8^/a + 5Vx-7i/a+ V{4.a)-3y{Li-) + iV{dx) ;
/.c;+3/(2.f) - Zy'{dx)+y^{ix) - v' (S^j + v'(12..) :
APPENDIX,
251
4.\/{a^x) + 2^(b^x)-Sj/{{a + h)^x\;
^/{{a-b)^x}+^/{{a+by-x\-V{a^:^^ + l/{{l-a)^x}-Vx;
^/ (a _ 0) + ■,/ {lQa-Wb) + i/{ax^-bx^)- V { 9 {a - b)} ;
l/{a3+aV>)—i/{b^+ab-^);
j/{a^+2a^b + ab^)- ^(^a^ -2a"'b-Jrah^) - i/{AaJr-).
7. In the following quantities, perform, as far as ]DOSsible, the
indicated multiplications and divisions, expressing the results in
their simplest forms :
^/2.|/o; a/3, a/12; |/14. a/S-j.-j/IO; j/a.y'[3a);
l/c.A/(12c); V{Gx)y.{8x); Vy^Vu^ ; -^y' -V^u' I
fa.^a^yb: iA.a/(^); Va.vi}]; l/-3^'- y (£) '
|/r/"+i.-i/«"+^ ; f/b^^\^lr-^^ : v/12-- A^3 ; ^/{Qx) - x/(2a;) ;
{a^x)~V{a-\-x); (a2-a;2)_=.v(a-aj); (.^^ _i)_^^y(a-^l)3 .
(3v/8-6s/2+>/18+\/32+x/72-2v'50).i/2;
(7i/2-5 A/r)-3v/S-}-4 a/ 20) '/18) ; (v'5+i/3)(^5-v 3) ■
(i/2+1)(s/6-a/3)- (3-V2)(24-Bv/2);
(5V3+v/6)(-V2-2) ; {^n- Vb){ya-\-y^b) ;
(«y'/;4-?,yrt)(/; Va-aVb) ;
{ ^/(:/;-hl) + v/(a;- 1)} { a/(^;+ 1) - n/(x- 1)[ ;
{A/(3rt-/>) + |/(3&-a)}{v/(3ff-&)- A/(36-rr)};
V(<' + ^/b).V{a- Vb); V{^/x-\- ^ ij). V^x- Vy) '.
252 APPENDIX.
{a + y\l -«2)13 . !y^a + b-.r)-y'{a-h + T)}^ ;
[{V(«+^)(x-Z>)} + v/{^z-a:)(;.+i)}]2;|4i^)-4|^)['
W {{a-^x){x+h)] + ^ {{a-x){:c-h)]Y^ ',
{V^V10+1)-V(V10-1)}2;
■\y{a-\-^{a"- -a:-')}4- V{^/ - V(«2 _a-2)l j^ ;
(V./;+Vy)* + (V.^- v/.v)4 : [a^+ah^2+h2){a^ -ah s/'l-^-b^);
B. Find rationaliziug multipliers for the following expressions,
and also the products of multiplication by these :
a-{-Vh, \/a-\-l\/c, aVb — bVa, a + V{a^ —x^),
V(a-x) - V(ri+x), V(a^ + Vr)+\/(a2 -Vc),
V{8 + V(24 + Vo)}-V{8 + V(24-V5)}, V-v + V^ + Vc,
3 + V2+V7, v'6 + V5-V3-V2, ^a + J-b + ^c + ^d,
y{l+a)~^{l-a) + y(l + b)-^{l-c), f/a+f/c,
yx + l + ^x-^', y{aI-')-^{a-'b), f-^ + fS-fB,
fu + f'b + ^c, a + ^b + fc.
APPKNDIX. 253
«
9. Eationalize the divisors and the denominators in the follow-
ing, and reduce the results to their simplest form:
l^(2-^/3), 3--(3+i/6), 5-^.(■^2 + V7),
(,/3+v/2)-=-(v/3-i/2), (7|/5+5|/7)--(|/5+V7),
«-=-(s/rt4-a), {x — a)-^{-i/x — i/a),
aVx + bVii 2V6 1 + 3|,/2-2a/3
TVsT-^^' 1/2+1/3- V6' 1/2'+ 1/3+1/6'
a/6 -1/5 -1/3+^2 2
• V'6+i/5-a/3-V2' i/(fl + l)- A/(a-l>»
2c a.+a;+\^(a2+a;2)
|//(rt-j_c) + i/(a-c)' a+x-v\a^-\-x^y
^/{a-\-x)-^V^a—x) 1
V(a+a;)-V(a-a;)' ai/(l + h^)+b ^Jl'^aFy
\/(l-62)+ i/(l-a2)' av'(l-c-^) + cV(l-a2)'
l/{(l+a)(l + Z>)}-A/{(l-a)(l-6)}
. l/{(l + «)(l + ^)}+l/f(l-«)(l-i)}'
(a-x)i/(&^ + //2)-(6-y)i/(a2+a;2)
(«+a;)i/(62 + 2/2) + (6 + 7/)i/(a2+^2)»
A/(l + a)-l/(l-a)+i.l/(l+//)-vXl-^>)
l/(l + a)+ v'(l-a) + s/(l+/>)+V(l-^)»
%/(x + fl)-N/(y-a)- ^/{x+b) -\-^{x - ft)
l7(u; + «)+ V{x-a)-\-y{x+h)+ V{x-~by
V6"^l/a' ^U-W ^U+W' Na-V-C' ^Vj^Vy'
1 1 a/« \/a;
•4r-i/(a3-l)' 1 1 ' A/a v/a;'
\/j; •«/ V \/ic A/a
254 APPENDIX.
10. Find the values of the following expressions for n=l, 2,
3, 4, 5, respectively.
V5\ \ 2" / - \ 2 / [•
1 f(2+i/6)"+^-(2+V6) (2-\/6)"+^-(2-VG)]
2761 1 + 1/6 ~ 1-V6 I
11. Show that
2(x — 1)
is a square for n=l, 2, or 3 respectively.
12. Extract the square roots of
x+y-<2.A/{x}j), a-{-c+e+2V{ac-\-ce),
a+2c + e + 2^/{{a+c)(c+e)}, 2a+2V{a^ -c2),
2{a2-|-63-^/(rt4+a2/>2+64)}, x-^ + x'^,
^x-\-2 + Vx-^, x+3x^ +x^+2x^/x-^2x^ y/ X,
x^-xi/ + iy^-hV{Ax^y-8x^-ir'+xy^), 2x+^ {?,x^ -y^),
5-2i/6, 10+2\/21, 9+4a/5, 4-v/15, 7+4 v'3,
12-5V6, 70 + 3;/451, 4-x/15,
9 + 2;/6+4(i/2+v'3), 15.25- 5 V.6.
13. Find the value of
• ^^¥+te^'gi-ent.l=:j^^^^^andy=^,^,
y(a;2+i/2), given x=^{a^c) y = ^{a^e)-y
V(l+x)-V(l-a;) . _ 2ah
V, r+;^V(l -a;)' ^^^^ ^ - «3 +/;2 J
2«V(l + a^) _J |l_ l£l.
APPENDIX.
255
14.. JiV{x+a-{-b)-rV{x + c-td) = y/{x-i-a-c)+V{x-b-\-cl),
*(1.4-V5)a;-2 ^(l-V5)a;-2
15. Simplify :,y^^i^v5)x+l'^^-h[l-Vo)x-^l'
Complex Quantities.
Quantities of the form a+bV — l in which neither a nor i
involves \/ — 1, are called complex quantities. The letter J (or i)
is frequently used as the symbol of the ditensive unit •/ — 1, so
that rt + ij/ — 1 would be written a + hj. So also V —x^JVx,
y/-x.V -y ^p_ V {xy) = - Vxij, and p = -j
EXEKCISE.
Simplify the followmg, writing j for >/ — 1 iu any result in
which the latter occurs :
1. y_4, |/-3G, 1/-81, V-8, i/-12, 1/-72, ^-8,
v/-5.i/-G, V-QV-8. V-8.VI-2, ^/-8.lr-8,
l/-5.j/-20.
2. /-x, i/-a;^ v/-a3, ^-a^"*, V{-a)^ ■/(-a)^,
■«/5. \/ — rt.
Q o2 ;3 i4 iS ,;9 -;i5 ,*1G ,-17 ,'18 ,'4'> ,'ln+l -/te+2 „'4n+3
4. cj-bj, jVx.Ji/y, 5j, j-^i/5, ji/~a,ji/-a^, J-i/a.-i/ -a.
6. V-r, V-p, i/-j\ l/-i^ -/-i^", i/-p\
^_6 a/-6 |/G i/a |/ a 1
6.
V 3 ' V-3' i/-3' i/-/.' -i/-6' ■/-!'
a a^ \/{—ax) —V-1 a^
y/ —a v/— a^ y' — x V — a ' f/ —
a^'
256 APPENDIX.
J_ J_ J_ -JL 2_ _J_ ^iL ^
^' j ' P ' F' i ' j' ' i*"+'' i^"^'' i^"-''
2
8. A/(a-Z,)V(/-'-a), a/(3*-4?/)V(47/-3x), (3 + 5;-)(7 + 4j),
tS-9i)(8-7;), (7-jV5)(7+jVlO), (v'3~iV6)(V2-jVU),
(/a4-il/6)(l/a->^/c) (a+/i/'j («-/;;'), («;' + &)(«y-&),
( v'"+y|/^)(/a-iV'6), {(iVl)+GJy/x){a]/b — (jVx),
l/(l+yjV(l-yj, V(3+4i)V(3-4i),
v/(12+5;jVil2-5;-), (l+i)3, (ya-jV6)3, (5-2JV6)",
{a + hjf+[a-hjy\ ^a + bjy^-ia-bjr-, (a+l>jy + {aj-by,
{^/(4 + 3Jj+v/(4-3i)}^ {^(3-4;-)-v/(3 + 4i)}^
[i{y(30-6N/5)-l-i/5} + i;{N/15+ ^/3+ ./(10-2|/5)}]»
tor all positive integral values of n.
4 64 21 5 1-20;V5"
1+JV3' 1 -iV7' 414- 3;- V6' V2+Jv3' 7-2;V5 '
i+yv3' 1-/ i+i' i-i' (i+j)3' i-y' u;-^/
a+jy/x JV(i+V-b a — hj (i-j-jv'(l-£2)
a—[y/x \' -<t-j\/b aj-{- 1) a-jV\^ 1 - x - )'
9.
APPENDIX. 257
V{x-y)-V(>/—x) 1 1 l+i 1—,;
v{y^x)+v[x~^y 1+7+ r^'' i-j+i-h/
1,1 1 1 x-^yj ^' — vj
(l+^-)2-^(l-j)3' (l+ij_4 (i__^-)4' «_|_j;-r^, _ij
a; + yj a; — yj Vx +JV1J Vjz+jV^
a -\-hj a —hj Vx—j^/ij ^y—jVx'
y(l+a)+7V(l-a) _ y(l_--«.) -|-^-y(l+^)
y(i+«) -Ml - a) y(i -rtO -jV(i +rt)'
10. ^/(3+4y)+|/(3-4i), ^(3 + 4i)-A/(3-4y),
v^(4 + 3i) + v/(4-3i), |/(l+2;V6)±v/(l-2;V6)
V(5+2;V6)±a/(5-2jV6),
v/(2,/15 + 3q;)±v/{2v'15-30j),
A/{x/3+iv/105)+ A/(i/3-ii/105),
11. Prove that both i(-l+j|/3) and i(-l -jV3) satisfy the
a;^-l
equation . _^ =0,
that {x-\-icy -^-w-zY = x^ ^-y^ -\-z^ + ^{x + wy){y\- wz){z-{-n-x)
and that (a^ + ^ +«)(«+ it;^-}- it; ^z) (a; +m;^ ?/+«;«) =
x3-j-?y3_}.23 — 'dxyz, vo. which 10 represents either of the pre-
ceding complex quantities.
Hence, prove that
(i) {2a-&-c+(5-c];V3}3 = {2&-c-a+(c-«)j/V3)}3=:
{2c-a-6+(a-%V3}3;
(ii) u^-\-v^-\-w^ —Quvii:=-{a^ +h''^-\-c^ — ^ahc)x
(a;3 4-7/3 ^gS -3a;!/2), if u = ax-{-by+cz, v = a^-f &2'+c3«j
u; = az + bx-\-cy, or if w = rtx-f-c?/+/;2,
t; = ca;+% + rt2, w = hx-\-ay-\-cz.
258
APPENDIX.
12. Prove that i U/^ + t+j v' (10 — 2 -/S)} satisfies the equation
x^ + 1
x+l
Writing v for tlie preceding complex quantity, prove that
(7+'r+?<'2_|_:3„,3)(7_ti.4_„;3„3^(.2) = 71^
and {x ■}- >/ -\-z){x+w^y — ir3z){x - iv^jj — icz)(x - icz-{-rv*^z)
(x-^w^ij + ic-z) = x'^ +1/''' +z'' - 5x'''yz+5xy^z^.
Prove that {4« + (/;-,-)(,/5 - 1) + (6+c)jV(10 + 2 \/5)}5 =
{[{a + b){-l+jVii/5 + 2)} + ia-b){V5+jy'(V5-2)}]
X y5-4c}^
Section III. — Pure Quadratics.
Examples.
J x + S(a-b) _ a{Zx + %a -lb)
x-S{a-b) ~ b{3x^a+\9b)'
m p m-\-n X''^^ .
Apply, if — = — , .". = 5
^••^ •" n q m — np — q
X 3.r(ff + ^>) + 9«2_i4,,?; + 9J2
'*• B[a-h)^ 3a;(a-6)4-9(a2Tr^
Dividing the denominators by 3(a — b)
.-. a;2 = 9a2_ 14^6 + 9^3
'a;_2ff-f-4M 2 5a;-9r7+3&
x+4a-26/ bx + 'da-Sib
m p n — m q—p
Apply, if — = — , .'. — - — = — — , and factor the numerator
(a;+4a-2i)3-(a;-2fl + 46)2,
APPENDIX. 25i)
mx-\-aA-h){a-h) \^a-h)_
(a;+4«-2i)3 -5^+ 3a- 95*
x+a-\-b .T4-'4a-26 ^a-'h)
:. 7^ = T- — ,—7^ Ki = "a ^. oy taking differ-
•• x-{-^a-^h 5a;+3a-9Z; 4ai-a-lV ^ °
ence of numerators and difference of denominators. To the first
and third of these fractions, apply if
m p m p ^
n ~ q ' " ** — m~~ q—p
x-\-a-\-h ^a — b)
''• S{a - b)^'4x--4:a — A¥
.-. x2^x^i(^a + b)"}+d{a-by^}.
l/{Sx-'-l)-\-V{B-x'- ) _ a_
3a;- -1 a-{-b'
4
3_a;2 -a-b'
• 3a;2-l (a + b)^
" 3-x' ~{a-by'
S{a + h)^ +{a-b)^ a^^ab + b^
•■• ^^ = '(^j^Ij)2 ^ s^a-by-^a^ -ab + b^'
4. 7nV{l+x)-nV{l-x)=V{m^+n'i) (1)
Square both members and reduce
.-. {m^-n^}x-'imnV{l—x-)=0. (2)
Transfer the radical term and square both members,
.-. (m3_^3)3a;3=4^2„2(l_a;2) (3)
.-. {m^-{-n^)^x"=4:m^'fi^ (4)
The above follows the usual mode ol solving equations involv-
ing radicals, viz., make a radical term the right-hand member
gathering all the other terms into the left-hand member, square each
260 APPENDIX.
member, repeat, if necessary, until all radicals are rationalized.
This method is convenient but it does not explain the difficult^
that only one of the values of x in (4) satisfies (1) viz. ~jT~Va
— 2m7i
The otber value, „ , ^ satisfies the equation
w \/(l 4-a;) +?i a/(1 -ic) = l/(m2 + /, 3).
The exi")lauation is simple. Squaring both members of (1) is
reully equivalent to substituting for (1) the conjoint equation
{wV(l+x) + /.V(l-a;)- V(/?t2+n2)|^0 (5)
which reduces to (2) above.
Treating (5) or (2) by transferring and squaring is equivalent
to substituting for it, the equation
{m V {'i- -i- x) - 71 V {1 - X) - V{>ii'^+n")^- X
{7«|/(1 +a;)+» \/{l -x)- v/(m3 +^2)} x
{mA/(l + x)+W|/(l-a;)+i/'(m3+w3)} =0 (6)
^s'hich reduces to
Urn^ -n^)x-%nny/{l-x^)] {m'' -n^).c-\-2mnVil-x-} =0 (7)
which further reduces to (3)
Thus the whole process of solving (1) is equivalent to reducing
it to an equation of the type ^ = 0 and then multiplying the
member A by rationalizing factors. Thus instead of solving (1)
we freally solve (6), i.e., a conjoint equation equivalent to four
disjunctive equations. (See page 140, Art xl ) Now the values
given in (4) will satisfy (6), the positive value making the first
factor vanish, the negative value making the' third factor vanish,
while no values can be found that will make either the second or
the fourth factor vanish.
APPENDIX.
261
Hence, if one of such a set of disjunctive equations is proposed
for solutiou, the conjoint equation must be solved, and if there be
a vakie of x which satisfies the particular equation proposed,
that value must be retained and the others rejected.
(This process is the opposite to that given in Arts. XL. and
XLV. : there a conjoint equation is solved by resolving it into its
equivalent disjunctive equations. The two processes are related
somewhat as involution and evolution aie).
Further, it should be noticed that just as there are four factors
in (6) while there are only two values in (4), it will in general be
possible to form more disjunctive equations than there are values
of a; that satisfy the conjoint equation, and consequently it will
be possible to select disjunctive equations that are not satisfied by
any value of x, or, in other words, whose solution is impossible.
This will perhaps be better understood by considering the fol-
lowing problem.
Find a number such that if it be increased by 4 and also dimin-
ished by 4 the difference of the square-roots of the results shall
be 4.
Keduced to an equation this is
^(a;+4)--/{a;-4) = 4 (8)
Eationalizing this'becomes
{4--/(a;-H4)+v/(a:-4)}{4-/(a;+4)-i/(x-4)}x
{4+v/(a;+4) + i/(a;-4)}{4 + i/(a;+4)-V(;c-4)}=0 (9),
which reduces to
{24-8v/(a;+4)}{24+8i/(a;-f4)} =0
ie. 9— (a;-+-4) = 0, oric = 6.
Now »= 5 satisfies (9^ because it makes the factor
4-|/(a;+4)-A/(a;-4)
vanish and it is the only finite value of x that does satisfy (9), or,
in other words, there are no values of x which will make any of.
the factors
262 APPENDIX.
* 4-i/(a;+4) + |/(a;-4), 4+ V{x^^)+ \/{x-4.),
or 4 + A/(a; + 4) -V (^ - 4)
vanish. There is, therefore, no number that will satisfy the con-
ditions of the problem.
[It will be found that as x increases, ■j/(.'c+4) -|/(a;— 4)
decreases, hence as 4 is the least value that can be given to x
without involving the square-root of a negative, the greatest real
value of -i/(x+4) — -/(a; -4) is -j/S which is less than 4. We see
by this that our method of solution fails for (8) simply because (8)
is impossible] .
5. V.{{a + x){b+x)}-i/{{a-x){b-x)} =
^{{a-x)(b+x)} - V {{a-hx){b-x)} (1)
Collecting the terms involving i/{a-{-x) and i/(a-x) respec-
tively the equation becomes
{l/(a + x)-y/{a-x)}{^{h + x)-i-V{b-x)}=0 (2)
This is satisfied if either
l/(a+x)-i/ia-x) = 0 (3)
or Vib + x)+\/{b-x) = 0 (4)
The rational form of (3) is (a + x) — {a-x) = Q which is satisfied
by a; = 0 and this also satisfies (3).
The rational form of (4) is {b + x)-(b—x)=0 which requires
x=0, but this does not satisfy (4). Hence the second factor of
the left-hand member of (2) cannot vanish.
Therefore the only solution of (2) and /. of (1) is a; = 0, derived
from (3).
6. f{a+x)-]-^ia-x) = f{2a)
Cube by the formula (u+v)^ = u^ -\-v^ -\-Snv(u-{-v)
:. {a+x)-h{a-x) + 3f'{2a{az-x^)}='2a.
.-. 2a{a^-x-^) = 0,
:. x=±a.
Both these values belong to the proposed equation.
A1>1'ENDIX. i
The rationaliziiog factors of
f/^a+a-)-{-^{a-x)-f/{2a) = 0
are -^{a + x)-\-o>]V{" -x)-(^)^-^{2a),
and i/(a4-a;) i-o^-f/{a -x) -(^^y{2a). See page 257.
The remarks on Ex. 4, will apply wwfaiis mutandis io equation •;
of this type.
^' f/(a+x)^ -f{a2-x^-)^+^{a-xy "' ^^'
Assume -^{ft+x) = w and ]^(a— a;) = v
.'. u^ +v^ = 2a and u^ — v^ = 2x,
and .'. o , ^-0 = —- (2)
Also (1) becomes
ll^ + W + V^
= e (3)
m2 —UV+V^
U — V
Multiply both members by ^— ^
^3 — y3 u — v . ,^^ X u—v
Again adding and subtracting denominators and numerators
in (3)
u^ -\-\v^ c+1
uv ~ c — 1
Adding and subtracting 2 (denominators) and numerators in this
ti^-2uv4-v^ 3-c /M-r\ 2 B-e
or ' ' "~
M2_j_2Mu + t;3 3c— 1' \u-\-vl 3c- 1
a-2 3-c
.'. substituting by (4), -^ = c^q^_ i*
|8-r
264 APPENDIX.
8. ^V{x + a) + V{x -a)}^ {r{x+a) - \/{x--a)]^1c {\)
Assume u=\/(x-\-a) and v='^y{x--a), and (1) becomes
(?i+?.-)3(M-r) = 26- or (?t+i-)2(?i2_r2) = 2c (2)
Also ?t4 -r4 = 2a or («3 + y2)('u3 - ^2) = 2a ■ (3)
and u'^-\-v'^ = 2x. (4)
From (2) and (3), {u - v) 3 (nS _ t-2 ) = 4^ _ 2c • (5)
.-. (2)X(5), (l(.2_t-2)3(^2_^3)2or (M3-t;2)4 = 4c(2«-c) [<6)
Also (3) 3-1- (6),
|(M2 + ,.3)3_^(jt2_^,3)2|(ji2_^3)2=4(«2_|.2aC-c2)
or (?(44.^.4)(^i2_^,3j2 = 2(a3 + 2ac-c2)
Substituting by (4) and (6)
^a;|/(2ac-c2) = «2_|_2ac-c3.
Exercise.
1. (a;+a + 6)(x- a + &) + {x + a - h){x -a-b) = 0.
2. {a + bx){h — ax) + (/; -(-oa;)(c— hx) + {G+ax){a—cx) = 0.
3. {a-^bx){ax—h)-\-[b-{-cx)[bx — c)-\-[c + ax){cx — a)
= ^{a^-{-h2+c^).
4. (a+a;)(i-a•)-4-(l-fa;«)(l-^*a:) = (a + ^))(l+•^•3).
o. {a-rx){b-{-x){c — x)-{-[a-\-x){b — x){c+x) + {a—x){b+x){c + v)
+ {a-x){b-x){c+x}-\-{a-x){b-^ x) (c -x)-\-
[a-\-x)(b — x){c — x) = 5abc.
J. {n + x){b + x){c + x)-\- {a +x){b + x){G-x) -^{a+x){b - x){c + a;)
+ [a—:x){b + x){c-\-x)-\-{a+x){b — x){c — x) + {a — x)[b + x){c — x)
-t-[a-x){b — x){C'\-x)-\-{a—x){b — x){e-x) = Qx^
7. (a4-5&+a;)(5a+6+a;) = 3(a-|-i+a;)3.
8. {a + llb+x){lla+b-\-x) = %a+h+xy.
9. (9a-7i+3a;)(9i-7a-f3a;) = (3a+36+a;)^
APPENDIX. liuj
ab cd x — a x-\-n
a^ — b^x" c-^ — d^z^ x + 1 x—1
V
a + x x+b ax+b cx-\-d
a — x~x — b' ' a-{-bx~ c+dx .
a—x 1 — bx a — x -k—x
14. z = -, . • 15.
22.
25.
26.
1 — ax b — x' ' 1 — ax 1-bx
x+a + 26 h-%(-\-1x a + 46+a; 'db-a+x
^^' a; + a-2ft^6 + 2a-2.c' . a-4.b^x^ 2>b + ^'^x'
x-\-6a4-b x — a-\-b a—lbA-x a+5b-\-x
^°- x-3a-{-b~a-x + Sb' '^' Ta-b-x 5a-tb-\-x
3a~b—x 5b-3a+x Sa-2b + Sx x-a+2b
20. "771-— = .— oT^.. 21
32.
a — '6b-bx oa — 'db+.r' ^ ' a — 'Io + x 'dx — 'da-j-'Ab'
3a-2b+Bx x-7a + 8h^
a — '2>b-\-x ~ 3a;— 5a 4-46
5a-6b + x da — 5b-\-3x a + b~x S(a-h+x)
23 ! — 24 = ^^ '
a-^x a+b+x ' ' Sa—b — Sx a — ^b-^-x
la + b — x 3{a — h+x)
5a-\-db — 'dx a—llb + x
5a-b + x 2(2g— Z)+a;)
2(a + 26-a;)'^^ll/>-a; *
7a — b-\-x a{a-{-5h-[-x) x + a — h a(x + a-{-5by
27- Tb-a-{-x^b{5a + b+xy x-a+b"^ b{x + 5a+'Fj'
.5a-Sb+x\ 2 7a-9b + Sx
29.
30.
5b-da + xl 76 -9a 4- 3a;
la + 5b+x\ "- a+17i+a;
\5a + b-\-xj \la+b-\-x
na-h-\-x- 3 ^ lla + b-x
31- [rjJZ-^x] rfb + a-x
lla+b-x a^(a+nb + x)
a+llb—x 62(17a+6-i-aJ)
son Arpr.Nnn:.
33.
{5x + 'Sa-llb){:v-a + nb} bx+la — bUb
g^ (1 + Sx+]5x^){x^ + 3a;+5) _ ^
(l + 2.c+3a;2)(a;2-|-2^ + 3)~ 4"
35.
8G.
37
38.
39.
40.
y(l+^2)_j_^(l_a;2) ,,
V'(l+.f-^)-l/(l-xf)~ ^
lf(l+a;2) + ^(l-a;3) q.
4/(l+.,.2)^_4/(l_a.2) ^
t/(i+«2)-V(i-a;-)~ir
V(l4-x^)+!y(a;3-l) _ jt.
'y(i+x^-)-'^{x-'-i)- b"
y(a;2+l)+y(a;2-l) a
41. -,/(4rt + 6-4.i;)-2V(.r + Z/--2x)=v''^.
42. -,/(3n-2/>+2a;) - %/ (3a-26- 2.^) = 2 \/a.
43. A/(2a-/)4-2.r)- A/(10a-96-6a;) = 4v/(a-6).
44. i/{Sa-4:b + 5x) + ^/(x--a) = 2V(x+a).
45. V(3a-46 + 5.ri + |/(a;-fl) = 2v'(2.«-2&).
46. i/(5x -- Sa -{-ib) + -^/{5x-Sa - Ab) = 2V (x -{-a):
47. i/(2a+/^ + 2a;) + i/(10^/+9/>-Ga;)=2v/(2a + 6-2a;).
48. 2'v/(2r/+6+2.r) + ,/(10rt+6-6«)- N/(10ff+96-6.'c).
49. y'{2a-13b+Ux)+V {S{b -2a-{-2x)} =fiV {2.a-b + 2x)
60. i/{3(7a+6+a;)-N/(a+76-x) = 2i/(7a + 6-a:).
51. V{{a+x){x+b)} + V{{a-x){x-b)}=2i/(ax).
62. ^{(a + a;)(a;-l-6)}-i/{(a-a:)(a;-6)[ = 2|/'(6a;).
68. ^(aaj+ic^j- -t/(ax-a;2)= ^{2ax-a^).
APPENDIX. 207
1 1 _2_
x+V{ax) a + y^(«.c) x — a
56. 7- — .-{- 77 — r = ■•
a — V [ax) X — -i/{ax) a
l/{{a+x){x-^b)}+ V{{a-x){x-b)} U
''^' V{{a-{-x){x + b)} - V{{a-x)i:x-b)}--<b '
tSa-U + 'i.e {^/a-^^/(2ft-26)}"
58. ^3,f_2i-2.<;~ 26-^*
59. ^{a-{-x) + f{a-x) = 2^a.
62. ir(H-x)--'+iril -x)^ = ^h^{l-x-').
63. i^'(3+*-) + lf^3-.c) = ^6.
6i. -,K(i+^)='+r(i--^-)'=5{i^(;i+^)+f/(i--*^)i'-
65. f{U+x)^ -f^{196-x-^) + f'{U-xy^ ^7.
66. {if(9+x-)+i^(9-a;)}if{81-a;3) = 12.
67. {f{u-^.ir-f^{u-xy-}{^{u+x)-f^{u-x)} = ie.
68. {irt57+.«)^+#/(57-x)3}{if(57-a;) + iK(57+a;)} = 100.
69. 5{t/(-il+'-^) + V(41--^-)}'=8{^/(41 + '^-)+v/(41-^)}-
70. {t/{x+5)+t/i'^-5)}m/(x+5)-t/{x-5)}=±
71. {V(-*^+i)+M^-i)}{-/(-«+i)+v/(-«-i)} =
26{V(^+l)-t/(a;-l)}.
72. r\^^ + r\^^ = a. [y+y-'=a].
73. 2{f'(l+a;)3+^(l-x3)} = (c2+l){ir(l^''^) + #'(l-^)}^
208 APPENDIX.
74. f^(a+x) + ^{a-x) = fc.
75. {^(a + x) +f'(a - x)}f'{a"--x^-) =c.
76. ^ia+x}-=^{rr- -x2)^^(a~x)- = fcK
77. {f'{a + xy'-f/{a-xy}{f{a+.v)-f\a-x)}=c.
78. {f/[a+xy-+f{a-xy-}{f{a+x) + fia-x)}=c.
79. [a-^x)-^{a ~x) - ia-x)f{a+x) = c{f/{a + x)~f/(a-x)}.
80. (a + x) f/ia+x) - {a-x)f{a-x) - e{^(a + x) - |f (a - u;)}.
81. {^ia-^xy^-f{a^~x^') + f[a-xr-}^- =
c{f'{a+x) + f/{'i-x)}.
82. {V(«+^) + M«-^)} '=(« + !){ \/(«4-aj)+A/(«-.c)}.
Sectioi? IV. — Quadratic Equations and Equations that
CAN BE RESOLVED AS QUADRATICS.
Examples.
1. x^ + (ah^iy = {'i^+b^)ix^+l)+2{a^-b^)x-{-l,
.-. x^+aH^' = {a^ + b^yx- + 2{a^-b^)x+[a-by
:. X* + 2abx^- +an^- ={a-{-by-x- +2{a-2 -b^)x+(a-h)^
.-. x^.^ab=±{{a + b)x+{a-b)},
or x"+{a-\-b)x + ab-±{a — h),
:. x^^{a+byx-{-^ia + by = ^a-by ±{a-b),
.-. x^i{a-\-b)=W{['^~byziz^{^a-b)l.
APPENDrX.
269
{a-x)^l/(a-x)+(x-b)W(x-b)__^_f^
2k. '
{a - x) i/{a ~x)-\-)x-b)^^{x~b)
Writer — 6 in the form {a—x)-r{x~h) and multiply by the
denominator of the left-hand member,
.-. {a~x)^yia-x) + {x-by \/{x-b =
{a - x) V(« - ^) -^ {ci-x)(x-b){^{a-x)+ ^{x - b) } -L
(x~hyW{x-h),
:. (a~x)(x-b){V{a-x)+V{x^b)\ = 0,
:. {a~x) = 0, or x-b = 0,
or \/{a — x)-\-V(x — b)=0,
x^ =rt, .fo = ^•
The equation V{a-~x) -\-V[x -b)=0 has no solution for the
Bum of two positive square-roots, cannot vanish.
The solution x = ^[<i-\-b)) belongs to the equation
V{a - X) — V{x — b) = 0.
ax+b mx—n
3. T—~ =
bx -{- ci nx — m
Add and subtract Numerators and Denominatorg
{a+h)(x + l) _ {m-\-n){x-l)
{a - b){x — l)~ {m-n){^x+V)
^_l/ {a+h){m-n) ""
•'• ^1 = s"Zi' ^-3 - .s4-l*
b-\-x a —x
Square both members, subtract 4 and extract the square-root.
270
APPENDIX.
/fg —X
a — x
" b^x
= 63
1 - ^3
••• x=h{{a~b) + {a+b) j-^^3^
Or thus, cube both members,
a—x b+x „
.-. - -- + 3c+ = c3
b-^-x a—x
{a-xr- + {b+xY
= C3-3C
•• ^a-x){b±x)
({b+x)-{a-x)Y g^-3c-2 (£+1)2 (c- 2)
2a;- (g- 6) c + 1 |c-2
a +"6 ~ c - i Nc + 2'
Wrt-a;)-V(6-a:) ^/{{a-x){b-x)]
■ +
(Prove that ^~ 3
V(<i-x)-j-V(6-a;)
{v/(a-.r)-V(6-a;)}g v{(a-3;)(5-a;)}
{a—b) - c '
{ V(a —x)-V{b-x)] - a-h
V(a-a;)|-V(i-:c)) 3_ a-6
. Eationalize Deuom.
(-1)
" iV(a-a;)H-V{Z»-a;)) a-i+4c'
V{(a-a;)(^-a;)} _ I _a-b^
c ~ ^c7^b + Jc'
Also from (A),
a+b — 2x a — b-\-2c
V{{a-x){b^^'x)} "^ c '
(^)
APPENDIX. 271
Multiply (B) and (C) member by member
;.{„+,_(._, .+2.) 4--
h
■ 6. a;4-4=-5^^^;a;e.-2.7;4- 5a;3-12 = 0.
aj- — 2 '
Find the rational linear factors of the left-hand member by the
method of Art. XXVII., page 90.
.-. (a:-2)(x-f2)(^* + 2a;3+3)=0,
.-. x-2 = 0, ora; + 2 = 0, or x'^-\-2x^+3 = 0.
The last of these equations may be solved as a quadratic giving
a;2 = -l±2A/— 2, :. x=±l±V-2,
,'. x^=2, x^ = -2, x^ = l + V-2, x^ = l-\^-%
a;5=-l + V-2, a;6=-l-V-2.
I^.B. — In solving numerical equations of the higher orders, the
rational linear factors should always be found and separated as dis-
junctive equatiom, before other methods of reduction are applied.
Such separation may always be effected by the methods of Arts.
XXVIl. to XXX., and unless it is done the application of the
higher methods may actually fail. Thus, if it be attempted to
solve as a cubic the equation,
a;3-9a;-10 = 0
the result is aj= {5+ V- 2} +{6-V-2} , which can be reduced
only by trial. The left-hand member can however be easily
factored by the method of Art. XXVII.. and the equation reduces
to
{x+2){x^-2x-5) = 0,
which gives a; = 2 or l±y'6.
272 APPENDIX.
7. {x-2y -x'-\-2' =0.
Factor, (See No. 20, p. 89), rejecting constant factors,
.'. x=-0, ov x-2=^0, ov x^--2x + 4: = 0. *
The last equation gives a; = Izt \/ —.3.
Exercise.
' Solve the following equations :
1. {x + a + b)^=x^+a^-hb3. 2. (x + a + b)^ =.r:^ + a'' +b».
3. {a—b)x^ + {b-x)a3 + {x-a)h^ = 0.
4. {a-b)x^+{x-b)a^+{x+a)h^ = 2abx.
5. (x-a)^ + {a-b)^ +{b-x)^ =0,
6. (x-ay -\-{a-by+{b-x)' =0.
7. {a^-b)x'>^+{x^-a)b^-\-{b»-xyi* = abx{a^b^X' -1).
8. {x-a){x-b){a-b) + {x-b){x-c){b-c) +
(x — c){x^ a){c — a) — 0.
x^-1 x^^-1
^^' x^-1 x^ — 1
13. a;'' +5a;3-l6x^4-20,x-16 = 0. (See Art. XXII.)
14. a;4 - 3.6-3 + 5.C2 + 6.C + 4 = 0.
15. (x-a)4+a;4 + «4=0. 16. 2.r--' = (.r-G)».
17. x{x-2y{x+2) = 2. 18. (4;.-3-17)x+12 = 0.
19. a;* + (ai+l)= = («"-+^')(^' + l) + 2(«'-&^)-i^ + l.
20. a;2(x-169)2+17x = a;3-3540.
APi-ENDIX. 273
21. 6x(a;2 + l)3 + (2^'-+5)3 = 150x+l.
22. 2x{x-l)^+^ = [x + l)^. 23. x^ = 12x+5.
24. 5x^ = 12x^+1. 25. (x+4)3 = 3(2:«-l)=.
20. V(.c2+w2)-fv[(7i-2;)2+m2}=V{(a;-4«)^ + (i«'^3-wO}-
27.
31.
33.
34.
35.
86.
(x + iy . m 28. 1-^+1)^ ''J^L
(.6-" + lj^.t;-l)~ n x{x^+l) n
(^3 + 1)0:3 + 1) m . (:,3 + l)(,;."+l) m
-^- {x + l){x^ + l)~ W "^^^ {x^-l\x-^-l) n
x2(u;+l) ~w* ''"■ x(x'-^ + l)(x--l;-~ ?t
x{x-\-l)' n{n—m)
(a;3 + l)(:c-l)2-2»i(2m-?2)
(a;3 + l)2 4m2
.<;- - 1)- 7/i- — ?4''
t>-l)(a;2+l)2 2(»?- -n)3^
(j;3-l)(a;+lj'"~ '"^'^
a;^ - 1 2h?.
(.c+l)(a;^- l)'"2m-
71
^x^-l){x + l)^ _ in+n (x+l){x^ + l) ^ m + n
39. x^ =7 •• • 40. x^ = 7
bx-a hx-a
ax-b ,^ , ax3 + ?»ar + c
41. ^^=i^i::^- 42. a.^=^:;-j^qr^-
43. a;3 = (a;-l)2(x2+l). 44. aS^jS = («_a;)2(«2 _^2).
45. x^ = {x-ay{x^--l).
46. aA/(2;2 + l)-a;/(a;2 + r72W^,
274 APPENDIX,
47. l^(a3-f-.,.-3)4-^(,(3_a;3)^^(a6_a.6)3.
48. tn{x+7n~7i){x — m + ln)^ =n{x — 7n+n){x-\-lm.—n)^.
49. vi^{x + m+nn){x-m- 5n)^ =u^{x+nm+7i){x- 5m+n)^.
50. )7i^{.r-i-m + n7i){x-7n + l7i)^ ^n^{x + nm+7i]{x+lin - ii]^.
V{x-a) + i/{x-b) _ Yx-a
V{x-a)--i/{x-b)~^x-b
V{x — a) + V(x — b) \a-x
52.
53.
V{x-a)-V{x-b)
]a — X \b4-x \a — x,\h-x
\r^~-J =c. 54. , +V = (
'vj'^+x' ^u — x ^b — x ^a — x
oyCi—x o/h+x 3 / la — x\ ^ lb—x\"
b -\- X a—x ' \b —xj ^ \a — x!
^,a-x b + x 4/«-a; 4/i-a;
o+u; a — a; b — x a — x
.,a — x . b + x riA—x .,b—x
59. y.-— +^ — =c. GO. V^ — -e^ ^c.
b+x a — x b — x a — x
61. V7n— +v'^-=c. G2. V, -V = c.
w+;<c ax b — x a—x
^^ ^{a-xY-^sf(b-xY __
^{a-x)-\r^{b-x)
^/(a-xy^+i/{b-x)^ _
^^' {^^a-x)+y{b-x)}^-''
V{a-x)^+l/{h-x)^
-\/{a—x) — V{b—x) ~
_ W{a-x)+^/(b-x)y
'^^' ^/(^a-x)-V{b-x) -"'
y^{a-xy+V{x-by
^'' V{a-x)+/{x-b) -"'
APPENDIX. 27/)
68.
70.
it^.
76.
■x/{a-x)^ + V{x + h)5 _ ((? + />) 2 _
~~V{a -x)+V{x + b} ~4:V{{a — x) {x + bj}'
x-+(a-x^)V(a-x~)
a;+V(a— a;^)
72. ^ , 77-0-—- o-x '' ^ ^-^"^i^' --x^).
x+V{a^ —x^)
73. if(«-x)-^-ir{(«-.T)(.r-/>)} + |3/(a:+6)2 = ^/(a2-a5+i')
74 b^/{^-x) + aV{x-b)
75 ^\/(fl-a;) + ^l/(a;-/^) _
l/(a-aj) + i/(a;-ft) ~
V{x- a) + -/(a;+q) - .7(2^) _ y^'+f
v'{x — a) — y(.x-H-rt)+v/(2a) ~ a;-c'
77 -y(^^ - a:) 4- y ■ _ 4.^ -a:
78. ■|y(rt. - a:)2 - fV | (a -x){x + h) } +-^/{x+h)^ = iy{a^ - ab + h^).
79. {^{a-xY-^^[{a-x){x-h)-\ + ^3/(a;-/.2)2p =
(^,-i){r(«-.^) + l>n:« -/')[.
80. {x1/(rT-a;)2+ir(/;+.'^)2l2 = (a + 6){^3/(„_.^)4.^/(/,^^-jj
81. f/{a-x)^^{x-h)=f/c.
82. -,3/(a+^)2-T^/(^'-^')'=if(2^a;).
83. f/{a-x) 2 + 1^/ J (a. _ a,) (/, - a:l } + f{h -x)^= f/c^.
84. ■,f(a-a;)2--,f{(«-x)(a: + o)}4-^(a;-fi)3 =
276 APPENDIX.
85. { ,3/(a - x) + f/{x + h) )f{{a-x){x-\- h) } = c.
86. f/{a-xY +f/{x-hy^ =c{f\a~-x)-{-f{x-h)]'^.
87. x+f/{a^-x^) = "' .
89. (a+x)V(fl+^) + («--''')M«--'') = «{t/('-« + -'^) + M«-2j)}.
90. (a+x}^/{a-x) + {a-x)t/{a-\-x) = a{'^{a+x) + t/{a-x)].
91. 4/l^G-»^) + t/(^-10) = 2-
93. {a- x)t/{(Ji'-x) + {x-y)t/i:^ -^>) =
{a-b){J:/{a-x)+^{x-h)].
95. {^{a-x)-\-if{b-x)]{s/ (a-x)+ ^ [b-x)y~ ^
c{t/{a-x)-if{h-^x)}.
90. rt\/(l+.r=)-3;i/(a;2+rt3)^g.
97. {a-x)^{x-h)+{x-h)-^{a-x:)=e{^(a-x) + f/{x~h)]'-,
98. {Ty(a-^)+ir(&+a:)}^=c{ir(«-a?)2+r(/>+^)2}.
99. {^{a-x) + f{h+x)]'^=cf^{{a-x){h-^x)].
100. ir(a-a;)^-i«/(?>-rc)2=r|r(fl + /;-2a;).
101. V('^-^) + M-'^-^) = ^''-
102. ^{a -x) + '':/{x-h)= ^c.
{a-x)^{n-x) + {x-lAt/(^--h) _
^^^' (^.-a;)i/(a;-&)+(^—*)^> -•'«)"''•
104. ly^a-x)^i/{h-x) ^■''•
105.
V(a-a;)4-V(->--fc) ^ g
4/(a-a;)- t^(3;— i) a+^— 2a;'
APPENDIX. til
«
107. [a-xyi/{a-x)-{x-hy^{x-b) = c{:i/{r,-x)-^{x-h)}.
108. {a-x)^{x + b) - {x+h)^{x- a) = c{^y^a -x) - i^[x+b)}.
109. {^{a-x)^ +i/{x-b)^}-l/{ia-x){x-b)} =c.
110. {:/(a-a;)-y(^-i)|3{5^(a-:c)^-M^-5)2}=c.
111. {^y{a-xy- ^{x-by-}^- {y{a-x) + -Ifix-b)} =c.
112. {^(«-u;)^+:/(x4-Z^)3}2=c{.V(a-a:) + y(a;+Z^)}.
Section V. — Quadratic Equations involving two or more
VAUIABLES.
1. (x + y){x^+y^) = a, I.
X^-^j/+XlJ^ =c. II.
I + 2TI. .-. {x + yy = a + 2G
.•, x-{-y = -^{a+2c). (Any one of the three cube-roots). III.
ic^ c ' * ' U+2// a+2c'
Tj TTT ■i/(a-2c)
By 111. a; — y = -'— ) — -; .
Ai . i/(«+2c)
^^^° "+•'' = ^/I;h:2^)'
. ^ ^ y(a+2g) + y(a-2c)
26/^a + 2c)
= A/(a+20 --!/(« -2c)
^ 2V(" + 2c)
(Not any one of the six. sixth-roots of a -f 2c may be used indiffer-
ently in the denominator, but only any cube-root of whichever
equare-root of a-t-2c is used in the numerator. Thus if the radi-
278 APPENDIX.
cal sign be restricted to denote merely the aritlimetical root, if it-
be defined by the equation k^-k+l = 0, and if m and n indicate
any integers whatever, equal or unequal, the value of x may be
written *
{l-"'s/(a+2c) + k"'-^ v^(a-2c)}-4-2V(« + 2c).
2. 8x^-5x>/+d>/^=9{x-\-y) !•
llx^-8xii+5y^=lB{x+7/) , il'
1st Method. Eliminate {x+y).
.-. 104^2 _65.r//+397/2= 99x2 -72x!/ + 45;/».
.-. 5.r2-f 7a;// -6^/2=0,
.-. (5x-32/)(:/:+2»/) = 0,
.-. a: = §?/ or -2?/,
Substitute these values for a: in I.
.-. 72?/2 = 360.y or 45//2 = -9^
.-. ?/ = 0, or 5, or — ^,
anda; = 0, or 3, or f .
2nd Method. Take the sum of the products of I. and 11. by
arbitrary multipliers h and /,
k{Qx^- -5a;2/+3z/2)+^(llx2-8:r?/4-5?/2) = (9/,- + 130(x+?/). HI.
Determine h and /. so that the left-hand member of III. may,
like the right-hand member, be a multiple of x+y. This may
be done by putting x=-y in III. from which
16/C + 24^ = 0, .-. 2k=-dl
.«. if ^ = 3, /=-2.
Substituting these values in III., it becomes
2.^2 +a:7/-?/2 =«-+•?/
.-. {x-¥y){'ix-y)=x+v, or (.r+?/)(2ar-u-l) = 0.
.-. either x+y = 0, or 2a;-?/ - 1 = 0.
.. t/= -ic, or 2a;-l.
APPENDIX. 279
Substituting these values for y in I., it becomes
16^--' =0, 01- 10a;--' - 7a; + 3 = 27a;- 9,
.'. x = i), or 3, or | ;
and y = 0, or 5, or — ;^.
a;^ + ?/3 ~(^3 -j. ^3 ' 1.
a;* —x'^y + x^y^ — xy^-{-y*
I.~I1., .
{a-i + b^-y^-a'ib- '
x^y+xy^ a^b-\-ab^
• * (x^+ir)- - x^-y^ ~ (a3+/.3j2 _ ,,2^2
Write 2 for -;r-,^ — 5 and k for
z k 1
I^i- ••■ 1372 = 1313. ••• 2 = /'^or— ^
a;;/ ab ct" -^-b'-^
or i — >
v/(II.-3IV.) &nd^x-y=±{a-b),
M-+ab + b^
,*. 3;- ±a, +6 or
III.
a;;/ «6 a^ -{-b^
IL, .-. x^v = a5,or («2 + i2)-__X_ iv
n/(IL+IV.), .-. a;4-?/-+(a-fi)
orV(2a3_«5 + 263) y^^' +'^ -r^' .
280
APPENDIX.
y=+b; ±« or
.a"--{-ab + b^
Putz =
xy
1-z
l-^.
lU
lY
a
T
: %tz^-bz—{a-b)^0
: Aaz'^=b±:V{8a^ —S(ib + b-)^b-\-y say.
xy _ b+r
X' + ?/3
. x + y
4a
[2a 4-^ +
--Nl'ia-/;-
x-y
. x_ _ |/(2c/-f/>+r)-}-v/(2«-/'-r)
" ~y ~ ]/(2a + /;^+r)— v'(2rt-i — /•)
2(6 + r)
{x^^-y^--^1xy){x''-\-y^Y{{x^+y^)-xyY^a
Ua+2b + 2r ] I 4a \ ^ (ia-b-r^ 3
M'
b+r
I Aa
b + r
— a-
.1 0
X
1
II.
in.
IV
32u2{2a+b+r){4:a-b-ry) ^ ^
. (b+r)^ ' ~
X'
- [yl t32(2a + /'+r)(4«-/>-r)2|
JT/(2a + 6+r) + V^ (2<< - 6— r)} \'
102i(2.t-f A + '•)(•!'.< - i-r)2
APPENDIX. 2S1
_ V{''^'-i-\-lj + r)+ V{-2,a — b-r)
in wliich r^±s/{Qa--iDab-\-b-).
The value of y may be deriverl from that of x by the first form
in lY.
6. X'* =ax — bij, 1,
?/4 = ay - bx. II.
X.I. — 2/.II. x'^ —y^ =a(x^ — y^)
y.l.-x.Il. xy{x^-y^) = b(x^-y^),
.". either x — >j = 0 from which x = y = 0, or ^(a — 6) III.
or x^+x'^y-^x^y^ -\-xu^+y* = a{x-lry) IV.
and a?^(ie2-[-a;//+y^) = i(.«4-^) V.
(IV.+V.) (^^ + y)2^^'^+y2'^ = a.{.b VI.
V. » (a,^^J4_(^-2 4._y2)3^4/,(^^.y) VII.
l/(VII^ + 4.VI). (^+^)4 + (^'-'+^2).^2|(u; + y) VIII.
in which t= y{ia-\-bp -j-ib^}. IX.
i(m.+VIIL), .. ^x+y)^^{2h + t){x+u)
:.{x-{-y)^ = 2b+t
. :.(x+y) =f''{2b+t) X.
VL-.X^ •■-^+^^-rPT7) ' ^^-
o VT V / ^•^ 2(a + b) 2a — t
x-y =
y{2a - i)
282 APPENDIX.
in which t= V{'-i'^ + '^ah-\-5h^),
6. x^—G*'=m{x-\-y)*y
Letz= -, .'. z-{-l= audz-l = — ^ III.
x-y' ^ x-y x-y "^*
I. + 11. x^-\-y'^ = m{x-{-y)^-^n{x-y)'^
,'. (2 + l)4 + (z -1)4 = 16(W724+W)
{3 + ]/{9-(8/ra-l)(8TO-l)f ,
•*• * - >J . 8m- J ~ J^«
11. & III. (?-l)*(x--7/)4+lGc4-16w(.c-^)*
2c
•■•'^~^~i/li6/i-{2-i)4} V.
2f2
Kz+1) c(g+l)
' C(2-1)
and 2/ = 4^ ( i6« - (zHtTT' ^^^ *^® ^^^"® °^ * ^^ S^^®^ ''^^ ^^•
7. a;2 +2/2 = 1 (2»7.+n2),
and {x-\-y) 3 — o.(://(^- + ?/) =•. mn.
APPKNDTX. 283
Let u = x-\-y aud v = xy, and the equations become
U^ —3llV = 17171.
Eliminate r, .-. u^ — {2m+n^)u+2>nn = 0,
.'. w* - {2m+n^)u^ +2mn ti = 0,
.*. u^—%nu^-\-m^ =n'^u^ — 2innu-\-m'^ .
.'. u"^ —m = ±{uu—in),
.-. u = n, (the value it = 0 was introduced by the multipiica-
fcion by m),
or u^ -\-nii — 2m ~0,
,'. u=l{ ~n± \/(n^ 4- Fyni)}
.'. v=\{n^-m) or l{;j3+8/« + 3;?|/(w24-8w;}
.% tt and V are completely determined.
Also x-\-y = u, x—y=V(u^—4.v)
r
If ?« = 7 and n = 5, the above equations become
x-+y^=^13, and a;^ +^3 = 35.
Solving, as above, gives
u = 5, or 2, or —7,
2j; = 12, or -9, or 36,
.'. x+y=:5, or 2, or —7,
x-y=±l,or±V22, or ±ji/2n.
,:. a; .-=3, 2, |('2±s/22) or K-7+/\/2B);
_7 = 2,, 3, i(2+v/22) or i(-7q=,;V23).
284 APPENDIX.
8. x^-\-y = U;
« +2/** = -
4"
•■• H-^/=(f-?/')^
Testing tliis for rational linear factors it is easily reduced to
li/-l)nz/^+2z/+4-) = 0,
.-. 2/ = l or^(-2+v/2);
a; = i or i(-l±4V2).
;«. (2.r— 7/+z)0r+?/+z) = 9; , I.
{x + 2y-z){.r + >/+z) = l', n.
(a;+7/-2z)(x+^ + z) = 4. III.
Let s = x+!i+z and the equations may be written
(s + a:-2//)s = 9' IV.
(s-\-y-2z)s = l ' V.
(s-32)s = 4. VI.
IV. + 8.V. (4s +X+.V - 6z)s = 1 2, or (5s - 7z)s = 12 VIL
R VII-7.VL {(15s-21z)-(7s-21z)}s = 8,
.-. 8s2=8, .. s=±l.
Substituting in I, II. and III. they become
2x-y+z=±9, x+2y-z=±l, x-{-y-2z= ±4.,
.-. x= ±4, y= +2, z= ^1.
10 x^+y^ = a;
u^-{-v =b;
xy+uv = c;
xu + yv = e.
Let t = xy — nv.
:. {x + yY=a + c+t, .'. x = \{^{a-\-e^t)+y/{a^':-t)}
{x-y)''=u-c-t, y=lU/{a+c+t)- V(a-i-t)\
APPENDIX. 285
(u-\-vy = b + c-t. u = l{^/(b + c-t) + ^/(h-r-{.t)\
(u-v)^ = h-c-\.t, v=l{y'{b + c-t)~V{b-c-{-f)}
Also 2{xii +yv) = (a; 4- ?/)("+■'') + (•«— Z/)(« —■*-') = 2e,
••• ^^W'+<' + t){h+c-t)}+^/{a-c-t){b-r-\-t)}=2e,
.-. {4:e^-\-{a-c-t){b-c + t)-(a + c+f){b+c-t)}^ =
IGe^a-c-tXh-r + t).
,'. {(a-i)? + 4r3}i2-2(a2-i2)rt-f
(rt4_/,)3c2-4e2(a6+c3)+4^'4=0,
(a^-62)c-j-2cv/ [{ab - e^){{r, -by ~4:{c^ -e^)\]
•*• * = (a-Z))'-^+4^ " " ■
11. xy = uv I.
aj+y + w+t-^a II.
jc9^y3+u^ + v^=b^ III
a;S+y5^„6+^;5^c5 IV.
het x+y = k{a+z). :. u-^v = l{a-z). V.
Also let r = xy = uv VI.
.'. a(302+a2) = 4(fc3 + 3rt,.) VII.
Also (a; + //)-^ =^\+^'^+6u:.v(a;3+,y3)^10^2^2(a.^^)
.-. a(5z4 + 10a2z2 4a4) = 16{c5 + 5'j3,-4.io.ir2} Vlll.
Eliminating r between VII. and VIII,
45„224 _30a(fl3+2J-^)52_|_„6 _20a3/;3 _80/;6 + i44,,cS =0
.-. Ir./22_5(a3+2^j^)=±2v/{5(a34-5^3)2_i80,,c5} IX.
. a ^ 4 2b^±2y [^{(gs + 56^)2 - SQac^ }]
2= V- 3a
X.
286 APPENDIX.
VII. & IX. 12ar = a^ -U^ + Saz^
5(a^ - b^)±V{5{a^ + 5b^)^ -ISOac^}
''' '' ~ ^ 30a ^•
X. aud XI. give the values of z and r which may now be treated
as known iu V aud V.
^+l/~^(a+z), and xi/ = r
.-. x-y = ^v{{a+zy-16r}
x=^{a+z±V{.{a + z)Z-16r});
i/ = i(a+zqiV{(a+z)2-16r}).
The values of u and v may be obtained from those of x and y
respectively by changing z into - z.
Exercise.
1. 6{(7 -a;)3 +2/2} = 13(7 -x)?/, x^- +Ay = y^+4:.
2. 10x3-9?/3 = 2a;^ '8x^ -6y^ = ldx.
3. xy={S-x)^ = {2-y)K 4. x^ ^y^ = 8x + 9y = U4.
5. x2_|.^2=a;+y+12, a;?/+8 = 2(a:+y).
6. x+xy+y=5, x"" +xy + ij^ =7.
35 28
7. x^+y^ = 7xy==2S{x-^-y). 8. a;2+cc^ + ^/» =^^-^2 =— •
10. {x-]-7j){x"-+y'-) = 17xy, {x-y){x^ -y^)=9xy.
11. 25(a;3 +2/3) = 7(ic_f.^)3 = 175a;?/.
12. 2x^-y^ = U{x^-2tj^) = U{x-y).
13. 2a;2-3a;_y = 9(a;._3//), 3(a;2 -St/^) = 2(2a;2 - 3a;!/).
14. 2x^-xy + 5y^ = 10{x-\-y), x^+ixy+Sy^ = U{x+y).
15. (2x-32/)(3a;+42/) = 39(a;-2i/), {3x+2y){ix-Sy = {99(x- '2y)
APPENDIX. 287
16. {x-\-^y){x+^) = ^x-Vy), {%x^y){^x+y) = 1Q{x+y).
17. x+y = ^, a;4 + i/* = 706. 18. x+y = 5, .t^ + z/^ = 275.
19. x-\-y = % 13(a;S + ?/5) = 121(a;3+2/3).
20. a;+^ = 4, 41(a;3 +i/5) = 122(.c4 + ?/4).
21. x^ — 5xy-\-y^ +5 = 0, xy = x-\-y — 1.
22. x^+y = 5{x-y), x+y'^ ='l{x-y).
23. 3(x^+2/) = 3(.:c + //2) = 13.c//.
24. 10(a;3+//) = 10(.>;+.y-^) = 13^x--^+^3).
25. a;3+.v=y. ^J + y^^V- 26. 9(a;2 4-2/) = 3(z-+^3) = 7.
27. x4-^//+// = 5, a;3+.r//+//3 = 17.
28. .«+i/ = 2, (^-}-l)-'+(i/-2)5 = 211.
29. 3(.-i)(,+i) = 4(.+i)(,-i), ^;,::^:^i = ^(^._^;i)
1 •
80. x-+//= — , x-y = xy.
31. x--F// + 1^0, u;6+i/^+2 = 0.
32. .^+^ = 1, 3(u;«+^«) = 7.
33. 4.f//a=5(5-.«), 2(a;3+2/3) = 5.
34. 'ilxy = 17, 9{x^-\-y-^)=-8.
35. (.^.■•^+//^)^ + 4x2^^=5-12r/, y(aj3 +^2) 43 = 0.
36. x-\-y=--xy, x^+y" =x^-\-y^.
37. x^ Cxx-W{y'-x^) -16y^ = 9x^,
{x^~+-A)^=4.{2 + x^]/{y"+x"-)-y^-}.
38. x{y-^+Sy-l) = tiy^-\-2y-\-d, y(x-^ + ?> r -l) = 2x^ +2x+S.
288 ■ APPENDIX.
39. i^ , ^ ^ 2c3, ^ + 1- . .(-^ _ r\.
a^ ^ b^ a ^ b \a b j
«
40. x^+xi/^=a, y'^ -\-x^y = b.
41. x+y = a, J. ^ = c.
b-y X
42. a;^+«!/2='^, aa;2+^2 =(a3 -1)^.
a—1
43. a; + T/2=aa;, x^-^y = by. Ai. x -{■ y- = mj^ , x^+y = bxS.
45. a;* — y^=a'^(x—y}^, x^ -x^y-\-xy- —y^ = b^[x-\-y).
46. (a;+2/)(a;3 + 3?/3) = »i, (a;- ;/)(a;2+3//2) = n.'
47. x^y^ =y(a—x)^ =x{b — y)^.
48. x3(6-^) = 2/3(a-.c)^(a-a;)2(6-y)3.
49. a3(^t:+^2) = i2(^.+^)2^ a^-iy^+e2) = c^{x + y)^.
50. x3-2/3=a(a;2-2/3), a;3+^3 = 6(u;+«/).
51. x+y = a, x^-\-y^^bxy.
52 I— - I— = ^~^, ^(^1+^) ^ ^3
53. a;+?/ = a;2/ = a;2+2/3. 64. a;-2/= — = a;^ -^^
55. x^{l-¥y^)(l-Vy^)^a, x^{\-y^){l-y^)=h,
_^ a;'^+a;i/ + 2/2 x^+y"^ xy
Ob. =- = = —. —
x^-xy+y^ a b
57. x''y+xy^= -^^, x''y+xy^=b.
58. x^y-\-xy^=a{x^-+y^), x^y-xy^ = b(x^ -y'^),
59. (^ + JL\ix+y):=a, fl ,+ HI ^ t,
\y ' xj y i'-
AT^!- vilX. 289
60. x^ +.'/'^ = nx^]!^ = a;//(.r+?/).
61. abxii = (i {x •' + // 3 ) = A {x + // ) ^ .
62. xi/(x+y) = a, x^i/^{x^ +1/^) = h.
63. (1 ^ l)(x-3-yS)=.^, [1 _ lj(.rs+^3^==ft.
64. a;*+_?/4 =7?7(.r2+7/2), x^ -\-xy+y'^ '—n.
65. fl6(a; + y) = .r;/(a+i), a;2+v3=a2 + i^
66. x3+?/3 = a(a; + t/), .r4+?/* = 6f:r-l-?..)*
67. x^+v-=o., x^A-ir^=b(x^-^V^\,
68. xy = a, x^-^y^ =b(x^+'i/^).
69. (x-?/)(:r=5+?/3) = (a_j)(fl3+53)^ a;2_^2=(^2_^.j_
70. x^—y^ = o, x^-^7j^ = h{x-y).
71. a;+?/ = «, x*+//*=6. 72. a;+^y = a, a;^+i,'5=ft.
73. a;+2/ = o, x^-^y- =h^-x-y^.
74. a; + ?/ = a+^ (a-i)2(a;*+?/4) = fa;-7/)2(a4 + 64).
75. x^ry = a, c{x^->ry'^)=xy{.c^+y'').
76. (a;4-?/)3=rt(,i;2-f-?/2)^ a;.?/ = c(a;+i/).
77. x^y+xy^ =a^, c^(x^-\-y^) = x^y^. ^,
78. x^ = a{x^-\-y^) — cxy, y^ = c{x^-\-y^) — axy.
/ 1 1 \
79. a;2-?/2 = fl2. 3.3 _y3 ^^4 |____ 1 .
80. ic4-y''=«2a;?/, (a;- +?/2)3 = ^2(3.2 _2^2\_
81. [x+y)x^y'^ = a, x^-hy'^ = b.
82. {x+y)xy = a, x^+y^ -hxy.
290 APPENDIX.
83. x^-i y^ = a{x+y)'^, x^+y^ = b{x-\-y)^.
<
84. x*+x-y'^-\-y^=a, x^-xy+y^ = l.
x'^iX+x^y^) a 1 + xy
85. {x"-+>i^)xy = x^ ~2/^ yi{l+xy)^ =X ' r^y'
■j/(x^i(/^)— a;
86. x+y = {x-y)V.{^y), V(^5y7)+^ = «-
87. ^ + ^ = ~ i/{l-x)-V{l-y) = b.
88. «-+,'/-=«(»; + ?/), a;4 + 2/'^ = i(a3 + 2/3).
89. x3+y^=a, {x + y){x^ +y'^) = b{x^+i/^).
90. ('a;2 + (/2)(a;3+,^3) = rta;;/, (:«+?/)(«* +//'^ ) = ^'-^.V.
91. ix+yy'{x-'+y^) = a, {x^--Jry^)\x^+y '>') = (>,
92. (x-2/)(x2_^3)(a;4_j/4) = 4aa;y,
{x+y)(x^+y''^){x^ +y^) = K^ - ?/)•
93. x'^y +xy4 = a(x3^ -hxy^) = h{x'^ +*/*).
94. a{x'^ -f ?/5) = ai(a; +?/) = hxy{x^ +?/^)-
a;3 - y^ a^ - h^ x^ -y^ a^ - &*
97. .r^ = 2a.'<; - &?/, y^ = 2a?/ — /)»;.
98. {x+y){x^+y^}=^'', {x-y){x^ ■-y^) = h.
APPENDIX.
201
100. {x + y){x^-*- ij^) = axi/, {x — y){x^-y'^)=bxy.
101. (^ + 2/)(^2+i/3)-a(a;^+.V^j, ^x-y){x^-y^') = h{x^-^y-).
102 {^+y)^{x^+y^) ^\.,
[x^+xy + y-){:x'-\-y-)
'(x-y)^{x^~y^y ^ ^^
{x"'-xy+7j^){x-'+y-')
105. xy{x+y)[x^ + y^)=a, xy{x-y){x^ -y^) = b.
107. ^{x-x]f)^V{y-xy) = a, y(^x-x^)^V\y-y-') = h,
108. (.c4-l)(2/-l) = «(a;-l){2/ + l),
{xo + l){y-lY=b^{x-lY{y-^+l).
109. a;+t/ = rt, V(iC-^) + V(2/-'^) = <^.
110. x+y = a{l+xy), {x^yY = b^{l±x^y^),
111. a;+?/ = «(l+a;?/), a^^+y^ = 63(l+x'i/5).
112. (a;+l)(w-l)=a(a;-l)(y+l),
{x^-l){y-l) = b--{y^-l){x-l).
11^- (l-x)(l-y}-"' (l-a;3)(l-^3)
(c+a;)(c+2/)_ {c^^x^)(G^ + y^) _^ '
{x + m)iy + n) _ ' (a;^+m^)(y4 4. ?i4) ^ ^^
292
116.
APPENDIX.
119.
120.
121.
(a;+l)(y + l) a_ (a;^ + l)(y^ + l) ■'•
117 i'K- • -- / c/yx-rx-;
•'"'• :.(l + ^^)=«' ..3(^+^4) = *.
118. ,^±^ = « \^, yil+^±^)_^
414-y-)-''- x^(r+yio) = 6.
122 ^£±J()M±J:)__«'^+^^ :%2-fl)_a-6 •
(^-2/)(^^^^^ 2(<6 ' (/(a;2-lj=.7+d .
123. ^^-^^^^-^\-aHb--l) ^(i-^"-')-fe
124 (^+y)(l+'a-y)^ (a;^+2/3)(l + a:V)_.
125 (^+2/)li+^/)^ (a;3+y3)(i^^.3^3^
126 C-^' + ?^*)(l+^V) {x + y)(l + x>,)
127 (ii±jO(l+£//)^ (a5^+Z/^)(l+a;V)^/,
128 (^!±^^l^l+a:^+^^2/=')_
{^->jY{i-xyr~ ••-*•
129. x^ ^x-^u + oa^x-\-y"=0, y--x^~y^-2a'^r = 0.
APPENDIX.
293
130. 2x(y^-2x)^=a, yiif- -2x)-^{y^ -4:x) = h.
(Hence deduce the solution of x^ — 5x^-\-2 = 0).
131 2xij(x^+y ')''-= a, {x- -y-){x' -^y^)- =h.
132. v^x-+//2) + V{(a-x)2+^3|=V{(>V3-^)- + (ia-a;)^}.
Q{x- -//^) = «(6x-2^vo + -0-
ExEltClSE.
1. (2./-+7/-4z>(.r+7/+c) = 24, 2. x^-iz=l,
{x + 'hj -2z)(:c+.V+z) = 6, y^-xz = %
( — 2a; +3^H- 52)1^0; + 2/4- z) = 30, z^-xij = '6.
3. (x-+2//-83)(a;+ // + 2)-2(a;y+i/s+2.i-^ = - 12,
(2a; - %+2)(a;+.7 +z) + (a;(/ + 2/Z + 2./.) = Gl,
{;dx-y + '2z){x-Ty-tz) — o(xij+yz+zx) = 5.
4. a!2_^2,^^ 6. (a;5^^5^.2o)3_j_(a.^.,/)2^31.
a;+2/+z = 7, (a;^+^^4-2^)3 + u-+^./ + 2)3 =729.
x3+i/2+22 = 21. (a;4-i/;2-}-(a;+y/4-z)3---=31.
6. a;2-(,'z=0, 7. x+yz = U,
• a;+.y-f2 = 21, ^+za; = ll,
(a;-2/)2+(i/-s)2-i-(2-.tj2 = 126. z+xy = l(i,
8. a;+|/ = Sz, 9. a;+?/ = 5z,
a;=*+i/3 = 134.'23, x^+y''=S9z,
a;2+-y- +2^ = 134. a^ +^3 = 105^2.
10. x+y = 7z, 11. a;+?/ = 78,
a;2+^3=25z2, a;3+?/3 = 25z3,
a;4+7/* = 674z3. a;» + t/« = 20272z.
12. aj4-2/'.V+2:3+i<'-:^:^.<5» 13. '■x+y:,/-{-z:z-\-x::a:b:c,
{n + b+c}xyz = 2,, {a + b + c)xyz = 2{x+y-\-z:)
294 APPENDIX.
14. ax = b)/ — rz— — _j_ — -f. — 16. zl — _[_ —
Z X
— 1.
15. {x + y — z)x = a, ^Jx
{x — y+z)fi = b,
i-x+y^z,.=:c. ^(f + ~) ^'*
17. (y + z){1x-h!f-j-z) = a, 18. x{^j+z):y{z + x):z{x+y) =
[z+x){x+'Iy-'rz)=b, b-\-c:c-^a:a+b,
{x+y){x-{-y + 2z)=c, xy+yz+zx={a-^b+c)(x+y-rz).
19. {a + b)x+[b+c)y + {c + a)z = {a-^b+c){x + y-i-z),
a{x+y) = c{y+z),
{x + yy' + {y + zy- + {z+x)2=4:{a^^b-+c-^),
20. c{x+y)-^b{x-z}-a{y+z)=0,
b{x-z) = {a-c)y,
21. x-{-y—az = x — by-'rz= —cx + y-\-z = xyz.
22. {a+b+c)(x-y)+a{x+z)-b{y-\-z) = 0,
{a + b-\-c){x-z)+a{x + y)-c{y-\-z) = 0^
ax^ by^ cz^ ^
(6+c)3 "^ (c+aj3 ^ (a + by^
23. xy + — = a, yz + 1- = b, zx + ~ = c
z X y
24. y-\-z'.z-\-x:x+y::b+c:cJra:a^h,
{x+y-\-z){xyz) = {a+b-\-c){xy+yz+zx).
25. n^—yz = a, y^-xz = b, z^-xy = c.
26. a;2+(i/-«)3=a2, y^^^^.^^. =/,2^ z'^ +{x-y)^ =c».
APPENCIX. 295
28. x^+y^ —z^+Sxyz = a(x+y — z)
x^ —y^-{-z^ +Sxyz = b[x — y-\-z),
— x^ +]/^ +z''-{-3xyz= c{ — x+y -^z),
29. x+y+2az=--0, 30. x-^y-az = 0,
«"+?/" +2" = <-•". x^-\-^J^=c^.
31. a;0/-l)(2-l) = 2a, 32. x{y-l) = a{z-l),
a;3(y3_l)(23_l) = Cv22 a;3(2/3_i) = c3(z3_i).
33. x{y-l) = a{z-l), 34. a;(2/-l)=na(z- 1),
35. a;(2/-l) = fl(2-l), 36. {x-y^- =nz{x+y),
x^{y^-l) = h^{z"--l), {x^~y^=bz{x+y)^,
x^y^-l) = c^{z^ - 1). (x-y)^ = cz{x^+y^
87. x—y — af 88." a;-|-2/ = <*)
w — i; = 6, w + ?; = 6,
xy = uv, x^ + H^ =c^f
x^ —y^-\-u'^—v^ ^c^a + h). y--\-v^=e^.
89. xy = icv — a^, 40.. xy = uv = a^,
x+y+'u.-\-x = h, x-^y+u-\-v = h,
a;3 ^^3 4.,t3 _J.i;3 = c». a-4 +7/4 ^„4 _,_j,4 =(.4.
il. xy = uv = a^f 42. a;^ = w'y = a3,
296
APPENDIX.
43. xy = uv = a^, x-\-y-\-K-\-v = h, (.c + u)^ ->r{y + vY =c^.
44. xy = nv, 45. xy = uv,
x + y-}-ii-}-v = a, x-*-y-\-n + v = a,
X- +?/"+"- +r2 = h^, X^ 4.,y2 +„2 ^^,2 ^ ^,2^
x^ +2/^+"'^ H-*^^ =c^. X* -\-y^-Yu^-\-v^ =0"*.
46. xy = uv, 47. xy — ^tv,
x-\-y-\-n-\-v — a, x + y + u-\-r = n,
iC^4-?/^ +M^ +1-^ =C^. X^+y^ -f-W^ + f* =C*.
48. a;//-?<?; = 0, 49. a:2+?/2 = fl,3^
xii+yv — a- , ii--^v^ =71(1^^
iJC-\-y + v+v = h, iix+vy = c^f
x^-{-^^-\-7i^+v^ =c^. vx + uy = n^.
50. x+y+7( + v = a, 51. 7/{l+x^) = 2x,
xy + uv = b", ' u{l+y^) = 2y.
u^+v^=n^. x{l+v^) = 2v,
52. x + y-i-ii+v=:a, (x+y)" +{u+v)^ = b^,
53.
X la — u y 2b— u z ' 2e—u
y + z~ a — 2n z + x fc — 2i*' x+y~ c—2u
ANSWERS.
Exercise i.
1. 9. -P.O, 1, 0. 1206, -29, 1|. 2. -160,106,41,108
8> -^,i-h -25, 125, -.v, -31, -4^V0. -1- 4. 9,8
7,-^1^. 5. 176, 82, 254II-, -37-=-7^3. 6. 18 each.
7. 146,14,-72,-270,896. 8. Eacii = 0.^
Exercise ii.
1. -1. 2. -166542. 8. 100. 4. -2967511.
6. 968. 6. -162. 7. 10. 8. -8. 9. 0.
10. -20. 11. 706440254900. 12. 0 each. 18. Each 0.
Exercise iii.
1. 0, 16a4. 2. a, a^/B. 3. 2a, 0. 4. 26^«, -26«^
6. 0. 6. 4^7*. 7. 6a4. 8. #. 9. c. 10. 0.
11. a-f-(a+6). 12. a2c{b-h2c) -^b^. 13. a2+/,2_|_c2.
14. 0. 15. (12a2i-24rt^2_j.2863)-f-(3/>-a)3. 16. 0.
17. 0. 18. -b^c. 19, 20, 21 and 22, each 0,
25. 2{b-{-l)h, 4:x^. 82. <Z2=3i3. 33, i= y'^^ojg).
85. wr^ <;-+r')(r-r')-
Exercise iv.
1. 2{bx+cy). 2. 3(aa;-6?/).
3. a2{x-z)-ab{x-i/)-b^{y-z).
4. (a;+2/+2)(a + 6 + c). 5. (a + b+c){z^ +y^-i^z^).
6. 2(a;+?/+g)x{«-+63+c2-a&-6c-ca). 7. 0.
8. 2/'7;r4-i3/ + c?). 9. a^+^s+c^.
10. 2a;"(a-2fe). . 11. a-[-b-c.
11 ANSWERS.
Exercise v.
1. 2(r? + 9//4), 4rt2ft2. 4. 4(«.3-S3)2.
5. x^+4:x, -3ia;4-4a;3.72_|.3iy4. 6. a2.
8. a;2 -6.1-3 +9^4+ 2x// .-6x//3 -6a;3y + 18x3//3 +7/3 -6»/3 +9^/4.
■ 9. ixu{x^-y^), 2(1 + 12a;2 +16.1-4).
10. yV^^- 11- ^'2-2*2, 8«6(fl + &)2. 12. 2(a-c)(i-d),
13. ia;2+i//= + i2= + K.^7/ + Z/« + ;^a;). 15. (l+x--)^.
16. A(x;/+yz + zx)-2.{x^+y"+z^-). 17. .x^.
18. (a'-^+2Z.2_2c3)2. 19. 16a;2;v3. 20. -4rt&.
21. 4(« + A-fc)2. 23. 4(1+^2 +a;i+;c^^).
24. (rt^X-- +63^2)2.
Exercise vi.
1. l-4:x+10x^-20x^ + 25x^ -2ix- -f 16a-.^,
l-2x+3x^—4:x^ + 3x^-2x''+x^.
2. 1-4:X + 8.r3 - 14.c3 ^ 14.^*-8.r5 +o.x6 +6x'' +a;%
l_|.6:c + 15:c2 +20a^3 r l^x^ + (3.c^+x«.
3. 4a2-+??- + c4 +1 -4aZ/ -4af3 -4a + 2&c2 + 2/) + 2c3,
1 +a;2 + ,/2 ^23 _ 2^ 4- 2^ + 2z - 2;cy - 2xz+2y2,
kx^+y- + 362- - k^y + 6x2 - iyz,
4. a;6_2x-'^^ + 3x%3_4.,.3^34.3^3^4._2x7/5+;/6,
a2a;2 + 2ai;«3^.(2ac + &2)a;* + 2(rti + 6c)a;^4-(26(Z+r2)^8^
2cc/.«^+(i3a;8. 8. 3{a^ +b^ +c^')-2{ab+bc+ca).
11. 4rt.-+i?>2a;2 ^__i^c3x- +4:d-x'^ - 2abx — acx + 8adx+ ^icx* —
2Wa;2-c(ia;3.
Exercise vii.
1. (&2_63)2. 2. ix4 + ?/4. 3. rt4 + 3fi253+454.
4. a;4_.^4. 5, ^2^ 6. IGr,?. 7. U.
8. 4a4-9&4_16c*4.24^2c2. 9. 5* -9c2 -4n'2+12ac,
Qc^-4.a^-b^+iab. 10. a;.8 -^/S. 11. ajS+x^^^+t/'.
ANSWERS. "^
12. a-'-a^b^+b^-l. 14. x*+.v* + iV'Z/'-
15. a;S+2.c«+3.K'» + 2.^2+1.
16. 4:a^x^ -Aa^xy + a^i/^ -a^x^ -2a'''.}^y+2ax^ +^ax^y -x^.
20. (a:2+i/2-2x//-z2)3. ■ 21. x»-.v«.
22. i-6«2 + 27a4. 23. (w+^)3-(w+^)2.
2i. 2.i;-+a;^+2x°-a;«-l. 25. 6*8-^,16.
F.
XERCISE VIU.
1. x^+ix^+Sx^-2x-l-2, a;-+i/3-2.r// + 8.c2-8i/2+1622.
2. x^ + 12x--i-49.t-2 4-78u;+i0, x''+6.«3 _a3_|_3fli_2/;^
3. a3+8a6-10a4-104«3 + 105, a;^4-2.i;S -x^— 2.
4. a;44-5x3^^'- 12.^2//^ +5a;?/3+?/4.
a;2 1/2 2a; 2y
6. a;2»-2x"-a2-16fl-63, —+^+— + —-1.
(/- X- y ic
6. 7i2a;3_^2na;?/ + 7/2 + 10»a; + 102/ + 21.
7. (^. + a)3 + 27/(.i; + a)-3y/3. 8. u;4» + 2a;3» + a;2»(l-a-6) -
a.«(a+i)+a6. 9. ia-« -.C^/y^+i/* -a;^ + 22/a -8.
/I 1 \ 2 /I 1 > 5
11. a;4-8x3+19a;2— 12a;+2.
12. (u;+6)4-(a34-c3)(a;+^')^+»^<''^- 1^- «^+c<i.
Exercise ix.
1. 2(1+3j;4), ^xy^{dx^+x''y^). , 2. 96(a» + i2+a63),
b{21a^-21ab-i-lh^). 3. (a; + ?/)3. 4. Sa^. 5. Sx^.
6. 8a;3. 7. aS. 8. 27x3. 9. {2+x)^. 12. 8{x'+y-')^
14. (a3 4-i3)(;x3 + 2/3). 15. 0. 16. 0.
Exercise x.
1. l-3a;+6x^-7x^ + Gx^-dx^+x^, a^ -h^-c^ -3a^{b-\-c)
+ 3b^{a-c)-^Sc^{a-b)+6nbc, 1 - 6a:+21a;3 -oCx^ -h
lllx* - 174x-5 +219x« -204x^ +144x8 -64x'''.
IV ANSWERS.
2. ~(x^ 4- iar« + 27.r' + 2f)x<' - lix^ - 36a:* + 5a;' - Sx^ - 2). .
5. 0. 6. ■i5x«+16Ssx*//^-432a;3^22a. 7. (^ax + bu + cz)K
ExERcisF, xi.
1. a;«4-6^^?/+15a;*//3+20a;^2/^+15a;3y4 + 6a;y'+2/^,
a;^+7a;G?/+21a;-5?/2 + 35a;4?/3 + 35x3.y4+21.i-27y«+7ic?/«+i/%
a;8 +8x^v/+28j.-6^2 + 56s;5y 3 4. qOx^yi + 56a;3yfi4-28a;2^6 +
8^//' +;//», a;i 2 4.12x1 i^+66a;iOi/2 + 220u;'^//^ + 495a;Sy*-h
792x-^^^+924^6;y«4-792^"//'+&c.
2. The signs will be alternately positive and negative.
3. a^ - 5a^h + IQaH^ - lOa^b^ + 5ab^ -b^,
a* — 8a3^-[.2ia-62 — 32tt/;3 4-iG/>^, same as last, terms in
inverse order. 4. l + Giii-{-lo)tt--\-'2{)in^ + 15m*-[-6m^ ■+■
m^, )n^+5m'^ -f- lOm^ + lQm-+oin + 1, 6-i/M^ + 192;u--|-
240;«4 + 1G0m3 + GUw- + 12^ + 1. 5. 120.
6. x^ -4:X^y + &x'^r -ix^ij^+y^, a^ -10a462_f.40a3^4_
160a9^9+240a6/;i--192a3i;i5^ti4^is^
7. 49i5a8^*-792rt^i-'.
Exercise xiL
1. l+x3+a;*+-^•«+a;l^ 2. 1 +x + a:''+x3+a;4+.BG_|.a.7.|.
a;«+a;»+a;i«. 3. x4 + 2a;3-85a;2-86^+1680,
2^3 -3^6 +4x5+^4 4-^^ -2^- -x+2. 4. x^-57x^ +
266x2-1. 5. 18x8 4-21x' + 8a;6+a;5-f-63a;-^+96a;2 4.
43ic+6. 6. l-i^--ix-4. 7. 6a;i3_4^9_5^8_2a.7 +
9a;°-10x5+a;4-5x3+5x-2 4-:c-f-4. 8. x^ +dx- -irlQx+ll,
0 x4 + 3a;3. 10. x'^-'dx^. 11. a;4-f-8.c3_8a;.
12. (1), -1, (2), -1, (3) -4. 13. -1.
VNSWERS. y
Exercise xiii.
1. Sa-S-2a;*-4r+2. 2. 5x* - 4x^+Bx^ -2x+l.
5. a3+8„2^+8.,:^.3 4..,.3. 6. 4:x'' -\-8x-\-7, -13a; -20.
7. 10^-3 + 5a:-+l, 10.^; + 10. 8. x^-xy + T/''.
9. a;2-«3. 10. x^ + {l~a)x^ + {l-a+b)x^+(l-a]x-\-l.
11. 3a;3+2.c3+a;+Ur3i(a;+l). 12. 5a;2+13.ry+12(/3.
13. 6x^-x^-x^+x^-x+Q, -1.
iJ. 2xi-3a;3+4a;2_5^ + 6. 15. a + b. 16. «+!/+*,
17. 10x3, 10(a;4-20). 18. wrc^ 4.^^.2 _}.«.
19. i+x-oix^-dx^+Ox^. '20. 33. 21. -4.
22. -2U. 23. 15^4. 24. 85x+8. 25. 755
Exercise xjv.
1. ?/3-2y2-4i/-9, if?/ = »-l.
2. ?/3 + 3?/+5, if ?/ = a:+l- 3. y^ + Sl, i: y = x-2.
4. ^^4^1,^3 _432/3+92y_67, if7/ = a;+2.
5. 3.v''+30y4 4-ll9v/3 + 238^3 _|. 249,7 + 106, if 2/ = a;-2.
7. V^-'-iy+W^ i!iy = x-h
8. (x-2?/-'')-3//(a;-2y)--18i/3(a;-2r/)-24yS.
9. (^-//)^-107/2(a._,y)3_207/3(a:-2/)'2-10^4(a;-_j/).
10. (2a: + i/)3 + 2//2(2a;+2/) + %^-
11. 5l2(/3-3//--ri/4, ifi/ = ia;-TV-
12. ■*-24(/-^+492/-28, if2/ = ^j;+2.
Exercise xv.
1. a^b, +ab^-\-n^c+b^c + bc^+ac'>,
(a-6)3 + (i-c)2+(c-a)3, a{b-c) + b(c~a) + c(a-b),
ab{x-c)+bc{x — a)-^ac{x-b),
abc{a^b-\-a^c + b-c + ab^ -tac^-^bc^),
VI ANSWERS.
{a + h){c-a){c-b) + {b + c}{a-b){a - ^r ■■i-a){b-c){b - a).
(a+(;)--63 + (6+a)3 -c2+(c+&)^ -a^
a(b+c)^+b{c + ay+c{a + b)^.
2. cibc-{-bcd-\- cda-\-dab,
a^{b+c+d)+b^{c + d + r,)+c^[d-[-a + b) + '!"(a + h+c),
{a- b) +(a-c) + {a-d) -\-[b -o)-\-{h - d) + {c-d),
a^{a-b) + b^{b-c)rho-{c-d)+d2{d-a).
13. X and y. 14. ax and i?/, x, y, z. 15. / and h,
16. a; and y, also ic and — z, and y and —2. *
n. a, ^ and — e. ^18. ic^, — ?/- and s-. 19. 6 and c,
20. a and c. 21. a and 6. 22. a^ and 2a&.
23. a-o and aic. 24. a^/^^ abc. 25. a;^, x*?/, and x'^y-\
same ; a;^//, u;^?/-. 26. Not symmetrical.
28. uS u^o, a^bc, abed; u^, a'^u^. 29. a^, a^b.
Exercise xvi.
1. 4(a3 4./,2^c2). 2. 3(a24-62_|_c2)^2(ai+k+ca).
3. 4(a3 + fe2+c2+rZ3). 4. ^(a2+63+c3).
6. 4(x2+?/2_i.22+7t2). G. 2(a3+63 4-c3) + 62a3&-12a6c.
7. 14(a;3+2/3-fz3)+2(a;(/+?/3+2a;). 8. ^Ubcmnr.
9. 2ak(a+i+c). 10. a263+i2c2+c2a2.
Exercise xvii.
1. 115. 2. ?m»-3<7a2+3m-s. 3. 2. 4. -17-3533.
5. 1, 2(3a2+l). 6. 0 or 27/", 2?/", 0. 7. 36.
8. -(i2+a2)3_(3&2)3. 9. -15a*. 10. StiSSa^b*,
11. a2fe2(« + ^,), 12. 0. 13. 2a3-3a6(a-6),
263-i-6«6(a+Z/), 2(«3 + i3).
Exercise xx.
1. 3. 2. 1. 3. -l±2v/-2. 4. 2. 5. 36. 6. 11.
7. -l^f 13. ;^=-?, 9 = 6. ■ 14. /'=-46, 5 = 14.
ANSWERS. Vll
Exercise xxi.
1. ^=_3, r = 8?5, ^=-24. 2. c= -20i, f7=-13-^, ^=GOf.
8. /)= -3, (■= -10. 4. a = 3, /> = 0, ' = -57. 5. a=-2,
c = 24i, ^ = 0. 6. c=- 100^, (1 = 202^. 7. a = 200,
6= -810, c = 639. ^ 8. a = 4, c = — 27, d = 7, e = m.
9. 399. 10. x^-{p + -3)z^+{2p-{-g+3)x-{p + q-^r + l).
11. x3-(/?-3)a;2-(27J-g-3)a;-(^-g+r-l).
12. ra;3-(3r-g)u;2 + (3r-2g+7j)a;-(r-5+;>- 1).
13. x^-qx^+prx-r^. U. x^ -{p2-2q)x^ -t{(]' -2pr)x ~r-.
15. x^ — 2qx^+{pr+q^)x + r^—j)qr. 10. ra;^ — (y^y + 3r)a;2 +
(p3_2;>g + 3?-)a;-(/)?-r), 83. -i. 34. 1. 35. -1.
3G. 1. 87. -1. S8 -And Bd. a + b + c + d. 40.-1.
Exercise xxiii.
1. 5h*+15c^. 2. 6. 3. 3. 4.- {y}(?;+r+(0 + --- + --- + ...}.
5. 0. 6. 5i4_30fl/,3-f80rt3^a-5«36. 8.0. 9.0. 10.0. 11.1.
12. {a + b-{'r + d). 18. -1. 14. a + b+c + d.
15. (a+^>+c')(r/2 4-i3_^c"+rf& + ftc+m)+^//;6'.
16. {a + b+cy{a^+b^-{-c^) + 2abc{a-{-b + c). 17- a + 6 + r+(i.
18. (a + A + c+rf)2. 19. {a + b-{-c-{-d){{a-{-I) + c + d)^ -
(ab-{-ad+ac + bc + bd-\-cd)}+abcd. 20. a + 6H-c. 21. 3.
22. -1. 23. 0. 24. 0. 25. 0. 26. 1+lx-^x^ -{-T^t^x^.
27. l-ix-ix^-^'-sx"", 28. 1+x+x^+x^.
29. 1— 2a; + 3a;2-4a;3. 80. l+lx-^x^+^\x^.
Exercise xxiii. (a)
1. (p-pf+q)^ = {p + l){p2-pp'-q).
8. 9(p3 - .7)(r2 -g«) - (pr-t)^ = 9{3(^3 -^)(gr-p«) -
(j99-r)(pr-0}x{30)g-r)(r3-90-(??/--?;)(9r-;7/)}-
9. af{4:X^-[-Spx^ +Sqx+r) -r- (ic4+4i;a;3 + 6^a;2+4ra;+ij.
VIU ANSWRRS.
10. -ip, (ip)^~2(Gq), -(4:p)'^-^B(ip)(eq)-^(ir\
(4;^) 4 -4(4/>)--'(6y)+4(4p)(4r)+2(6^)3 -4^,
-(4/;)^ + 5(4p)3(6y)-o(4y>)2(4r)-5(4j9)(6?)2 + 5(4p)<+
5{6q){4r), (4p)6-6(4;j)4(6^) + 6(4,v)3(4r) + 9(4;^)2(G(/)2 -
6{Ap)H-12{Ap){Qq){4r)-2{6q)^ + 6{6q)t+S{4r)K
11. .Sp.s^ -4si.s3+a'?^, .f^Sp -e.Sj.s^.+lo-Sj.s^ — lOs^, where So, -"i,
&c., are the coefficients of the terms (taken iu order) of
the quotient in No. 10.
12. a;»(4,x3 _2Sx + l) -^ (x^^ -Ux'^-\-x-S8) ; .s, =0. Pj^SS, s,
= -3, .54 = 544, .s'5=-70, 6-g=8G83; 2(a-/.j4 = 4526,
S(r/-6)6=2G4122.
Exercise xxiv.
1. (3m4-2)-', (o--l)2. 2. (y3 _,^3)2, 4,^?(2.r-f-7/)3.
3. (3ai + 2c)2, 4?/3(3«-.7/)2 4. (^a;^ -47/^)3, (^a^ _i^2c2)2
5. (., + /' +c■)^(3.^•4-l,/3)2. 6. (2-a;+2/)3,|(-) -(--]} ,
7. (ar2-22)3. 8. {x-7j)K 9. (a + M-, (fi-'^—k-^)-.
10. (x-//)3. 11. 4(a:3 +7/2)2. 12. (x-+^)*.
15. («2_^3_c2)3. iG. (2rt-2c)3. 17. (2a2_3S + 4.)s.
Exercise xxv.
1. (7rr + 2/*)(7a-26). 2. {Sa+U){^a-U).
3. (3a-2i)(9.^3+4/;3)(;v, + 2&). 4. (10.r-6/y)(10.r+G.v).
5. 5i(a + 2x;/)(«-2x2/) 6. (3a;-^-4.y3)(3j534-4//2).
7. (ic + l)(fc-l). 8. (27/2-t:f2)(2.v= + §x^).
9. (3a-l)(3fl + l)(9r/3-f-l). 10. (rf-2h){a-\-2b){a2 + U^).
11. (a-Z-)(a + Z>)(a2+Z/3^(a4 + 64)(a8+&8^.
ANSWERS. IX
12. (a^h-c)(a-b-\-e). 13. {a + 2b -•Sx + 4:y)(n-}-2b-'^x + 4:y).
14. (a;3 -7/3)3 15. (^x+y-\-2z){x+y~-2z). 16. XG(.r+l)(l -a;).
17. {x+y+z){x+y-z)(z—x+y){z + x-y).
18. ixy{x + y){x-i/). 19. (a;— 24-?/)(a;-2— i/)(.c+2+2/)(x+z-?/).
20. 4(a + c)(i+^). 21. 24a;(H-2a;2). 22. 8«i(a + 6)2.
23. (a + b+c-\-il}{a+c-b-d){a-h-~c + d){a + b-c — d).
24. (a;+j/+2)(a;-?/— z)(x+^-z)(.r-.V-l-2).
25. 8a3Z)3^aG_3a3?>3 + 66). 26. {a^ + bS){a^ -b^)».
27. (a;2+2/2+«2)(^2_|.^2a-22--2x^-2?/z-22x-).
28. {x+2z){x-2y). 29. (« + i-c)(rt-6+6-)(i/ + c+«)(i + c-«).
30. (a;-2/ + z)(a;+2/-2)(d; + 2/-f2)(a;-</-2).
Exercise xxvi.
1. (a--7)r.r+2), (a;-7)(a;-2). (a;+4)(a; + 3).
2. {x-3)(x~5}, (a;-7)(a;-12), (a;- 12)(.r+5).
3. 2(2a:-5)(a; + 2), 3(3a:-20)(a;- 10).
4. -^(a;+l2)(ia;-3), 5(x+l){5x + S), {Sx^ -4)i3xS - 5).
o. (ia;+4)(ia;+3;, 4(4a;-5)(a;+l).
6. (x-a){x+a)ix-b)(x+b), {2(a:+?/)- 11} {2(a;-!-y) + 9}.
7. (a;^ +7/2 -a2)(a;2 +7/2+^2). s. {a-tb-3c){a-^b+c).
9. (x+y)(l-\-x+2/){x+y+(x-y)^}.
10. (a-{-b){l-a-b){a+b + (a-b)S}.
11. (a;^+x//+y-+2a;+2/)x{.c2+a;//+2/3-(a;4-2y)}.
12. (a-86 + 3c)(a + &-c). 13. (x--'+y2 + aS)2 ^b^ =Scc.
U. (a;- - 10.-c-l2)(a;^ - 10a;+8).
le. (a;2-14a;+10)(a;-9)(a:-5). 16. {x^--y^-)2,
17. (z+l)i2-l)(z2-2), (.«^'-3)(ar3 + lj,
ANSWEKS.
(3.c* + 5./y2)(3a:4-2?/2). 18, (L-'"-f 2)(c"' - 1),
19. {x"'-ufj{x"' + by"].
JdjXEEciSE xxvii.
1. {x-hy){l>x-v). 2. 3(ic+2//)(2.c-?/).
3. A[Ux-5y){x-y). 4. 4(14.c4-5^)(a; - i/).
5. (14.c-^v}(x— 20//). 6. 4(7ic-5//)(2a;-2/).
7. 2(28x + 7/)(x-10?/). 8. A{14.x-5y){.r^y).
9. (8x-5^)(7x-4y). 10. (8x+5//)(7«-4?/).
11. 2(3x+?/)(a;-3?/). 12. (3x-2//)(2a;+3?/).
13. 2(28a;+?/)(^c+10i/). 14. 2(28.6- -5//)(;c-- 2y).
15. 2(28x+5.v((.«-2//). 16. {5Qx-5y)[x-4:y).
17. 2(4,.r-2/)(7u;-10i/). 18. ^{l4.x-\-y){;£-5y).
19. 3(3.i;+^)(4.c-5i/), 20. (8u;+.jy)(9x' -8^).
ExEKcisE xxviii.
1. (5:i;-7)(2a:+3). 2. (6:c + 3)(2x-7). 3. (5x--3)(2.c+7).
4. (2a;-5)(3x-ll). 5. (4a + l)(3a-2). 6. (dx-T)>yix-o).
7. (8a; + 7)(4.c+3). 8 (.5.,3 _46-2)(3a2 + 56^').
9. (4a;+l)(3a;-l). 10. S^^^x— y/)(:J.c+2/y).
11. (2a; + 3!/)(2aT4-?/). 12. xH^b-^x){'lh--dx),
13. (3a;3 + 72/3)(2a;3-.5?/3). 14. (2a;3-9)(j;3 + 5).
15. (2a;+?/)(22;-j/)(a;-37/)(x+37/).
16. (2.c+4+y)(2x4-4-?/)(.c+2-3?/)(a;+2 + 3?/).
17. \mxy. 18. (19//2 + G0a;//-6a;3)(3oa;^-12x?/4-yU//3).
19. 2(4.i;y-3x3-32/3)(61a:2-49a://4-6l2/2).
20. 2(5^2 + 4.,-y + 10;/3)^x- + 10.cy + 2(/2).
ANSWEKS. Xi
Exercise xxix.
1. {7x-^ey + S)(x-y-z). 2. (5a;-5i/-22)(4.r4-?/ + 4).
3. (3x2 +4y3 + 13)(x2 -2/3-1). 4. (4a-_|_52^)(5a;-4?/-«-7\
5. (9x+8?/-20)(8a;-?/-l). 6. {x+Sy){x-i>/-5).
7. (4x + 3;/-z)(2x+3//+0). 8. (3a;-22/-2z)(2a;- S^z + Jz).
9. (3x2 — 27/2 +5z2)(2x3 _|_5y2 _ 5),
10. (15x-2+8//2 4-5z2)(a;a_2?/2+3a2).
11. (2a.-5i-7c)(2a + 3/i-f3c).
12. (rt - b-\-c){a + b- c){a + b + c){a-h - c).
EXEECISE XXX.
1. x-+-J+fi/5, 2a;2 + |±-||/5. 2. x^ +|^2-^|^/2 ^C ■
j\(6x2-r5y^±y^vlS). 3. i(4a;2 + 6± v/13),
T-V{6(x+y)2 + 523±a2yi3}.
4. (a:2 + iy2)(a;2_|.6^^3), (.^2 +ii,/5)(a;2 + |^2).
5. (2.f'^+4i^2)(2^3 + iy2), if4(V(+/;)2+5ci>/13}.
6. ^V(6^^+5^2)(6x3+ll//2), (6x2 4-5)(6x2 + ll).
7. J(5x2+10 + 3v/10), i2a2^3-t.2j/2).
8. {2(x+yi2 + (3±2v/2).s^} ;
-H 10x2 + (10±3v/10)!/2}{10x'^+ (20- 61/10)2/2 }.
9. J(9.c^+7+v/13), 4{2x2 + (0.L--i/16)(2/+2;^}.
10. ^(2x2+6± a/6), 1(7x2 + 20±v/; 85).
11. i{4x2+(9±y23)//3}.
12. i{7(^.-fe)3 + Sr2+rV29}, i{-)a^±b-]/n}.
13. i{3x2 + (3±v/3)^2|^ ^{3(^a-^by + {S±^/S){a^hy}.
14 {7a--'+(G±v/14)6-}, (5»i3+9/i2)(5m--}-3,i3).
Xii ANSWERS.
Exercise xxxi.
1. (x^±^xj/-\-^if9), (x^±xy-y''), {x^ +xj!-\-y^).'
2. (x^±2x]/ + 2y^), (4a;2±3x?/+?/2), (Xx2±xy+7/'').
5. (x"" +i/2x+l), {x''±VGxy+Sy^), {l±2y-Ay^).
4. {x^±Sx-\-l), x^±V6x + 3, ^,r2±2j-?/ + //3.
6. {2x^+y^±?ixy), (x^ +y^±ixyVS9), {2x- + l±2x).
7. (a:2'" + 8?/2'" + 4a;"'?/"'), x^'^ + 2y-^'"±:2x'"y"'),
(^a;2 _ |jy2 ^;;£c^/ V5).
8. (2d:2-l + 2a;), -(^icS _6?/2 +x?/) (^.7-2 -62/2 -a;y),
(«- +a^y^ + axyi/2).
9. mig^—ny^±xyyp), a;2"«-f 2'"-»y2m^-2'"?/'».
10. 4x3 -3±a;, 2a;2-2±2a;^/2,
-(3x2 -2i/2+a;y)(3a:2 - 2;y2 -xy).
11. 2^2 ±4.1-7/ -3?/ 2, a;2 + 22- + 5.
12. 2{a2 + ab + h^)^, {2a^-\-a + l)2.
13. {(a:+2/)2 + 3(x+2/>+z'}{(ic + y)^-3(x4-?/)~4z'}
14. {a + b)^+^^dz^G^V5.
15. {4«3 + 6a(6-c) + 2(fe-c)2}{4a3-5a(6-c) + 2(6-c)2}.
16. 4(a2 + 5«i-2/>2)(62_|_5«i_2a2).
17. {ix^+y^-xy)^±S{x^+y»-xy){x-[-y)
+ {x+y)^}.
19. (4a2 + 2a+l), a;2+7a;+4.
20 (a;2±9x// + 9//2), {l±:Sz+^z^).
21. 4(3a;2_2x+l;(a;2_2a;+3).
Exercise xxxii.
1. (a;2 + 3)(3: + .Sl(.r-l). 2. 2(.)-2-f 3)(r2 + .r -f?)
3. {x^ + A)(x+i){x-i). 4. (a;+2)(x-2j(3a;2+x+12).
ANSWERS. Xm
5. (a;2-B)(5.f^ +4.^+15). 6. {x^ ■\-G)(10x^ + ^x-G0).
7. (ix2+TV)(i^^+-^0-«-To)- 8. (5^2 _i)(5^2_ 8^+1).
9. (5i;2-8)(7ia;3-6x-12).
10. (3x2-4)(21.k2_13u;-28). 11. {18x^-\-l){A5x^ +^x + ^).
12. (ll.'«2 + l)(22a;2-8x-2). 13. (^^^ -f)(^a;--^+ia;-f§).
14. 8(;r2-2?/2)(10x2-4a;?/ + 20y3).
15. (2x^ -5?/^)(12a;2-6:r(/ + 30//2).
16. (x^-16y^){2x'- + l^y + ^^y^}-
17. (a;2-|)(llx2+10a;+Y)- 18. 10(.t2+2)(4x2 + 3.^-8).
19. (a;2-6//)3(13x3-12a;y + 78y3).
20. («2+4?/2)(3.,.2+3a;y_i2y2).
21. (x2-3?/2)(5a;2 +4x^ + 15^2).
22. 2(x2-2//2)(2a:2 -7x?/+2//2).
23. (a;2 4-i.!/2)(ic2-t-80xt/-^//2). 24. (x^ -6y^)(2x^--xy ^12ij"-).
Exercise xxxiii.
1. (x2+3./: + 27)(:«2_9.^+27). 2. x2 +a;(l±,/3)+4.
3. {.j;2+l+i(l+ a/5)4. 4. x^+l-x(2±^/5).
5. 2x2 + 2-3x±a;\/23. 6. (x^ + lSx- 5)(.x-2_a;_5).
7. (la;2-2)(4x2-6x-2). 8. {x^-}-8x+4:)l,x-\-3x+^).
9. (x--^+7x-2)(x2-a;-2).
10. (a;3+5x// + 3i/2)(a;2-a;2/+3y2).
11. (x2+.10x^l)(a;2 + 2a;-l).
12. ix^+l.ry + y^){x^ -Sxy+y^).
13. 2x^ + xy-5y^±xyi/4.6. 14. [x^-^.'Jxy -y"-){x^ -xy-y^),
15. a;2 + 2?/2+3.r?/±a;yi/3,
10. (3x2+10a;?/-2?/2)(3x2 -4xy-27/2).
17. T-V{lla;^+22//'-' -h5x(/±ffx!/v/ll}-
XIT ANSWERS.
Exercise xxxiv.
1. {y-z){T:^-y). 2. {hy + c){ax-^hy-c).
3. (z-+a){x-\-a){x-a). 4. (2a;-a)(a;- 2/*).
5. (.T+3a)(a; + 2^). 6. {x~b-^){x-a){x + a).
7. (a;-6)(a; + ?))(x-fl)(x2+^cG+a2). 8. (2a;-l-3«)(4.'c+5/;).
9. {a-\-hx){a-hx-\-cx'') 10. (a-&s)(« + ^^x+c.x-2).
11. (aa;-(^)(5a;3+rz-/). 12. {px-q^x"' -x-l).
13. („_6_c)(« + 2A + Br). 14. (ic + «)(x^ +x + 1).
15. (wx-n)(;9x2 4-(?a;-r). 16. (.'B-«)(.'c~/>)('a:- (■).
17. {x-\-<i)(x-h){x-c). 18 (xH-ri')(^4-/;)(>c-c).
10. {a^-{~z){x-ay){x,^ -y). 20. {abx+nly -c.fz)(ax+hy). ^
21. ('/ic + c)(^'a;3-/«+c). 22. (x-y){x-i-y){nix-iiy-\-rz).
23. (i»x-w?/)(rta;+/'// + ''z)- 24. (;/'a^ + «)(«.i-'-6ra-4-'f).
25. (c2-a;3)(/>2-|/2)(«--a://). 26. (x^ -jwSa;^ -«)(.r3 -« + »-M,
27. {l + x-x^){l-ax + hx^—cx^y^
28. (fla:-'/'/M^-«— ^.'A)("^+c?/). 20. (?7?.r+9)(;?a;4-")('"^-»- ")■•
30. {mx-\-Hij){:iix-ny}[p'x'^ +q-y-){x+l).
Exercise xxxv.
I (^a+x){a — b). 2. {ax-{-hy){bx ny).
3. {x-n){x+a){x- +ox + rr-), -L .r(n+x){a^ -^ax+x^).
5. (aa;-6)(ca; + f?). 6. (5a;2 - l)(5a;2 -.^ + 1).
7. (fl-/*)(rt + /^+.i--c). 8. (« 2 + 7,) („ + /;).
9. (x-?/)(a;+?/)^ 10. (a;-;y + l)(a;2 +^;_v-^-_y2^
11. (fc-2.r)(2 + 6.v). 12. (;r-l)(a; + 2)2.
13. {p-q){p--2g^)- 14. (a-l)(a2 + 2a + 2).
15. (a/.2_l)(3a/;2 + l). 16. ^y-l)'-^y + 2).
ANSWERS. X'V
17. (a^h)(2a2-nah-{.'lh^). 18. (Z^"'-l)(/>2'"4-2?>'"4-2).
19. (^" + z"j(y-'«_8v"z"+,2-"). 20. {u-b)(a^-i-ab-'db-),
^1. (a"'-c")(a"'-2c''). 22. (aa;-6)(a;- -aa;-/>),
.28. (5a;"-3rt2)(7:c" + 3rt2). 24. (a6+6c-c«)(aft-ic + 01).
25. (w-6)(w + M(a-»t). 26. {^ -Sa-){l-Sa){l + 3a).
27. (a;-y-z)(x3-2.c^-}-//2+2). 28. {6m-7n){4:m^ + n^).
29. (.6-"' +/)(.*;" +y"). yO. (.c-+.c// + ^;x-+(/2)(a;-^+sr^ -ax-y/)^.
Exercise xxxvi.
1. {x~y){x-{-7/)(x^+x;/ + !j^){x^-xy+!/'), {x -l){x-' +x + l),
{x+2}{x^-2x + 4:), (2a--Sx){ia'+(iax-\-dx-),
{2 + ax){4:-2ax+a-^x^).
2. {x-a^){x'^+.r'^a-+x2a'^-\-xa^+a^),
(3f(-4)(9aa-[- 12^ + 16), {a^ -b-^){a^ + b-'){a<^ + b^),
(a;3 - 2//)(u;» + 2x-«//+4.i;4^iJ + .-:Jx-' //3 + 16//4).
3. (rt-/0. 4. a:+4.v. 3. (.c+//){ic- +2/3)(u;4 + //^)
7. i'(a; ?/)(i/-hl). 8- {x-a)[x^+ax+u-)[a r b).
11. {a^+bc){a^-^a^bc + 7b2c-2).
12. (x-a + ^){(.r-(0^-(^-«> + ^^}-
18. (a;2 _2x-// + 4]ry3)(.c+2?/+4.r,y).
!4. {2x+S,j){2x-3y)^. 15. (1 - 2.r)(l+4.t3).
LG. (a-^ -^ubc + b^c^){a+be){a^ -abc + b^c-'i).
Exercise xxxvii.
1. d(x-\-y){y + z)(z + xJ. 2. {a-b){b-c){a-c).
3. 3(aa-/>2)(>-!-6-3)(c2-(,3). 4. ^^_j.^)^^_|.^)(2_j.a;).
XTl ANSWERS.
5. 8(r;f + /;)(/; + o)(c + a). 6. («+fc+c)(a-?v)(&-c)(c-a).
7. (a+//)(ft + c)(f+rt). 8. („3-Z»)Z.2_(.)(t.3_«).
9. (a + A)(/; + f)(c + a). 10. (a-6)(6-c)(c-a).
11. (.r3-f?/2)(//3 +23)^23 4.3,2).
12. («3 _f.52 .|. c^^ab~bc - ca)ia - b){b-c){c- a).
13. (a3 + />2^c2)(«4-& + c).
li. (C-i3)(a-c3)(6-a3). 15. (^2 _2/2)(y3 _22)(,c3 _2-2).
10. (x + ?/-f2:)(a;-//4-z)(.7-c+.'c)(z + 2/-«)-
17. (a-b){h-c){a-c). 18. 8(a + i4-c)3.
2i. (a-6)(6-c)(a-c)(rt2 + A3 + c2+<76-f-/;c+ca).
ExERcisK xxxviii.
1. (r,-2)(a2-7« + 2). 2. {x-2){x-S){x-'i).
3. (x-3)(a;-2)2. 4. (a;-2)3(.r + 4).
5. (x+l){x^ +2x-rS). 6. (a;2+2a; + 3)(x2+2x + 3).
7. (x+2)(a;-l)3. 8. {x^+2x+3)(x^ -2x+'d).
9. (»t-«)(m2-2wi»— 2?i2). 10. None.
11. (M-n)(»/-2u)3. . 12. (6+3e)(i2-26c + 13c2).
13. -(m-w)3(w2-wn + >(). 14. (rt+26)(fl— 26)(«3— 7ai+46-';
15. (x-5)(.-c-3)3. 16. (.«+2)(x3-|-3.c+l).
17. (r*-l)(a2_2a-195). 18. {p + 2){j)—l){p+4:).
19. (a-l)2(«+2)(a+3). 20. (fl.3"-l)(a=''-2)(a2"-3).
21. rt2 + 463+7a&. 22. (a-i)2(a3+2a6 + 263).
23. {jj-2){2J^-2j>+-2). 24. (.c"-l)(a;'-» + ua;" +5).
25. (y/-2)(,)/5 -37/2 +2^+4). 26. None.
27. (fl-/^)('t*- + 2ai+362). 28. (a''+l)(2«3»_3rt» + 2).
29. (x-2)(x-3){x-6){x-7). 30. {x-t/){x-2y){x-3i/)^.
ANSV/F.R3. XTU
Exercise xxxix.
1. 2(x-l){x"--9x-\-10), {x-2i/)^x-3y).
2. (ix + 3i/)(3x'-i-x!/ + y''), (x-l)(4c-2)(2x+3).
3. {x-5ci){3x^+a''), {2x + oy){x^-{-3xi/-y^).
4. {b+c){b-4:c){2h^~bc-\-c2}, (5a+4i;)(3a3+76i//-3&3).
5. {2p+q){2iJ + 3q){p^+q-^).
6. (lOx- 9v)(15^+ 16//}(x-3 - 5xtj-\-8y2).
7. (2./:-3//){2x--f3//){3a;+47/)(3a;-5?/).
8. (5x - 2z)<^2.cS - 3^2^ + B^^a _^ 12^3^.
Exercise xl.
4. (^+2^)(a;2+Hv-). 5. l-2x-+3a;2.
6. {a~x)ia-\-x)^. 7. a;^ +z/^ +s3 +ie^+7/2-2x.
8. [a + />){da+b}. 'J. (a;-2/)(2.c + 3^).
10. aa_/,2_f.^3. 11. 7a2-Sab+2b2.
12. a -7. 13. {a-h)(b~c){a-c).
14. (x-a)2-ft(.a-_^,)-f/.2. 15. a;2_j_^2_j_22 ^1,
16. x{x--ax+b). 17. rc3 + ^3.
18. {x-y){x'+y^). Id. a^-b^+c'^+l.
20. a3-63_c3, 21. 'i-i-x. 22. (c-i)(a-f-6 + r).
23. ab — ca — hr. 24:. x^+y^-\-l-xy'\-x+y.
25. (x3-2)(a; + l). 26. a^+5a-\-B,
27. (2.<;-t/)a2-(x- + //)a.7:+x3. 28. a(a;2+a;+l)-(a;4-l).
Exercise xli.
1. j:2-3. 2. x-\-o. 3. a;3-.T+l 4. aa;^'4 6a^4-c
5. None. 6. c^+c*. 7. (a-i)(.cf«). 8. 6(a;+3/).
XVm ANSWERS.
9. (a-b){b-c)(c-a), 10. «.2'"+l.
12. 5(a-b){b-c)(G-a). 13. {y-l)(x-l).
15. (x+l)(x~ + l){x~l)^. 16. {x+l){x+2){x+'6)(x yi). 17.5
18. Same as given quantity. 25. (a — 6j((!>-c)(c-a).
29. a;^+a;2+2./;-M.
Exercise xlii.
1. (.r-l)--(a;2+4x- + 16), a;(3^-7)-^y{77/-3).
2. (^^-ax^a^)-^[x--a^), (x + 4)-^U- 1)^.
8. (a;-l)(,x+2)-^(a;2-f5.*; + 5), ix^ + 2xyd)~{x^--2x~3).
4. l-(6-2./:), l^(ar3_2.<;4-2).
5. 5a^[a + x)-ir-x{a'-'+ax-rX-^}, {4:X- +l^^{5x^ -i-x+1).
6. (a;-7/)-^(;c-L//). 7. [Sax^ -hl)~{ia^x'^-\-2ax^ -1),
[ax^bi/)-^{''X — bf/). 8. —1-T-abc.
9. -(.,+/'+')-H(a-6)(6-f)(c-«). 11. 5-7(»2 + a;2/+?/3)
Exercise xliii.
1. (4-.r)--(5-a-), [a^ +h-)^2ab.
2. X, 2a-(«--M). 3. r^(l+rO--(l+2r,+.3rt2), X.
4. b^-i-a^, {h + l)-^cib^. 5. («c--M)-=-(ac+^)c^), ^n-a.
6. l + 6x27/z(7/+2j--{y-3--s:3(v/ + 2!)3}. 7. (a^ +63 ^c2}-v-a6c.
8. 1. 9. -0/^+a^6^+o4)^rt6(a_6)2.
/I— a;\ 2
10. {a + b + cy-i-2bc. 11. |j^l , 4a2a;2H-(a2+a;2).
12. (a.-+2/)-j-{a;-y). ' 13. (a-6)S-=-(a + 6)3.
14. (a; + ?/)^(.<;-2/) 15. l^a;3,
m. 1--W. 1/. ±(l-i)-T-(l + o). 18. l-»-c
ANSWEBS. XlX
Exercise xliv.
1. (x-a)-i-5. 2. a+b. 3. 16a^x^ (a* -a:"^)*.
4. 0. 5. 1h-(5:+2). (y. lH-(n4-a;4).
7. 12^?/^(9.r2-4y3). 8. (4^2 +2) --.-'•; 10x4-1).
9. l~-{x + l)(x+2){x + B). 10, 4(a;4 + 4^2y2+^4j_j_(.c4__;y4),
11. (^a-b)3-i-{x+a)2{x+b)". ' 12. 2,',- --a;.
13. mG-77x)^18{llx- 8). 14. l-^(a-6).
15. 15a(8a-«)-^(9a4-2a:)(« + 3a;). •
16, {10x~7)-^{x~l)['2x-5)-l-^{-2x-l',.--4:). 17. Z.
18. y^y^'-K"). 19. (fl-/>r"+2.
20. 0. 21. 4.c^~ix'2_i).
Exercise xlv.
1. a;-y. 2. a + h. 3. 0. 4. 0. 5. 0,
6.- {{a-\-h){c+a)x^-\.2iah + >' +ca)ax-%i''hc] ^ •
(fl + //)(rY+r;)(x + a)x(a; + />)i>+c.) 7. 1.
8. «+i+c. 9. 1. 10. a;3-.v3 11. Q.
12. (ff-fc)(/>-c)(a,-c)-i-(a+6)(6+c)(c+«).
18. a;2^(a;-fl)(a;-6)(a;-o). 14. 1. IC. 0.
16. {h{;x + a-b)-\-ax] ^ {rt64-(o-rt)(x— o)}.
Exercise xlvii.
1. («-i,)2+4c.2^0. 2 8. 8. 10. 4. rt2+i».
5. 7«. = 2, ?i=l. 6. 2a;2, or 5. 7. 7?7=-5. 7? = 6.
8. +12, 9. (a2+/;3l(c2+d2). H. _ 3?;C - 4^2 4./,2^2 _ 4^2.
12. (a;2-4a; + 3)(x2-4), also (a;^ -3x-4-2)(.r2 -« -6).
13. i(-l±i/5). 15. a-^c = d-^ ^e-, a^b^p ^e^.
XX ANSWERS.
b^c = d^ ^p. 17. ac^ = b^d and 9ad = be.
19. ip^ + 27q = 0. 24. p = 27n^q±2mqV{m^-^l'}.
26. 4(jt)-3)=2.
Exercise xlix.
1. 5, 31, a, -3. 2. -4i, -a, 2, 10.
8. a + ^, c-rt, 6-c, 3. 4. -2,6, -5,12.
5. -14, a -3b, 2a- 3b, 5b -3a. 6. 7, 4, a, 6.
7. ic, 5-=- a, 0, 1. 8. -1, {(a + b)^-a}-i-b, a+b.
9. (/>-«), a + i. 10. 1-^a-h, l~(a-b}, l^(a-' + &£).
11. 2b, a. 12. a + 6, c-^(rt + 6), b-^{a-c).
13. (i-c)H-(«-6), i + c. 14. a+b, a^-^ab + b^.
15. «2_a&+/,2^ 1. 16. -1, (n-{-b)^{u-b).
17. (e + i)(e_/,), 2^15, 3---14. 18. -1-12, b~-ac, a-^b.
19. («-+;>2)-^rt-7>2)-^a2/ja, ^(63-|-c3J^5p,
20. 10, 12, 4, 4. 21. 1000, f, f. 22. 9/^, ab, bc^a.
23. h^^ac, c{n-\-b), b{a + b)^a.
24. a-f-i, {a-b)-^(a + b), -{a+b)^^{a-h)^.
25. -1, -1. 26. (a2_c2)--(a+6)2, 2, 3^.
27. ab, b^a, nc^b, 12. 28. 12, -ac-^6.
29. 9, 2. 30. 12, 1. 31. 3, 1. 82. (2fl -l)(2r, + 2), 0.
33. l-!-m. 34. 1. 35. {ab ^-bc-\-ca)---{a^ +bc+c-^).
36. {a^Jrh'+c-)^{ahJ^hc-\-ca). 37. a + b+c. 38. 1.
39, 1. 40. 1. 41. 1. 42. 15. 43. 16^. 44. 6.
45. 5. 46. {npqa+j)qb + qc + d)-^vmpq. 47. — ^.
48. 0. 49. -25H-136. 50. 1.
answers. xxi
Exercise 1.
1. 2, 3. 2. *, h 3. +2, li. 4. 1, H. 5. ±f,
d=(a4-*), «• 6. 4, S, 2, 2i. 7. -8 or 2 ; 4, -3 ;
2i, -li. 8. 1; for-l; i 01-3. 9. -f or |, ^ or 6;
4 or -|. 10. -1, 2, -i, 1. 11. 0, -Z*, 3ft.
12. a, +av'-l. 13. 1; l( — l+v/5). 14. ±rt.
15. +6c, -(/)+c). 16. rt + 2Z>. 17. 6or+rt.
18. -2ai, iab{lztVl). 19. «, /;, - (a+i). 20. a, b.
21. aorl-rt. 22. -a, -b, a-2b.
.2a a, 6, /;(1 -/;)^(l+«.-6). 2-1. a;^ -6.c2 - 37.1- + 210.
25. a;4-4fla;3-13rt2a;2+64rt3a;-48a4.
26. a;(cc-l)(a;H-2)(a;-4)=0. 27. a;* -4A-3+a;2 + 6x-f-2 = 0.
Exercise li.
«
31
1. 4. 2. -7f 3. -107. 4. 8. 5. 3./. 6
7. 50|o, 17. 8. 22, 46^. 9. 7, 3. 10, 10, 10, 11.
11. 0 or 11; 33. 12.3956-^3971. 13. i(15+s/190).
14. 3. 15. 3. 16. 4. 17. If. 18. l^-. 19. 3^.
20 4. 21. +3. 22. 11. 23. 2 aud -1±:|/ -3.
24. 2^. 25. 0. 26. 3a. 27. S- 28. \§. 29. 3,
80. 10. 31. 0, 1, or (-5±a/-23)-^8. 82. 102|.
33. (-ll±i/4681)-^20. 34. 2, i, |. 35. --4
86. 0 or ±:y (fl2+62).
Exercise lii.
1. {l-a)^(l+r7), rt(m + l)^(/»,-l), h(m-^l)^a{m-l).
2. a -6, 0, 0. 3. h, ma^h, h-^ca. 4. 1, -1, 0.
5. -|or-l. 6. (c-/;)(ft2_f_c2):^2a6c.* 8. 14, 4^.
9. 2; 6^295. 10. 73 -=-210, {a -^b+c + d) ^(m+n).
XXU ANSWERS.
11. h-^a. 12. h^a. 13. a or 0. ,
14. ±:|/a2+l-j-2. 15. |. 16. ||. 17. 0 or 4.
18. c-^ab. 19. 83|/(2a;-l) = 100A/(3x-3).
20. 75 --52. 21. 8. 22. 34^^?-
23. l^«(n-l). 24. ac-^{h-a). 25-. 4, 3i or 13^.
26. a2j2^(rt_6)2^ 3 27. 4a2^(l+a)2, 6(a + 6)2-7-(a-6)2.
28. (1+62) ^2a6. £9. ■,/(!-«) = 2 ^ (a + 1)3.
30. -a + aV{(l + & + i2)-j-26}.
U-1/ - U-1/
5
ExEEcisE liii.
1. 8. 2. 0. 3. 3. 4. (v/m+|/n)».
5. ^;,^(1-2n/&). 6. 4-f-7. 7. l-^(a-2).
8. 18962^12393. 9. V^-^ ( v/^v + 2).
10. (c4-2/;c2)-T-(2c-2_26). 11. i. 12. 18a.
13. .r2 = 80-^81. 14. ±V°n/^t- 15. +Av/-ll.
16.
hs/ !«=- l^^-^^^^'U 17. 0. 18. ^.
19. (c-a-6)-''' = 27a?'c. 20. a;2 =a2(«-l)2 s- (2?i-l).
21. 16.r/y=(»-4x--2/)2. 22. 0, -ff
23. / ""^ -1) ^ 0. 24. 2s/(l -w2) -i-m ^(4 -m2).
\2rt — 2 /
25. (rt3_i)|,,2+2+ V(a^ + l)}H-a2.'
26. (m-an+c)2H-/;(n-l)2. 27. ±5.
28. 2V(3x2+*10) = (17i/17-3^3)---7. 29. ±5.
80. ±i/(36-2a). 81 |/|(a2_fe2).
ANSWERS. XXm
32. {2i/ + 2z-2x)^-{-2Wx/jz^0. 33. |«-^/6.
34. a(^^2-4/t+_8) - (2«-4). 35. a^ +2a,
36. ±|/(3ft2+i2)^^3.
Exercise liv.
3. {(a-&)«2_2c(«2+a^ + 62)|^|„2_2t.(a3_/,3^j_
4. —6. 5. a + 6-fc. 6. ab-r-{b — a).
7. a;2 — 3«a;-<*2=0, &c. 8. a. 9. K« + ^ + <') 10. I^a6c.
11. l3;(a + 6+f). 12. {a-b)[ac-2b)-^{a + byiG. 13. -c.
14. (ifa4-^/>)3. 15. ±2. 16. c^{a-b).
17. {a-b)^{a + b). 18. |a. 19. ±2. 20. ±2, &c.
21. i(a + c)-^(«-c). 22. a, {Sab-Bb'' -a)^{l-\-'da~db).
23. «. 24. a, b, 2b. 25. a, {c- .+ Qab)^Gb. 26. i(c + 6rt).
27." ia. 28. r( + 6. 29. (a6 + k;+ca)H-(« + 64-c).
30. ±b, ±a. 31. |/{l-=-(a 1)}.
32. {6{a-b)-4:c{c~b)}-^{4:C-db-a).
33. (c-2-a/.)^(a + 6-2(;) 34. -^(- 29+ v/37).
35. (a; + a)2=2i3-a3. 36. v/(^^2._|«i), 37. ^(&- a).
38. 8h I- 89. x--6x = a. 40. 1±]/19. 41. Z*, b~a.
42. , («3 + 63)^(a + 6). 43. a;=-5-^2. 44. -1(5+ a/3).
45. -2a, -^a, |a. 46. -3a.
Exercise Iv.
1. bc-i-(a + c). 2. (a3 + 6_2a5)-j.(a + ^>2).
3. (arf-66)-^(«-i). 6. -^;X7- C- ^- 7. ^(a+6)
8. a + 6. 9. 0. 10. 0. 11. abc.
12. («2+62+c3)--(a + fc + c). 13. (a + & + c)-(a2 + 63+c2).
AXi^' ANSWEES.
a^b ,^ ah
14. {a^^h'-^-\-c^)M'tb + bc + ca) 15.^3^. 16. ^^•
17. 4*. 19. 4. 20. -140. 21. 17. 22. 10. 23. a.
24. ^!!^'7^l,. 25. 3i,0. 26.3^. 27. («6-6-2)--(« + Z>).
r^< (lb
28. -i, «, 29. 0, 0. 30. U'l-^b-c). 31. —-,.
32. d. 33. fl6--(a3-62). 84. -3|. 35. f. 36. -3§.
37. Infinity. 38. 10. 39. abc-^{ah + hc + m).
40. {ab-\-hc + ca-ad — bd-cd)^(a+b-{-c-Bd).
41. a{b + c)^-^(b^ + rJ-ab-]-bc-ca).
42. Z*c{f/-«) + (a-6)(&-c)(c-rf)-=-(«tH-ic4f'?-'^'d-&---c2)-
43. bc^—b^c-ac'^-\-b^d-abd+cicd^{ah-^bc-ac-h^).
44. _(fl_j_&.4.(.). 45. rt + /) + c. 46. {ab + bc + ca)-^abc.
47. _i(i4.c). 48. («^ + c)--2«. 49.9. 50.2. 51.7.
52. 4. 53. r,S(5±^/7S5). 54. 4, (am.-nb)-^{n-'m + a-~b).
56. ^Ij, b{a + o)-^{cr--]-ab-\-b^). 56. 0, -|, *. 57. 10
n(i{ap-\-i)ih) — wp{cq-\-nd)
58. («prig - cmjpq)-^{apn^ +cqn,2); ^^^^^^2^,nnH ^^n^f^^^^m^d'
59. rt6-(6-c), c{a2 + (6-0«-ic} + a(62_f2)^ .
((j2^i2_c24.rtJ_5c — rtc). 60. b-^{u + b),
61. mpcq-]-apnq-~{Cjtn^ —cqrn^).
02. {&m(a— c)4-CT«(6— rO + f(/)(c-5)}-^
{m((a-c)+n(i-a)+y>(6--i)}. 68. (a2-|-,62)^,,6, 0, ^.
d{u-q)-q{h-d)
64. (a;.-cm)H-(«7i-im), ^,, _ ^) _ ,,,(^ _ ^y ^ 55. i, 3.
66. 100. 67. 13, 111. 68. 11, 7.
a-2 +2ac+«rf+26c + 2«fe
69. (rt + ft-m-n). 70. ^3-^ ; •
ANSWERS. XXV
71. ^^^h) + i/{^(a-hy-^c^\,aoYb. 72.0. 73. rt + 6 + c.
74. ^^1!+!!^+^^ 75. a + fc + c. 76. a + b + c.
abc
77. (a/* + /;c+r'^)H-(a4-?' + c). 78. b^+a^-c^.
79. c-a-6. 80. 0. 81. 0 or 11.
ExEECiSE Ivi.
1. ^ = 0, orB = 0. 2. A = 0, or 5 = 0, or C'=sO.
3. x-0, or(i—b = 0. 4. x = 0, or;y = 0.
5. In the first case either .r — 5?/ = 0, or x— 4y + 3 = 0, in the
second case both conditions hold. 6. x = 0, or x~a.
7. x = 0, orx=—b. 8. x = a, ora;-- c^-&.
9. a? = 0, ora: = 3. 10, x = 0, or a- = a + h.
U. 3; = 0, or «=+«. / 12. x = 0, or x^b^-^ a.
13. :c=:0, orx = a. 14. x = 0, or a.
15. x = 0, or ic=rt + &. 16. a; = 0, ora + i^.
17. -{2ab)-^{<i + h). . 18. a: = rt, or 6.
19. « = a, or6. ore. 20. 5. 21. 1. 22. 21.
23. x = \, « = 3. 24. a; = 9, .r = 4.
25. x=l, orB. 26. (a/O ^ («' + ^').
27. x = a,orb. 28. a; = (a^ +/,2) h- («+i), x = /v+a.
'J.9. (2r/.ft)-i-(rt + '0- ^^- « = «, or^.
1. a;=i, or (l+a)-f-(l -a). 32. x = a.
34. a; = a - 6, or i(54-c). 35. x = «+??, or i(rt + c).
36. X = — " — , orl. 37. a-^b-c,
a^rb-\-c,
38. x = rt, or ^(46-a). 39. x= -c, or a^h + c.
XXVI
ANSWERS.
40. x = lf or
m—n
41. X
_ nc—pb
n—p
42. P{'i—^)-<^{m-'>^)
)n{c ~ b) — a[n — p)
•1:4. x = 2a — b, or 36 — 2a.
17(7-76
46. « = « + /', or
mc — ap
43. a; = i(a+6), or ^(6-4
45. x = a-\-c — b, ora; =
4a+4c-26
47. x = 4:a + h, ov a + b.
48. a;
8
• ' 0 a
- — , or_
b — c 0
49. {a - h){b - c]x^ - (^2 4-^3 ji^c- -ab-bc- ca)x+
{n-c){a-h)z=Q.
50. x=±d, or ±2. 51. a;=±G, or +2.
a-\-b
52. a; = 3, ori. 53 a; = ^^, or ^
a
r. 54. x — -r-, or
a
55. x = b — 2a, or a — 26.
• 2rt + 36 Sa + 26
56. a; = — ^ , or
5 ' "^ 5
57. x = {tnb+na)-i-{m + n), or (wa-w6)-7-(m+n).
58. a;=v/{(»i + 2/i)- A/(w-2«)}^l/{(m-|-2;i)+ i/(»i-2n)}.
59. a
[v/(c+i)+ A/rc-1))
60. a{y(3c-2) + i/(2-c)}H-{i/(3c-2)-|/(2-c)}.
61. a{v/(2c-l) + l}-=-{l-'/(2c-l)}.
62. H«+26). 63. i(a+6)-(a-6)v'(7n-2w)-=-y(w+2«)
64.
66. «
26
, or — 2 • ^^' 2a6-^(a+6).
3a 4- 56 86— 5a
8
-, or
8
67. « = 2a{v'(c+4)-i/(c-4)}--{v/(c + 4)+ \/(c-4)}.
ANSWERS, XXVll
68. ir = 4, 01-3.
69. ^{«±:|/a3-4w) where m = «2+^\/(''+«*); 3 or 1.
70. a, b. 71. x = ^[a + b±i/{{a + b)2—4.{ab-\-t)}],
72. a; = 0, or a, or ■^rt(ldr-i/-3) ; a; = 4, or 2.
73. x = 0, or «+&, or ^{{a+b)±lV{a-b)^-4:ab}.
74. a;2-(a-6)x+a6 = &c. 75. x= ^(3a-fc). or ^(3fe-a).
76. a; = 3a— 2&, or 3i-2a.
77. y^ -m^ =0, where y — m—x and 2m = a+b. See Key.
78. 2/^-»*^=0- 79. ^2_^2=o. 80. y^-m^ = 0.
81. i/^-m3=0. 82. t/3-m3=o. 83. ?/3-w»=0.
84. (?/2-^3)(52/2^.7;^.2)^0, (where also k=i(a-b),
85. k^-y^ = c. 86. A.- + 10^3y2^5^^4 = c(^4_y4)&c.
87. s?/ihA:N/(A; — 3c±r) = 0, where s^^g^ + c, and r2 = (A;— 3c)*
4-(^_c)(3/.-+6-). 88. -3±:a/(9 + 12/24).
89 — 102. Work with a variable w such that wx = x^+l.
89. w = {a±s)^b, where s = a^ + 262.
90. «j = (3a+26±i)-=-2(a-6) where s= ±:6/(a3+2a6+462).
91. M; = (3±:s)^(l±i>) where s = (6 — 4a) -^6.
92. (M; + l)3=a-^(«-i). 93. Mj3==2a-^-(Z»-«).
94. {x+l)~{x-l) = a^{a-%h).
95. (w + 2)-^(w-2)=^(l + sJ where s=(16rt+6)H-6.
96. «;3(4a-6)-^-(a-6). 97. ■«;3 = (4a-36)H-(a-fc).
98. M) = (6±s)H-2a where s^ = 63 -(-16a2.
99. u; = (a + i»±s)-i-2(a — 6) where s = (a + 6)3 +8(^a -6)..
•XXVlll A>'SWERS.
100. Wz^(a-\-b±s)-^2{a-b) where s^-^{a-b)^ =
101. (it;-f2)it(t«-2)=+s-=-(4+3s) wheres2 = 2a^(a + 6).
102. {w-{-2)-i-iv=±]/{5a-i-{a+U)}.
103. i{2a + b), i(a + 26j. 104. 2a-6, |(a+6), &c.
105. 1, 2, 4, 5. • 106. ±1, 2, 4. 107. 1, 2, 3, 4.
108. -^, -1 1, f. 109. -1, 3, 4. 110. -a, o^t, 5a,
111. 15, 20. 112. 2ia. 113. 4, -1.
114. 7, —1. 115. ^(6c-^-a + 6•a-^ft + rt/;^c). 116. ±a^m, &c.
117. 26(s-a)(s-6)(«-(-)^V^{s'2_a2j(s'3-6)(.s-'2_c2j} where
2s = a + />+c, 26i=a3+i2 4-c;
118. (2«6+2ac2 + 26c8-a3-6'--c4)H-4c2. ll'j. a, b, ^{a + b).
120. ±a or ±JaV^' 121. a, 6. i(ct+6).
Exercise ivii.
2. a;, 2; ?/, 1. 3. x, 8 ; y, 1.
5. X, — lOi; «/, 51 6. a;, -2; t/, ^.
8. x, -2; 2/, -3. 9. X, -f; 2/, i.
11. aj, 12; y,S. 12. x, 8 ; ^, -9.
14. X, 12; t/, 15. 15. x, 18; 2/, 13.
17. x,l\ y, 9. 18. x,l; y, -3.
20. a;, 2 ; y, 3. 21. «, 3 ; y, 4.
23. X. -3; y,\. 24. «, 12 ; //, 15
26. X, 8 ; y, 9. 27. a;, 3 ; y, 1.
29. a;, 11; ;/, 7. 30. x, 17; y, 13.
32. a;, -4^; ^, -|«. 33. a:, 13 ; y, 10.
84. a;, 4|;y, 3tV ^S- x,ll',y,Q. 36. x, 7 ; y, 5.
1.
X, 7 ; y, 9.
4.
x, 9 ; y, 5.
7.
«, -1; 2/, 1-
10.
X, -i; ^, f.
13.
«, 10 ; y, 12.
16.
X. -3 ; y, -2.
19.
a;, 7 : y, 3.
22.
«. r ; 3'» ^•
25.
»' Tff ; 2/' A-
28.
a;, 7 ; ?/, 8.
31.
a;, 5; y, -4.
ANSWERS. XXIX
37. x,2; y, 3. 38. a;, 5 ; y, 3. 89. Equations
40. x,d\ y, 1. 41. x,l;y,5. not independent.
42. x = 0 = y = 0. 43. 0,0. 44. a; = Oor 13 ; y/ = Oor ^f.
45. X, 17; y, 20; z, 5. 46. a.-, ^„ y, \%^, z, f|^.
47. 11,7,9. 48. 21,22,23. 49. -15,-0,-8.
50. 3, 4, 5. 51. 12, 15, 10„ 62. 5, 3, 1.
53. f, 1^, |. 54, 3, 5, 7. 55. 11, 13, 17.
56. 5, 3, 1. 67. 9, 7, 3. 58. Ik, 8i, 9^.
59. 3f, 2|, 1|. 60. 2-3, 3-4, 4-5. 61. 30, 20, 70.
62. 88 H- 59, 1098 h- 59, 1004 ^ 69. 63. 30, 12, 70.
64. 6, 12, 20. 65. 5, 2, 0. 66. 1, 1, 1.
67. 11, 9, 7. 68. 6, 3, 1. 69. 2, 3, 1.
70. 3, 4, 5. 71. h h i- 72. 5, 4, 3.
73. 7, 3, 1. 74. 2, 3, 1. 75. 1, 3, 6.
76. 0,1,2. 77. 1755-698, 360 -f 319, -15705^698.
78. ^,i, 1. 79. 5,4,1,3. 80. 4f , 3^*, 2^, U-
81. 31, 41, 51, 21. 82. 7, 4^ 4, 8i.
83. 20,10,0,30. 84. 11 --24, J-, 1 ^24, ^.
85. 270 -- 117, -52-- 117, 15-^-117, -126--117.
86. Eaok 210.
Exercise Iviii.
1. [a'c—ac')-i-{a'b — ah'). 2. b(cn- dm)-^(ad—bc).
3. b{d-c){d-a)^d{b-c)(b-a), c{d-a){d-b)-^d{c-a){c~b).
4. ?/ = cz+f^'^ + ^M^ + ^aJ) 2 = rfM-j-eif'-|-aa;+«t/,
u = ew+ax-\-by + cz, iv = ax + by-{-cz-{-du.
5. a; = ^m(a — ^4-c), &c. 6. x= {j:>(a3 — 6) — ?n(aA — l)-j-
XXX ANSWERS.
n(b^ -a)'^{nS + b^ — Sah + l}, &c.
7. x={l — am + abn — abc27-\-abcdr)-^{l-\-abcdf), &e.
8. l=a-^{l-i-a) + b-rr{l-{-b)-{-c-i-[i+c).
9. l=-ab + bc + ca-\-2abc,
ExEEasE lix.
1. (nc — bd)^(na — bm), {'mc — ad)-~-{mb — na).
2. (??a+6d)-r(aw + 6?n), (mc — acZ)^(6w+(m).
3. c{n — b)-i-{an — mb), c{jin — a)^{hm — am).
4. (6 — c)a-^(/>— «), Z/(a — c)-i-(rt — i). 5. a5-T-(a+5), t/, same.
6. a62-^(a2 + '^3)^ a,n^{a'^+b^). 7. ac-^(a4-6j, 6c-r;a + /:')
8. {a- —b')-T-{am — bnfy {b^ —a^)-T-{bm—an).
9. « + 6 — c, c+a— &. 10. a + c. b+c.
11. a(c»— <f?n)-j-(ii — ac), b{cn — dm)-T-{ad — bc).
12. t/={93(a2_c2)_o(6+2a)}^{(a-Z>)'^-i-s + 4ic}.
13. a + 6-c. a-b + c. 14. a+6-6-, c+a — 6.
15. (TO-a)(?i — a)-5-(6-a), &c. 16. l-*-(a-^)(a-c), &c.
17. (Hi-ic)(?--«)-^(c— a)(a — 6), &c.
18. x=p-i-{pl+mq-r'iir)+a, BO y and g,
19. p{l — (?rt4-w6 + wc)}-^jp^+m3+nr)+«, &0.
20. {m^ + '2a^ -b- -c^)-^Sa, &c. 21. y = a — 6-fc, &c.
22. x= {ab+bc-\-ca){b+c-2.a){2b-a~ c)-i-{{a- c){b +c ^ 2a)-{-
(^_c)(2&-a — c)}. Corrected equation, a; = i(i4-c), &c.
23. nui-^{a-\-b+c), &C, 24. npr-^{anp'^bmp-^cmq).
25. l-=-(6-c), &c. 26. -^(i+c— a), &c.
27. a«H-rf, &c 28. «=l-^(«+6-c).
29. a+fe, &c 80. l-^2a, &c.
ANSWERS. IXXl
31. (mS+nS-ZSI-^-Smw, &c. S2. i{a + c-b), &G,
83. l{m'^ +n^)-i-'imn, &G. 84. l-=-(ft + c- a), &c.,
85. bc-^[b-\-c), &c. 36. /> + c-a, &c.^
37. a, 6, c. 38. ^>^-c3,&c.
89. i(a4-26-c + 3fZ), &c. 40. i(4a+^ + 3c-2(/ + 5«).
Exercise li.
1. ,, + ?,. a-h. 2. i>.2+;,), ^(^2-6).
3. {8;^2+m2)-^-5m, (2^2 -w3)^5?n.
4, a_^(a_&), />-^(rt4.6). 5. l~{a~h), l-4-(a + 6),
8. a-f-6-c, a-h+c. 7. a + 6-c, o-fe + c
8. (a2+fi/j+&^)-^(« + &), (a2-a/j+6-')-4-(a-&).
9. (a&-l)^(«-l)f^-l), {a-Z;)--(a-l)(6-l}.
10. (l+a)--(aft-l), {l+h)-^{ab-l).
11. (« + l)(^' + l)-5-(«^-l). {a-b)-^{ab-l).
12. a(a+fc), b(a-b). 13. a{6(a4-o)-c(a-c)}-4-(a3 -6c),
a{6(a-6) + c(a+c)}-j-(a2_6c). 14. -(a + 6). at.
15. i(^+c), &c. 16. (a-26+3c)-f-38, &c.
17. 2-T-(/>+c), &o. 18. fl + 6, &c. (by symmetry).
19. b^-c^, &c. 20. b-2-c^, &e.
21. ia&c, (l-a)(l~^)(l-c), (2~«)(2-6)(2-cy,
22. 2a&c-4-(a&+fec — ca). 28. 1, 1, 1.
24. ar-{ma + 7ib+pc-\-qd), &c.
Id \ Id^ dn
25. w = 0, or(— -l)-(^-^|-
26. (5-|.c-fl)-f-(rT + 6+c), 2/ = (6-c-a)-«-(a-*-«).
27. ^(a— 6+w-w), &0,
XXZll ANSWEB9.
28. (4:a + 2c—d—Bb), y+z by symmetry-
29. —{a-b-\-c-{-d}, {nb-\-bc, &.c.), -{abd-\-&c.), abed.
80. ^(a — 6 + e — rf-}-e), others by symmetry.
31. x = (a — l.b-\- Imc — Imnd + Imnpc) — (1 + Imnpq), the others bj*
symmetry. ' 32. a; = 6 + c— <?, &c.
34. j^ = (a+56 + 3c-7rf+9e)H-22, &c.
35. z = ^(a+c), then symmetry. 36. 2 = -J+^ + e, &c.
• 87. a5 = a— 2fe+3c — 2(i+e, then by symmetry.
Exercise Ixi.
1. a;=(2a^ + a+fe+r)-=-2(a-/») where r = 4rt(&2-j.i + i) _j.
{da-h){U-a\
2. x=(ar + l)^(fflr-l) where r« = (J2 - 1) H-3(a3 -62)
3. a;={v/(l + a)(l+6)-i/(l-«)(l-6)}-5-
{l/(l+a)(l + ?.)+v'(l-a)(l-fe)}.
{^/(a + 6 + c)(« + ^>-c)-i/...[.
6. {a-\-b)^{l-ah). 7. x=(a^-ai)s-(a^+6a)
8. /^+^\ * _ («+»0(^>-f-n) ^
\a; — 1/ " {a-~ni)[b — n)' '
9. a;={aT/(l-6^)-6v/(l-a3)-^ ^(^S-js).
10. x={b + c-a) ~ {(Vb-^c-a)(c-\-a-b){a + b-c).]
11. x = {b + c)^^(a + b){b-^c){c+a).
12. aj= VC^+t + c) -j- rt, &c.
13. a; = {62.^.^2 _^(t+c)} - i/2(a3 + 6»-f-c» - 8«5c).
14. (fc+^-a), &n. 16. a or (a^— 6j -j- (i_ai).
ANSWERS. ZXXIU
x-y= V(a-i-b)(a — 2b) -^ V{a — h), &c.
17. ix + ,jr^ = l[3b- ^^, {x-yy- = il^^ - b'j .
18. (x ■}-]/) H- (x — 7/) = -/(«+ 36) -j- i/{a — b) = m suppose.
19. y^ = m -j- iam^ — 9??.+ 1) where m =
20. a, 6. 21. a;={^(a-c) + |/c[y-- {i/(a-c)-y/c}.
22. «= (a-|-c)?/ -;- (a — c), &c. •
23. x+y = {ab-l)^{a-b), &,G.
24. a;2/={v/(o + 2)-i/(Z»-2)}^-,/{(?<4-2)-i/(&-2)} =
p suppose, x-^y= {]/«4-2) + y («-2)}-j-
{s/(a+2)-T,/(a-2)}.
25. x'1j^z'^ = \{a + h-c)[h-\-c-n){c^a-b), &c.
26. x= (ai — 6c — ca)-^2s/a?y(;. 27. a — x'^ = ±m, inhere w is
the value of t; in the equation 4m — 4(r +</)»-+ ^f* =»
(ca - ai - 6c) 2 4- 46(ca -ab- bc)v + 462^3 .
/I 1 \
28. a; = i|/(<5i6c) \-t--\ I , y and z by symmetry,
29- a(624.c^)^(52_,_c2)_^(J3_}_c2)^ ^c.
30. c{V{a-\-b) + i/{a-b)}^Via + b).
81. 0 or a ib + c)^2bc, &c.
32. »= -1 or ff/(a«-l)-^v''(ft'« — 6'), &e.
xxxiv answers.
-Examination Papers.
I.
1. (a + h + r)(x + y+z).
3. x^-6x^ + ldx^-12x+i, x^+dx-^ + d,
X'
nln-
• 1) J.;^"!"— 3)_l.a;«(»— 3)
4. 2aj2'"-ix3m A_ — _ — . 5. -jhl4|/ - 19-^-27, 0 or ^.
'* 0 c a '
6. 16, 7. 199, 8, AB 37, CA 52, BG 45.
II.
1. {a + h]^. 2. (a+h + c)(a^ + J,S-^c^ -^ahc). 8. l-^3;3.
4. (a2'"4-2rt'" + 2)^(a"' + 2), (<(+i-|-c)-^(rt-6-c)•
7. -I, 4 or 6. 8. p'^q^{r-pq).
9. 584. 10. (am + b)i)^ {a -\-b), {by common rule).
III.
2. ,/4-fca;+ca;2, 3 — 4a;-h7a;- -10a;3
3. 2i ; 6, 9, 12. 4. 160 eggs.
5. 40, 35. 6. /|(l+«)+i/Kl-«).
7. 5 or J; 4. 9. (rt+6+c)(a— 6)(6-c)(a-c.
IV.
1. (a + h + c)9. 2. a-;-6. 3. (4a;2 -92/2)(4a;«— 4?/5). 1 + Va;.
4. L^3-|v^6, 17 5.4. 6.1. 7. Cor 4; 8,1.
8. 7i, 12. 9. 4 or 6. 10. i(-3± ^ -39).
V.
1. 8x3 + 1^125. 3. IGa^b^. 4. b-^8a. 5. 15, 12.
6. ia, ia; 4 or -9, 7. 6, 7, 8, or -6, -7, -*8.
ANSWERS. XXXV
Vi.
1. (r,3-f-ft2). 2. a^ -\-b^ + e* + ah + ac -he.
3. .c'+r±%,xy, {lx+6y-9){x-y-{-4:). 4. -20, 0.
5. borl^i. 7. x- = 4 01-9, i/ = 9 or4; 1, 2, 3;
a; = i(_7 + y^33). 8. -1, 0, 1, or 5, 6, 7, or -V. -4. ?•
9. xix'^d)[x^-tix-l) = 0. 10. x-^//-3^4.
vn.
2. --01. 4. a;2+x-l.
5. ^36x--^+l8a:+9)--fl6a;4-81), (u;3 -a/yj--(x^ +«6).
6. 9; 3i .t; = -^(a-Z*).
VIII.
3. (»3_y3)2(a;_y); a=y;4.ry, i=i?g.
5. <'.6-3 + ^/x + ca;-^ ; ^a;* + i^" - i«
6. x^+px+p''; 50{x+o){x-4:}{.c-6).
7. a=0 = ior«.= l, ^1 = 2.
8. 3 or — 43-J-7 satisfies the equation 2 - y . . &c.
IX.
4. axij+b. 7. +^/a6; A/(a4-6)-f-i/(«-6), andt/ = reciprocal
of this. 8. P(22a-216)--20a(a-6). 9. 7, 15, 48.
X.
1. aa;3 4-%3+2ca;?/. B. x-2. 4. 24. .
.5. 5x-^-Qux+ia^;y+-^-Vi- 7. 3;*. 8.4,6.
XI.
1. 2. 2.6,8. 8. (4x*-9i/V,TV^^+-5%*.
4_ ^3_^3 4.9a:-S x'^ -{a-\-b)x-c. 6. 0. ^
7. Oor -2i; i/ = di3or ±|/-9, &c.
XXXVi ANSWERS.
XII.
« «•
6. m^ —vi=p, q = 0.
7. a;2=::(a_2)-^(«4-4), y/iV= {b{n~ I) - cn\ -i- y'a[n-y}
a a6-{(a-l))i-l)-l}. 9. 87. '
XIII.
1, 2. 3. 14-rja;+Coa;^4-&c., where c^, Cg. <^c., repveseut tbt
combinations of rt|, (''21 • ■ ■ taken one, two, &c.. at a time,
4. §4000. 5. {m{b'c — bG')—n{ca'~c'a)}-i-{ah' - a'b),
V=P""c^' 7 (« + /3f+(ar-/i)'-2/ = 0. 9.66.
XIV.
1. 12rtie. i. Smiles. 5. «tm-i-/i; a;+^ = ±5 or ±1,
X - ^ = ± 1 or ± 5. 6. ■ 20.
XV.
2. A. 3. c2(r2-6d!) + ci!2(62_^,c)+a/r(rt(i-ftc). 4. 2^(OT-n).
If m— w is negative u; is neg. -wbicli shows that they were
together before noon. If m-n = Q, x is infinite, i.e., they
are never together.
5. x^(x^ -r,2)(x-2a) ; (x^ - a^)(;£^ ~ y^} .
6. (rt+6+c+3d)-=-(a-}-6 + t-l-c/): {x + y-^z)-^{x-y^z);
3a-T-(a + i).
XVI.
4. 2x* -3a; 3+ 4a; +3. 5. See paper XIX., prob,, 4.
1 i 1
& i{l±l/(«" + 4//)H-\/4ft". 8. (i,Ci-^C2)--(ai63-fl2&j)
AJ^SWEES.
xxxvu
1.
8.
1.
7.
1.
4.
6.
4.
5.
7.
8.
1.
4.
XVII.
8a^-2ab-10ac-Sb^+2c^ + 5bc. 6. ?/-»i^ = 4.
(w. + n){bq—pG)^niq -pn), {p-\-q)(j)ic - bn)-i-{mq —pn).
XVIII.
l-aj»«. 2.1. 4:. x-a-b. 6. {x- y){x-z).
1. 8. 9a; ±:1-=-a/2 or ±1^6. 9. 10,11.
XIX.
i(4a;-« + l)8. 3. 5;c2_i. (3.3+^^^4.^2)3.
(u + 2)-'^4(a2+a> 7. (76'-^ -a-')^12fl. 8. 18, 22, 50.
XX.
{l+w.)x+(l-n)y. 3. a,-2_i.
{a^b-2ab^ - a^ -^ab^ + b^)-r-{a^ -b^).
1. 6. a>6'-6c') + /*^(ac'-a'6-)4-c"(^'6-«ft') = 0.
l/{^*^
3m
; 2, 3, 4.
10-
a{b-c)-T-{b-a}, b{c—a)^{b-a). 9. 3. 10. 2000.
XXI.
8. 2. 24a&c; 2a^ +4:Sa^x^ + Q2a^x'^+A4:a^x^ + lSx».
0; rj7+16. 5. 16; x+^Jx^ -2a. 7. 3377 oz. of gold,
783 oz. of silver. 9. - (8±4-i/3)h-(3±2 \/3 ;
a;==fc2 or %/— 1, y= +1 or q:2|/-l ; 8, 4.
y = cost of 2iid bale = 60 ± 20 \/ 7.
XXXVUl
ANSWEBS.
XXII.
1. -02997, a^+aq+p^. 2. ab^'~h^+c = 0,
{a).{a-b){8a-'db). {b).x{x- l){u—b){b - c){(( - c).
8. b^=A'ia. (a), (a + b)^. 5. (a) {ax-bi/)^{i:x + by)
7. (a) U(^ + b + c), {b), f, -I, 2.
(c). b-c,c-a,a-b, (d). -1±V2.
XXIII.
1. ^x' + 4.r*-.T3 + ia;3 + ^^3^ + (95a;3+21a;3 - 40a:+42)--
(3x4-21a;3-j-9a;-6); -382. -4. x = a + 'Ic, y = b-[-Sc.
6. (2). ^. 7. (1). a+i + c. (2), 1, 2, 3, 4. (3j, 0 or ^.
8 5'j(^2_i(3yi-|-2). 11.
ayb^c^^d-^'
a^b.^c.-^d.^
a^b„c^,l.,
= 0.
12. Coll. to Newmarket 63 miles.
XXIV.
1. i^^x-y){y-z){z-x). 2. -d-rti—n; 0. 3. 5, 3, 4^.
4. (1). A = bc^2a, £ = ac~2b, C = ab-ir-2c; (2). a^+b-^=c^.
11^ y=+2, x=±5,&c.; x'-'-lOx=-lQov -16,
1 i 4pq
:. a; = 8or 3, &c.; (6 ' ±a )^
XXV.
1 %
1. (a2-3x)-^(a2-x2); 1. 2. — - — -i2(a;+a)H-a=a;3.
5. ^•2-2.<; = 2, a; = 3, y/ = 2; z/ = 53--24, &c. 6. 4 miles, 3 do.
7. (a;»-l)H-(x3-l).*;'
.»-i
8. fl-IP-fl+f- '■>■■'■
ANSWERS.
xxxix
1 1 1
10 1+ -j-+Y-2~^r'2^'^^^-~^''^'^^'^^ approximately;
{(5x)>3'"^^}{l-6-ll ... (5r-4)}^|jl. 11. f and -^.
XXVI.
i. a-^{nh-a^)', 4. 3. 2, i, or ^(-8± '/S) ; x-\-y-^z—
v/(a3+2^*2), .■.2 = o+i/(rt3+2?>2), &c.;|(_4±i/76). ■
4. 3. 7. )t>3H-4+r3-f-27 = 0. 10. (l+a:)-^(l-a;)*-
a;"-l{3.«-l+2«(l-a;)}-^-(l-a;)2 ; n-^(15u+9j, yV-
xxvn.
1. {Tm-nltY-. ' 2. (a + t+^)(rt-ft)(6-c)(a-c). "
3. ia-b){h~c)-\-{b~c)(c-a)-^(c-a){a-b); {a-b)^ +
(6-c)2 + (c-a)2. 4. a; = a^(«2+62 4.c2), &c.
5. N/a6; 18or-2; i + Sj. 7. i(« + />)&c.;
a;={l+rt2_63 + i/(l-a-6)(l-a + ?>)(l + a-i)(l+a+i)}
-i-2a; x-i-y = {l+n){l — b)-i-{l — ac), &c.
8. —8; (^4-4;j2^+87?r)-^(yj^-4g).
9. (m-}-n)'i-2>inn, (?i — mJ-5-2m»
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School teachers who wish to have a Cheap Edition for the use of their
pupils preparing for admission to High School.
Hints and Answers to Examination Papers
in Arithmetic. ''
By J. A. McLellan, M. A., LL. D., and Thos. Kirkland, M. A.
Fourth Edition, - - - - - $1 o6.
McLellan's Mental Arithmetic,---Part I.
Containing the Fundamental Rules, Fractions and Analysis.
By J. A. McLellan, M. A., LL. D., Inspector High Schools, Ontario.
Third Edition, - - - - SO Cents.
Authorized for use in the Schools of Nova Scotia.
McLellan's Mental Arithmetic. ---Part II.
Specially adapted for Model and High School Students.
Third Edition, - - - Price, 45 Cents.
The Teacher's Hand Book of Algebra.
By J. A. McLellan, M. A., LL. D.
Second Complete Edition, _ . . $1.25.
Teacher's Hand Book of Algebra. ---Part I.
Prepared for the use of Intermediate Students.
Price, ----- 75 Cents.
Key to Teacher's Hand Book of Algebra.
Second Edition, - . . Price, $1.5Q.
WORKS FOR TEACHERS AND STUDENTS, BY JAS. L. HUGHES.
Examination Primer in Canadian History.
On the Topical Method. By Ja.s. L. Hiohes, Iiis[K3ctor of Schools, To.
rente. A Primer for Students prejiaring for Examination. Price, 25c
Mistakes in Teaching.
By Jaj. Laughmn" Hughes. Second edition. Price, 50c.
M)0FTEI> BY 8TATB nNIVKllSITT OF IOWA, A3 AS 8LRMBNTART WORK FOR USE
OF TEACHERS.
This work discusses in a terse manner over one hundred of the mistakes
commonly made by untrained or inexperienced Teachers. It is desifjned to
warn young Teachers <>i the errors they are liable to make, and to help the
older members of the profession to discard whatever methods or habits may
be preventing their hiyher success.
The mistakes are arranged under the following heads :
1. Mistakes in Management. 2. Mistakes in Discipline. S. Mistakes In
Methods. 4. Mistakes in Manner.
How to Secure and Retain Attention.
By Jas. Lauqhlin Hughes. Price, 25 Cents.
Comprising Kinds of Attention. Characteristics of Positive AttentionI
Characteristics of The Teacher. How to Control a Class. Developing Men
tal Activity. Cultivation of the Senses.
(From The School and University Maoazinr, London, E^-g.)
"Replete with valuable hints and practical suggestions which are evident-
ly the result of wide experience in the scholastic profession."
Manual of Drill and Calisthenics for use in
Schools.
By J. L. Hughes, Public School Inapector, Toronto, Graduate of Military
School, H. M. 29th Regiment. Price, 40 Cents.
The work contains : The Squad Drill prescribed for Public Schools in On-
tario, with full and explicit directions for teaching it. Free Gymnastic Ex-
ercises, carefully selected from the best German and American systems,
and arranged in proper classes. German Calisthenic Exercises, as taught
by the late Colonel Goodvon in Toronto Normal School, and in England.
Several of the best Kindergarten Games, and a few choice Exercise Song^s.
The instructions throughout the book are divested, as far as possible, of
unnecessary technicalities.
"A most valuable book for every teacher, jnrticularly In country places-
It embraces all that a school teacher should teach his pupils on this subject.
Any teacher can use the easy drill lessons, and by doing so he will be con-
i ferring a benefit on his country." — C. RadciiIffb Dkarnaly, Major First
i Life Guards, Diill Instructor Xormal and Model Schools, Toronto.
Authorized for use in the Schools of Ontario.
The Epoch Primer of English History.
By Rkv. M. Creigiito.v, .M. A., Late Fellow and Tutor of Meiton College,
Dxloid.
Sixth Edition,
Most thorough.
Price, SO Cents.
AEERnF.rS' JOfRNAL.
This volume, taken with the eight small volumes cciitaining; the ac-
counts of the different epochs, presents what may be regarded as the most
thorough co\u-se of elementary English History ever published.
What was needed. Toronto Daily Globe.
It is just such a manual as is needed by pul))ic school pupils who are
going up for a High School lourse.
Used in separate schools. M. Stafford, Prik.st.
We are using this History in our Convent and Separate Schools in Lind-
say.
Very concise. Hamilton- Ti.mes.
A very concise little hook that should be used in the Schools. In its
pages will be found incidents of English History from A. D. 43 to 1870, in'
tercsting alike to young and old.
A favorite. London Advertisf.r.
The book will prove a favorite with teachers preparing pupils for the
entrance examinations to the High Schools.
Very attractive. British Whig, Kingston.
This little book, of one hundred and forty pages, presents history in a
very attractive shape.
Wisely arrang'ed. Canada Presbtterian.
The epochs chosen for the division of English History are well marked
— no mere artificial milestones, arbitrarilj' erected b}- the author, but roaj
natural landmarks, consisting of great and important events or remarkable
changes.
Interesting. Yarmoi'th TRiBrxF., Nova Scotia.
With a perfect freedom from all looseness of style the intei'est is so well
sustained throuKhoirt the narrative that those who commence to it.. .</
will liiid it difficult to leave off with its perusal incomplete.
Comprehensive. Literary World.
The special value of this historical outline is that it gives the reader a
comprehensive view of the course of memorable events and epochs.
m. J. (gage S: Co's. Jlcto eiurational moxks.
THE BEST ELEMENTARY TEXT-BOOK OF THE YEAR.
Gage's Practical Speller.
A MANUAL OF SPELLING AND DICTATION.
Price,
SO Cents.
Sixty copies ordered. Mount Forest Advucate.
After careful inspect on we unhesitatingly pronounce it tiie best spell-
ing book ever in use in our' public schools. The Practical Speller secures
an easy access to its contents by the very systematic arrangements of the
words *in topical classes ; a permanent impression on the memory by the
frequent review of difficult words ; and a saving of time and effort by the
selection of only such words as are difficult and of connuon occurrence^
Mr. Reid, H. S. Master heartily recommends the work, and ordered some
sixty copies. It is a book that should be on every business man's table as
well as in the school room.
Is a necessity. Phesb. Wit.ness, Halifax.
We have already had repeated occasion to speak highly of the Educa-
tional Series of which this book is one. The ■• Speller" is a necessity ; and
we have seen no book which we can recommend more heartily than the one
before us.
Grood print. Bow.MAiNviLLE Observer-
The " Practical Speller" is a credit to the publishers in its general get
up, classification of subjects, and clearness of treatment. The child wh«
uses this book will not have damaged eyesight through bad print.
What it is. Stk.miiroy Aoe.
It is a scries of graded lessonSj .containing the words in general use,
with abbreviations, etc. ; words of similar pronunciation and different spell-
ing- a collection of the most difficult words in the language, and a number
of literary selections which may be used for dictation lessons, and commit"
ted to memory by the pupils.
Ex«ry teacher should introduce it.
It is an improvement on the old spelling
introduce it into his classes
C'ANADIA.V STATESMA.N. j
book. Every teacher should |
The best yet seen. Colchester Sun, Nova Scotia.
Itis away ahead of any"speller"that we have heretofore seen. Our public
schools want a good spelling hook. The publication before us is the best
we ha\ e yet seen.
. J. (iagc S: aro'0. JlcU) @^urationnl ^orks.
The Canada School Journal
HAS RRCEIVED AX IIONOIiARLE MKNTION AT PARIS EXHIBITION, 1878.
Adopted by nearly every County in Caiia«ia.
Recommended by the Ministei'of Education, Ontario.
Reconimeiided b.v the Council of Pul>lio Instnictioii, (Jftelioe.
Recommended by Chief Sujit if Education, New Brunswick.
Recomniciidec liy Chief Supt. of Education, Nova Scotia.
Recommended by Chief Supt. of Education, Uriti.sh t:o!umbia.
Recommended by Chief Supt. of Education, Manitoba.
IT I.S EDITED BY
A Committee of sotnc of the Leading Educationists in Ontario, assisted
by able Provincial Editors in the Provinces of QucIjcc, Nova Scotia, New
Brunswick, Prince Edward Island, Manitoba, and T5ritish Columbia, thus
having' each section of Vm Dominion fully represented.
CONTAINS TWENTY-FOCR PAGES OF READINO MATTER.
Live Editorials ; Contributions on important Educational topics ; Selec-
tions— Readings for the School Room ; and Notes and News from each Pro-
vince.
Practical Department will always contain useful hints on methods of
teaching different subjects.
.Mathe.matical Department gives salutions to diillcult proljlems also on
ENamination Papers.
Official Department contains .such regulations a.s may be issued from
time to time.
Suhscription, $1 00 j-er annum, strictly in advance.
Read THE Followino Lettkr from John Greenleaf Whittier, the Fa-
Mois American Poet.
I have also received a No. of the " Canada School Journal," which seems
to me the brightest and most readable of Educationa Magazines I am very
truly tliy friend, John Greenleaf Whittier.
A Club of 1,000 Sifoscribei's from Nova Scotia.
(Copj) Em-CATiox Office, Halifax, N. S., Nov. 17, 1878.
Messr.s. Adam Miller & Co., Toronto, Ont.
i Dear Sirs,— In order to meet the wishes of oxir teachers in \arious part^
I of the Province, and to secure for them the advantage of your excellent
; periodical, I hereby subscribe in their behalf for one thousand (1,000) copies
, at dub i-ates mentioned in your recent esteemed favor. Subscriptions wiij
! begin with January issue, and lists will be forwarded to your office in a few
, days. Yours truly,
1 David Allison, Chief Supt. of Education.
I Address, W. J. GAGE &CO., Toronto. Canada
!
J
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
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WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO 50 CENTS ON THE FOURTH
DAY AND TO $1.00 ON THE SEVENTH DAY
OVERDUE.
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