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lU. J 6aci;c's glatbcmatical .Scries.
THE TEACFJER'S
HaND-BoOI of ALGEBRA;
CONTAIXISG
METHODS, SOLUTIONS AND EXERCISES
ILLUSTRATIXG
THE LATEST AIsD BEST TREATMENT OF THE ELEMENTS
OF ALGEBRA.
BY
J. A. McLELLAN, M.A., LL.D.
HIGU SCHOOL IXSPECTOR FOB ONTAIilO.
The object of pure Mathematics, ivhich is a'tother nainfifor Algebra, is the unfoUliiuj
0/ tlm laws of the human intelligence." — SYiiVEaTUR.
FOURTH EDITION -REVISED AND ENLARGED,
TOBONTO :
W. -l GAGE & COMPANY.
1881.
Entered according to Act of Parliament of Canada in the year
1880 by W. J. GagjE & Compamt. in the otnce ot tne jyiiiusior
of Agriculture.
\ 7/3
PREFACE.
This boot— embodying the substance of Lectures at Teachers'
Associations — has been prepared at the almost unanimous request
0^ 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 srreat number of exercises, many of them illustrating
methods and principles which are unaccountably ignored in
elementary Algebras.
It presents these solutions and exercises in such a way that
the stirdent 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 vnth which he is
quite familiar, may be applied to the solution of questions whic}>
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
to leau liim to investicrate 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 by means of it, should always precede
its 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.
Bv 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 interrupt 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 ar<^, 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.
Ai-ticles and exerci-^es havp been prepared on the theory of
substitutions, on Elimination, &c,, but it has finally been decided
to hold these over for P*-. ii., which will probably appear if the
present work be favorably received.
CONTENTS.
Chaptrr T. — Sdbstitution, Hoeneb's Diyision. Sec.
rxait.
Sect. 1. — Numerical and Literal Substitution 1
Sect. 2. — Fundamemal Formulas and their Applications 10
Sect. 3. — Horner's Methods of Alultiplication and Division, and their
Applications 21
Chapteb n. — Prixciplk of Stmjietp.t, &c.
Sect. 1. — The Principle of Symmetry and its Applications 33
Sect. 2. — The Theory of Divisors and its Applications 39
Chapter III. — 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
Sect. 5. — Factoring a Polynome by Trial Divisors 30
Chapter IV. — Measures and Multiples, &o.
Sect. 1. — Division, Measures and Multiples 101
Sect. 2. — Fractious 109
Sect. 3.— Katios 122
Sect. 4. — Complete Squares, <&c 130
Chapter V.
Simple Equations of 0::e Unknown Quantity 138
Preliudnai-y Equations. Kesolution by Factors. Fractional Equa-
tions. Application of Piatios. Equations involving Surds,
Higher Equations, &c.
Chapter VI.
Simultaneous Equations 170
Equations of Two Unknown Quantities. Systems of Equations.
Application of Symmeti7. Equations of Three Unknowns.
Systems of Equations.
Chaptkr Vii.
Examination Papers 207
CHAPTER I.
Section I. — Substitution.
Exercise i.
1. If a = 1, J = 2, c = 3, d = i, x = 9, ?/ = 8, find the
value of the followiug expressions : —
l_/l_(l_lZ^.)}.
a- {x—y) — {b — c){d — a) — {y — b){x + o).
a;-y^y-{y — a)\d + c(b-c)Ci.
{x + d){y + b -i-c) + {x — d){a — b — d) + (y + d)(a — x ~ d^.
{d-x)^ + {c-^v^*
la-b){c^-b''x)-{c-d){b^-a'-x) + {d-b-c){d^-^^\
d — a d + c _ c)d + b
d + a d — c d — b
2. If a = 3, 6 = - 4, c = - 9, and 2s = a + h + c, find the
value of the following expressions : —
s{s — a){s — b){s—c).
«2 + (s-a)2 + (.s-6)2 + (s_c)2.
g9_(^s-a){s-b)-{s-h){s-c)-{s-c){s-a).
2{s-a)[s-b){s — c)-\-a{s-b){s-c)^b{s-c){s-a) + c{s- a){s-b).
3. If a = 2, 6 = — 8, c = 1, X = 4^, find the value of the
following expressions : —
a2_i2 ^2+^2 (a-6)2 (a-by
c^^TP' a3^63' (a + 6)3' (a4:6p"'
a2 + r/& + 62 a2_ft3 a; |2x-3 _ 3a;-l )a;-l
^2^^+P' ^F:r^,2' 21 3 '~T~i~2~'
{a + b)\{a+by--c^\^
462c3-(a2-62-c2)2'
oa(&-c)-hfig (c-a)jH5^(a-6)
(a^'6)(i-cy(c-a)
2 SUBSTITUTION.
4. If a = G, & = 5, c == - 4, fZ = - 3, finti tlie value of th*
following expressions : —
y(62 +ac)+ ^(c^—2ac). yi b^ +ac+ v/(c2— 2ac) [.
a2-^/(b'^-^ac) c+^{d^ + c^)
5. If a; = 3, i/ = 4, « = 0, find the value of : —
{3«-v/(a:2+2/2)}2{2a;+i/(a:2+7/2+z)}.
(a;3 _2/3)^^3/|3a;3 + 3(8a;3 + 3a;j/+2/2)y}.
G. Calculate the values of {x-Vy+^)^-^x^+y^-Vz^) ^^^^
X'JZ
(a) <»=1, ?/ = 2, 2; = 3.
\h) a; = 2, 2/ = 3, 2 = 4.
(c) a;=3, 2/ = 4, « = 5.
(t^) a;=10, 2/ = ll, 2=12.
7. Given a; = 3, 2/ = 4, z= —5, calculate the values of
yx+y+x)^ -d{x+ij+z) (xy+yz+zx).
x^{y+z)+y^{z+x)+z^{x+y) + 2xyz.
(5«-42)2+9C4x-;s)3-(13a:-5s)2.
{3x + 4.y-i-5z)^ + {4:x + dy + 12z)^ -(ox+Bij + lBz)9,
8. If s = a + ft + c, find the Value of
(2s - rt ) 2 + (2s - i) 2 - (2,s + r) 2 , given
(1) a = 3, 6 = 4, c = 5, (2) a = 21, i = 20, c = 29,
(3) a = 119, b = 120, c = TG9, (4) a = S, b= -4, c = 5,
(5) «=5, fe = 12, c=-13.
9. If rt = l, 6 = 3, c = 5, rf = 7, <' = 9,/=ll, prove that
«/> 6c cd de ef 2 \a f
^+± + ±+±=111-1
a be bed ed« def 4 \ah e^
nhed hcde cdef 6 \a6c defj
SUJ^STITIITION.
^2 4- 1,2 .^c- - (lb -hc-ca~h^ +r^ +rf3 -hr-cd- db =
c2 _|_(/2 -j-fS _ cd — dfi — CC = f/2 -f c~ +/2 - rftJ - ^■— /^.
10. li a = l, b = 2, c = S, d = 4, e = 5,f=Q, </ = ?, prove that
ci+b + c = ^cd, a-\-b + c + d = ^de,
a + b-\-c + d + e=^ief, i' + l^+c+'^-\-o + f= ifg,
ab{a + b) ab[a-{-b)
a2+i3+,'' J.,/2 + ,2 +;-2 = M^_'+l).
«3+i3^c3+f/3 = (rt + Z> + r + ^)2,
«3+/;»+c3+r/34-e3 = (fl + i + c + ^ + 6-)2.
a^-\-b^4-c^ + d^+c^-\-p = {a+b+c-\-d-j-i+f}^
6c(64-c)
a4+/,4 4.c4+rZ4 = deid+e){cde-l)
«4 + /,4+o4+,/4+,4 = !^+^)Mi:i);
«4+/,4+,4+,/4+,4+/4= .K/-b'/)J^/Z±),
bc{b+c)
6-2+^2=^2^ (.3+,,/3+f,S=y3
11. Assume any numerical values for x, y, and s, ana calculate
the values of the following expressions : —
(a;» - 10ic3 -t-5a;)- + (5x4 - 10a;2 + 1)3 - (;*;3 + 1 )^
(a;+l)3_2(a;+5)3-(a; + 9)3 + 2(a;+ll)3 + (a;+12)3-fx + ]G)3.
(«2-2/3)2+(2a:^)2_(x3+y3)2
(«3_3x</2)2 + (3a:3r/-2/3)3_(a;2^.,,2,3.
{Zx^-\-4.xij+y^)^ + {Ax^+1xyY-[ox" + \xy-¥y^Y.
(K-i/)3 + ((/-2)3 + (2_a;)3-3(jc-?/) {y-z) {s-x).
Art. I. If it = any number, as, for example, 3, then x^
(which = x.x) = 3x, a;3 (which = x.x^) = Bx^, a;* (which = ic.a;3) =
3i:3, &c. Oi- 3 = a;, 3.^ = x3, 3^:3 =a;4^ Sx^^x% &c. Hence jn'ob-
4
SUBSTITUTION.
6 .-. x-^-'2x-d = 15-9 = 6.
lems like the following may be solved like ordinary aritlimeticaJ
problems in " Eeduction Descending."
Examples.
1. Find the value oi x^ — 2x—9 when x — 5,
x"-2x-Q
5
5x
-2x
8x
5
15 Exjylanation.
-9 x^ = 5x,
x^ — 2x = Sx=:lC, and
= 6.
2. Find the value of x^—x^ — Ax^ — 3.t — 5 when x = 3.
x*-x^-4:X^-3x-5
3
jOj Sx^
rj 2x^
3
7)3 6x^
-4x^
rg 2a;2
3 .-. a;4_a;3_4^2_3a;-.'> = 4
— ifa; = 3,
Ps 6r
— 3a;
/« 3a5
3
P* 9
— 5
u 4.
SUBSTITUTION.
Expl-aiuition.
:. x^-x^ = 2x^ = 6x^,
.: x* — x^ — 4x^ = 2a;^ = 6x,
:. x*-x^-4:X^-Sx = 2x=--Q,
:. x^ — x^ — 4.x^ —dx — 5 = 4:.
8. Find the value oi 2x'i + 12;t;3-f6a;3 -12.r+10.
Using coefficients only, we have
2 + 12 + G-12 + 10
Pi - -10
-fl2
'i + 2
- 5
P2 -10
+ 6
r, - 4
- 5
Ps 20
-12 ^
r, 8
-5
P4 -40
+10
r^ -30
.-. tlie quantity = — 30 if a; = - 5.
Art. II. If the coefficients, and also 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 rj,r^,r^, and the
result r^, the last example would appear as follows : —
SUBSTITUTION,
-5 ) 2 +12 -f6 -12 +10
2
-4
8
-80
Art. III. lii tlie above examples, the coefficients ave "brought
down" and written below the products ^^, ^?2, jfg, jp^, and are
added or subtracted, as the case may require, to get the partial
reductions 7-j, r^, r^, and the result ^-^^ Instead of thus " bring-
ing down " the coefficients, we may " carry up " the products ^^,
i>2f Ps' Pi^ writing them beneath their corresponding coefficients,
and thus get 7\, r^, r^, r^ in a third (horizontal) line. Arranged
in this way Ex. 2 will appear
11 -1 -4 -3 -5
8 +3+6 +6 +9
11 +2 +2 +3; 4;
and Ex. 3 will appear
2 +12 +6 -12 +10
-10 -10 +20 -40
2 +2 -4 +8; -30
Comparing these arraugementrf with those first given (Ex. 2
and 3), it will be seen that they are hgure for figure the same,
except that tlie multiplier is not repeated.
Art. IV. When there are several figures in the value of x,
they may be arranged in a column, and each figure used sepa-
rately, as in common multiplication. Where only approximate
values are required, " contracted multiplication " may be used.
4. Find the value of 8a;5 -160a;4 + 344a;3+700a;2 -1910^;+
1200, given a; =51.
3 -160 +344 +700 -1910 +1200
1 3 -7 -13 37 -23
. 50 150 -350 -650 1850 -1150
•7 -13 +37
result is 27.
-23
+27
SUBSTITUTION.
5. Given a; = 1-183. find the value of 64x* — 144a:+45 correct to
three decimal places.
1
1
8
8
64
0
Gt
6-4
6^12
. ^192
0
75-712
7-5712
6-0570
•2271
-144
89-5673
8-9567
7-1654
•2687
+45
-38-0419
-3-8042
- 8.0434
-•1141
64,
75-712,
89-5673,
-38-0419,
-•0036
•3156* -15792, for x
.'. result is —-004.
Exercise ii.
Find the value of
a;4- 11x3- 11x2 -ISx+ll, forx=12.
x4 + 50x3 - 16x2 - 16x-61, for x=: -17.
2x4+249x3-125x2 + 100, for x= -125.
2x3-473x2 -234x- 711, for x = 200.
6. x»-3x2-8, for x = 4.
6. x6 - 615x5 - 3127X-* +525x3-2090x2
= 521.
7. 2x-' +401x4 -199x3 + 399x2 -602X+211, for x=- 201.
8. 1000x4 -81x, forx=-l.
9. 99x4 + 117x3 -257x2 -325x- 60, forx=lf.
10. 5x'^ + 497x4 + 200x3 + 1 96x2 -218x- 2000, forx=-99.
11. 5x5 -620x4-1030x3 + 1045x2 -4120X+9000, fora; = 205.
Calculate, correct to three places of decimals,- —
12. x3 + 3x2-13x-38 forx = 3^58448, forx= -3-77931, and
forx= -2-80512.
13. 2/4- 147/2 +J/+ 38 for j/ = 313131, for i/= - 1-84813, and
for y= -3-28319.
Exercise iii.
What do the following expressions become (1) when x = a, (2)
when X = - a ?
1. X4 -4rtx3 + 6rt2i(;2 — 4r/3a;+a4.
2. i/'(x2-«X + rt2). 3. ■|/(x2 + 2flX + fl2).
4. (x2+flX + a2)3_(a;2_ax + fl2)3.
If x = 2/ = 3 = rt, hnd tue value of the following expressions:
H SUBSTITUTION.
5. (x-y) (y-z) {z-x).
6. (x+y)^ (y-^rz — a) (x+z-a).
8. -^ + Jl- + _^_.
y+z x+z x+y
Find the value of
r. X , X 1 abc
9. — + — when a; =
a b a-j-h
10. - + I + — ^ , when x- A (a-6+c)»
a(b — x) b{c — x) a{x — c) a
11. ^+ J^, when:«= ci^{b_-al^
a b — a b(b + a)
■ 12. (a + x) {b+x)-a{b + c)+x^, when a; = —.
b
13. bx-\-cy-\-az, when x = b-\-c — a, y = c + a — b, z = a-^-b -c,
14. <^3±^ - __«__, when x= -a.
a(l + 6)— Z^x a — 2ia;
,^ /a;+rt\ ^ a;+2rt+^ 1 w; \
15. — — — — , when ic=i(i — a).
\x+bl x — a — 'lb
16. (p-q) {x+2r) + {r-x) (p+q), when a; = '-1?^^.
17. a^{b-c)+b^{c -a) + c^{a- b), v/hen a-b = 0.
18. (a+b + c) {bc + ca+ab)-{a + b) (b+c) {c + a), when a= -b.
19. (a+6 + c)3-(«3+53_{.c3), whena + 6 = 0.
20. {x+y+z)'^-{x+y)'>'-{y+z)'^-{z+x)'^+x'^+y'^+rA,y7hm
x+y-rz = 0.
21. a3(c-62) + 53(^^(_(^^3)_|.t.3(5_rtaj^a/;c(a6c-l), when6-a>
~22. a^ i''l±^\ \b^ l^lll+^'l \ when a.^+6^=0.
23. Express in words the fact that
{a-by=a^-2ab+b".
■ 24. Express algebraically the fact "that the sum of two quan-
tities multiplied by their difference is equal to the difference of
the squares of the numbers."
SUBSTITUnOM. 9
25. The area of the walls of a room is equal to the height mul-
tiplied by twice the sum of the lengtli and breadth : what are the
areas of the walls in the following cases :
(1) length /, height h, breadth h.
(2) height x, length b feet more than the-height, and breadth
h feet less than the height.
26. Express in wards the statement that
{x-\-a) {x-\-h)=x^-\-{a-{-h)x-\r(ih.
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^ ■^^x]){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 Ihe cubes of two quantities be divided by
the sum of the quantities, the quotient is equal to the square of
their difference increased by their product ; express this algebrai-
cally.
31. Express in words the following algebraic statement ;
—= {x+yr-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 oi 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 contained 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 circle is equal to ic multiplied into the square
10 8UBSTITUTION.
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 i'u- height
into the area of the base, that of a cone is one-third of this, and
that of a sphere is two-thirds of the volume of the circumscribing
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 + h']x+{h-\-c)y.
2. nx—by, {a — b)x—{a-\-b)y, and {a-^b)x~{h —a)y.
3. {y—z)a^-\-(z--x)ab-\-{x-y)b^, and {x-y)a^ — {z—y)ah — {x
4. ax-\-hy-\-cz, bx-\-cy-{-az, Sbnd ex -\- ay -\-bz.
5. {a+b)x^+{b+c)y^+{a+c)z^, {h + c)x^ +{n + c)y^ + (n + b)z*,
{a+c)x^ + {a + b)y^ + (b+c)i^, and- (a + b + c) (tcS+t/S-fgS).
6. x(a-b)^ -{-y{b-cy+z{c-a)», y{a-b)^ +z(b -c)^-i-x{c-
z)2, stnd z{a-b)2+x(b-c)2+y{c-u)~\ '
7. {a-b)x^-{-(b-c)y^+{c-a)z^,{b-c)x^+{c-a)y^ + {a-b)z-^,
and [c-a)x^-{-{a-b)y^+(b-c)z^.
8. {a + b)x + (b+c)y -(c + a}z, {b + c)z + (c + a)x-{a + b)y, and
(a + c)y+{a + b)z — {b + c^x.
9. a^-3ab-:^*b^, 2b'^ -^b^+c^, ab-ifb^ + b^, and 2nb-ib*.
10. aa;"-36x", -dax'^+lbx", and -Sbx^ + lOax".
11. What will (ax — by + cz)-\-(bx-\-cy—i(z)~(cx-\-ay-^-iyz) be-
come when x — y - z = l ?
SeCTU)N II. FuNDAMKNTAL, FoERTULAS AND THEIK APPMCATION.
4. By Multiplication we get
(x + r) (x + s) = x'^ + {r + s) x + ri^ A.
(x + r){x-i-s)(x-\-t) = x^ + {r + s + t)x'^ + {rs +• st + tr)x -f rsf B.
From A we immediately get
(x±yY=x''±2xy^y- [1]
FUNDAMENTAL FORMULAS. 1 1
{x + y + z)'^=x^ + 2x>/ + 2xz + y^ + %iz j ga ["2]
(2<0^ = 2a2 + 2 £a& [3]
(a:+7/) (u; — y) =a;2— y3 [4]
From B we derive
(x + y)3=.f-^±3x2^ + 3.r//2±?/3 [5]
= x-^ ±y'' ±^x]) {x±y) [6]
+ 6-V2/2 [7j
= ^3^y3+,3 +3 (;,+_,^) i^y^^^ (^^^■J [-Qj
= . f3+ 2/3+^3 _,.3 ^ y _^^ _,. ,j (^.^ _|_ y^ + ^^.j — 3x^2... [9]
(2:.()3r=va3 ^ S^^s^^ 62a6c [10]
[The symbol £ means the sum of all, such terms as]
Formula [1] . — Examples.
1. Wo have at once {x -YyY + {x —yY =2(^-2 + t/2), aud
(x-YyY—{x — yY=^xy.
2. (a + 6 + c + tZ) 2 + (a — b — c + dY may be vrritten
{(a + d) + {b + c)}2 + {(a + d)~ (6 + c)}2, which (Ex. 1) =
2{{a + dy-±{b + cy-'} ; similarly
l^a — h-\ c — dY-\- {a + b —c — dY = {{a — d) — (b - c)}^ -f
{{a — d) + {b — c)}^ = 2{{a — dY + ib—cY};
.-. {a+b + c+ dY + (a — b — c-i- dY + {a — b + c — d)^^
{a + b -c- dY = '2{{a + dY + (b +c)2-f-(a-d)2 + (6-c)3} =
(again by Ex. 1) i{a2+b2+c''i + d2).
3. SimpUfy {aA-b-'rcY-2{a + b-{-c}c + c^ ;
This is the square of a binomial of which the first term is
(a+b+c) and the second — c; the given quantity .'. =
12 FUNDAMENTAL FORMULAS.
4. Simplify (a + b)^ - '2{a^ + b-i ) {a-^h)^ +2{a^ + h^).
By Ex. 1. 2(«.4 + i4) = (rt2+i2)2 + (a2-i2)2 ; .-. giyen quan-
tity = (w+Z/j* - 2(^<2 _^ i3) (rt + i)2 4. (,^2 + /,2)2 _^ (rt2 _ i2)2 =
{(a + 6j2_(,i2+i2jp + (,,2_/,2j3=rt4_(_2a2i2+i4^(rt2 +/,2)2.
Exercise v.
1. (a;+3?/2)2 + (a;_3?/2)3, (i„2 ^3i2j2 _ (l^s -3^,2)2.
2. Siiew that (??ia;+'2^^)^ + (»^ — w'i/)^ = (wi2-t-w2) ^2:2+2/2).
4. Simplify irt + 3i)2-f 2(rt + 3?;) {a-b) + {a-h)^] {a-b)^.
5. " (a; + 3)2 + (:t-+ 4)3- (a: + 5)3, and (|a;2 -22/2)3 -
(i2/2 +2x2)2.
6. Simplify («. + & + c)2 + (6 + c)2 -2(Z/ + c) (n + i + c)
7. Shew that {ux^byY -\-{cx+dyY -\-{ay - bxY-\-{cy -dx)^ =
(a2+i2+c2+^2) (a;3+2^2).
8. Simplify (a;- 3?/2)2 + (3a;2-2/)3-2(3a;2-2/) (x-3^2).
9. " (a;2+a;y-2/2)2-(a;2_a;^/_?/8)2^and(l + 2a;+4a:2)3
+ (l-2a; + 4.T2)s.
10. If rt + 6= -fc, shew that (2a-6)2 + (26-c)2+(2c-«)2 +
2(2a - b) {'lb-c) + 2{2h - c) (2c -a) +2(2c-a) (2a - ?>) = tV^-
11. Simplify2(«-6)2-(a-2i)2; (a2+4«?,+i2)2 _ (^2+^2)2,
12. " (r/ + />)2-(i+c)2 + (c + t/)2_(,^+,,,)2.
13. " (ia;-7/)3+(i_,y_^)2+(l^_a;)2+2(ia;-7/) (i2-3t^
+ 2{ly-z){lz-x) + 2{ix-y)(^y-z).
14. Prove that (a;- i/)2 + (2/-«)2+(«_a;)2 = 2(a;-2/) (^-i/) +
2iy-x){z-x) + 2{z-y)iz-x).
15. Simplify (l + a;)4-2(l+a;3) (l+x)3 + 2(l +a:4).
' 16. " (a;+?/+;s)2-(.« + ?/-z)2-(?/+z-a;)2-(2+a;-?/)2.
17. " (x-22/+3z)2 + (30-27/)2+2(a;-2i/+3z)(2^-3z).-
IQ, u (,(2+Z,2_c2)2_|.(c2_Zy2)2+2(i2_c2)(a2+^3_c2).
19. " (x+yy + (x-y)*-^r-y)^x + y)*.
FUNDAMENTAL FORMULAS. 18
21. Sbewthat(3fl-i)3+(35-c)2-f(3c-«)2-2(6-0a)(3i-c)
+ 2(3i-c)(8c-rt)-2(a-3c)(3a-6)-4(« + /; + c)2=0.
22. If z2 = 2a;?/, prove that (2x2 _2/2)2 + (z2 _2?/2)2+(a;2 -222j2
-2(2a;3-2/2)^22_2y2)_|.2(a;3-2z2)(22_22/2)_
2(a:3 -222) (2a;2 -y^) = (x+t/)*.
23. Simplify {1+x+x^ +x^y + {l-x-x^ +x^)^ +
{l-x+x--x^)"-\-{l-\-x-x^-x^y.
24. Simplify {ox-{-by)*-2{a^x^ -\- b^y^) {ax + by)^ +
^a^x* + b^y^).
FORSIULAS [2] AND [3] . EXAMPLES.
1. (1-2.C + 3x3)2 = i_4a,-+6a;2
-f4x2_19a;3
-\-V,x*
= 1 - 4a;+lUx-^ - 12x3-^-90;*.
2. (ab + bc + ni)^ =^a^h^ + 2ab^c + 2a^bc-^b^c^~{- 2abc^+c^a^ =
a^b^+b^c'+c^a^ +2abc{a + b + c).
3. {{x + y)^+x^+y^}^^{x + ij)* + 2[x + y]2{x^-^y^)-{-x^ + 2x^
y^+y^ = {x + y)^ + {x+y)^{{xi-y)^ + (x-y)^} -^x^ + 2x^ij^ + y^
= 2x+2/)'' + fa:2-2/2j2+a;4+2x2;/2+7/4 = 2{(x+y)4+x4 + 2/'*}.
4. (x2+X;/ + ?/2)2=x4 + 2x3;/ + 2x2^2 + .^2j^2 -|- 2x^3 + yi =
{x+y)^X'+x"y^+y^{x+tj)^.
5. In Ex. 3, substitute b- c for x, c -a for y, and consequently
6 — a {ov x+y, then since (i — c«)2 = (« — 6)2, Ex. 8 gives
{(rt-6)2 + (/>-c)2+(c-a)2}2=2{(«-i)4 + (/>-c)*+(c -«)*}.
6. Making the same substitutions in Ex. 4, we have
(a2+?/2^c2 ^ah-bc -cay = {a —b)^{b - c)^ + {b-cy{c- a)^ +
(c-a)2(a — i)2, or, multiplying both sides by 4,
|(a_i)2_j.(6_c)2 4- (c-rt)2}2=4(rt_i)2(i_c)2 + 4(i-c)2 X
(c-a)2+4(c-a)2(a-6)2. and .-. from Ex. 5, (a-Zy)4 + (6_c)4 +
(c-a)4=2(a- 6)2(6- c)2 +2(6 -c)«(c-a)2 + 2(c-a)>(a-i)3.
14 FUNDAMKNTAI. FOKMULAS.
Exercise vi.
5. Shew that («2 +62 +c2) (a;2 + ,y3+22)_((^a; _}. ^^Z^. .)3_
(a^' - &.c) 2 -}- (ex- - <72) 2 + (6; - c?/) 2 .
6. Prove that (a + /^)x+(i + c)2/ + (c + rt)2 multii:)lied hy (a — b)'x
-i-{h — c)y + (c — a)z, is equal to the dilference of the squares of
two triuomials.
7. Shew that {n-h) (a-c) -}- {b-c) {b-a) + (c-a) (c-h) -
H(«-Z>)2 + (&-c)2 + (c-«)2} = 0.
8. Simplify {a-{b-c)}^ + {b-{c- a) ] » f {c - (« - 6) } » .
9. Shew that {a^ + b^ -x^Y^ +{aj-{-bf~x-^)^ + 2{aa^ +bb^)^
= (a2 + rt2_;j;3)2+(^,3_j.i2_,^.2)3.^2(rtO + ai6J2.
10. Prove that {{a -b){b- c) + (/; - c) (c - «) + (c - ^0 (« - ^^)} ' =
(a-6)2(6_c)2 + (6-c)2 (c--a)2 + (c-a)2 (a-i)2.
11. Square '2a — ibx — ^cx + 2dx.
12. If a; + ?/ + 2 = 0, shew that x^ + //'* + s^ = (x^-yZ^i^
(//2-z2)2_(-(23_;^2)3.
13. Prove that a- {b + c)^+ b^{c + u)~-[-c~ {a + b)' + 2abc(a -\-b + c)
= 2{ab + bc + ca)'.
Art. V. To apply foriDula [4] to obtain the product of two
factors which differ only in the signs of aome of their terms : —
group togetlier 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 + Z> — c-f (Z by rt — 6 — c — tZ ; here the first group is
a — c, the second i+f^ ; •'• we have
l(a ^c) + {b + d)} {{a-c)-(h + d)}={a-c)^-{b + d)\
FUxNDAMENTAX. FORMULAS. 15
2. (1 J- 8a;-ff^a;3 +x^) (1 - So: + Bx^ - x^.) = {(1 + S.r^) -i
i3x-hx"^)} {(l + 3a;2) - (:3a- + a;3)} = (l + 3a:3)3 - (3ic+a;3)3 = 1 -
3. Find tliecontinuedproductof r/-f /'-}-c, b+c — a, c+a — b and
fZ + ^ — <^.
The first pair of factors gives {{b + c)-{-ii} {{^>+f^) — ci} ={6-|-r-)'
-a* = 63 4-2&c + c3-a3.
The second pair gives {a — {b — c)} {« + (& — '-')} =a3 — i3_|_2^c
--c3 ; the only term whose sign is the same in both these results
is 26c ; hence, grouping the other terms, we have
{2bc + {b^-{-:^-a2)}{2bc^{b^ + c2 - a^)} =
(26s)2-(62+c2-a3)2=2a262+262r34-2c3a2_^,4_J4_c4.
4. Prove (a^-^ab + by -a^b^ = {a^ + ab)^ + {ah + b^)^.
The expression ={n^+h^) {a^ +2ab+b^) = (a^ +b~) (a -^by =
a^ia+by -{-b^{a + by = {a2 +ab)^ -\r(ab + b^)2.
Exercise vii.
1. (a2 + 2ab + b2) {a^-2ab+b').
2. (ix-2 -x,j + yi)(ix^+y2+x7j).
5. {a^-ab + 2b^) {a^ + '(b+-lb^) ; (x*-\-M:y) (x'^-4xi/).
4. {{x + y)x-y{x-y)} {{x-y) x-y{y-x)\.
5. Simplify: (x+3) (a;-3) + (.r + 4) (a;-4)-(a; + 5) {x~5).
6. " (H-a;)4 + (l-a;)4-2(l-a;2)3.
7. (x3+j/3)2 - (2a:?/)2 - (a;2_2/3)2.
8. (2r72_362+4c3) (2a3 + 3J3_4c2).
9. (2a + i- 3c) (i + 3c-2«) ; {2a—b-Bc) (b—Sc—2a).
10. (x'^-\-y^) (x2+7/3) (a;+2/) (x-7y).
11. (x^+xy + y^) (x^-xy+y^) {x'^-x^y--'-\-y'>^). '
12. (a + /;-a/>— 1) (a-f /j + a/y+1).
13. Prove (a2-f/;2 4.^2)(//j_|_^.2 _ ^.a^^^a +a2 _/;2) (^2 _l/,2_ ^s)
= 4/;'*o4 when a^ = />4_j_,.4,
14. (a;2+y2_6a.^) (a.2+^2 + 6^.,^).
1/5. (a-4- 2x3 + 8x2— 2x4-1) {x'^+2x^-^Sx^-+2x-^l).
16 ' FUNDAMENTAL FORMULAS.
16. Multiply (2x—y)a^ — (a;-f-?/)«a; +x^ by (•2x-y)c''^-^
{x+y) ax -x^.
Prove the following :
17. (a^ +b^ +c^ +ab + bc+ca)2 - (ah + bc + ca)3 = (a+ 5 +c)^
x{aZ + h^-i-c^).
18. (a^-Jrh^ +c^ +ab + bc+ca)^ -{a^ +ah + ca-bc)^ =
{{a + b)[b + c}}'- + {(b+c) (c + a)}2.
19. 4(ai+cd)2-(a3+/^3-c2-r/2)3 =
'i+b+c-d) (a + b-c + d) (c+d + a — b) (c + d — n + b).
20. Find the product oi x^ +i/^ +z^ — 2xy + '2xz — 2yz and x^-^
r^+z^-2xy-2xz-{-27jz.
21. (a;2 +^3+^:^3/2) (x3 -a;.yi/2+y3) (a;* -^z*).
22. (l-6a + 9rt2) (L + 2a + 3«2).
2'3. {(»r+7i) + (jO + (/)} (?n-7+7)-n).
24. Obtain the product of l-{-x+x^, x^+x — 1, x^—x + 1, and
l+x — x^.
25. («-/>3)2 (a-fi2)2 (a2+&4)s (a^ 4.^,8)8.
26 Shew that {x^ + xy 4- y^)'' (x^ - xy ^ 2/')' - (x^J/^)^ =
Formula A. — Examples.
1. Multiply x'— a;+5_by a;^ — «— 7 : here the common term is
x^ ~x, the other terms +5, and— 7, hence the product = (x^ —x)^
4-(_7 + 5) (xj -a:)H-(-7x5) = (a;^-a;)2-2(a;2 - x)- '6o = x^ -
2x^-x^ + 2x-3o.
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^+3ax - Aa^){x^ + 3ax-18a^) = (x-+Bax)^ - (da'-^ + lSa^)
X(a;2 + 3aa;)-72a4 = &c.
Exercise viii.
■ 1, {x''+2x+3) {x'' + 2x-'k); {x-y + 3z) (x-y + 5z).
2. (a;+l) {x+5) (a;+2) (a;+4) ; (x^+a-b) {x^+2b-a).
3. (a»-3)(a2_l)(a3 + 5) (a3 + 7); {x^ + x-^ + l){x*+x^ -^).
4. {{x+yy -ixy)} {{x+y)^ +oxy}.
rUNDAMKNTAL POEMULAa. 17
C. (nx+y + S) (nx+y + l).
7. (x+a-y) (x-i-a + ST/).
8. (a;2« +a;" - a) (a;^" +x" - b).
11. Multiply together x- 2 + 1/ 2, a;-2 + T/8, a;-2-i/2, ana
x-2- v/3.
12. (x+rt + ?)) {x+b-c) (x-a + b) (x+b + c).
13. (a+i+c) (a + b-\-d) + {a+c + d} (b+c+d) - (a+b+c+cr)^.
14. Prove that
{2a + 2b~c){2b + 2c-a)-\-{2c + 2a-b){2a+2,b-o) + {'2b+2c-a)
(2c + 2a -b) = 9{ab+bc+ca).
Formulas [5] and [6] . — Examples.
1. We get at once
{x-hy)^ + (x-7jy = 2x{x^+3y'>).
{x + y)^-(x-y)3 = 2y{Sx^+y^).
2. Simplify {a + b+c)^ -S{a + b+cyc-\-S(a-{-b + ry^ -c^.
This plainly comes under formula [5] , the first term being n-\-h
+c, the second —c; hence the expression is {(a + i/+c)— c}» =
{a+b)\
3. Shew that {x- +xy+y-)^ -^{xy -x- —y^)^ -
6a;y(x* +x-y'^ +y*) = 6x^y^.
This comes under formula [G] , the first term being
{x^+xy-\-y'^), and the second- {x-—xy+y^) ; we have therefore
{{x''+xy+y^)-ix"--xy+y^)}^=^{2xy)^ = 8x^yK
' Exercise ix.
Simplify
1. (l-a;2)3+(H-a;2)3, (x^ +xy^)^ -(x'' -xy^)^.
2. (a + 26)3-(rt-6)3, (3a-6)s-(3a-2i)3.
18 FUNDAMSNTAL FORMt'LAS.
3. {x+y-z)^-{-Z(x-^y-z)'Z+z^-^-^x-^y~-z)z''.
4. (a-i)3 + (rt + i)^+6«(«2_Z>=i).
5. {x-yY+{x+yy+Z(x-yY {x+y)-^y-x) (x^y)^.
6. {1+x+x')^ -{l-x+X'Y -Qoil+x'' -^-x^).
7. (a_fe_c)3+(6+c)3 + 3(6+c)2(a_i_c) + 3(a-Z»-c)2(64-5).
8. (3a;- 47/ -i- oz)^ -(02 - 4v/)3 4- 3(5z - 4^/)^ (3:c - ^y -r bz)~
3(3x--47/ + 52)2(5s-4v/).
9. (l+x+a;2)3+3(l-a;3)(2-fx'-) + (l-a;)3.
10. Shew that a(a-26)3 -i(i - 2a)3 = (a_i) (^+6)3.
11. Shewthata3(a3_2i3)3_,.Z,3(2fl3_i3)3=(a3^/,3)/a34.i3)3
12. (x2+x^+7/-)3 + 8(x2+2/-) (a:4+:^*'+2/*)+(x3-a;2/+^2)3.
13. Shew that aS^^s + 263)3 4. ^3(2a3 + t3)3 a. (3a-Z/-')3 =,
fG+7a3i3_i.z,6)2_
14. SimpHfy {ax-h^yy+a^y^ -r-^^x^ —^(i'hxy{ax+hy).
15. What will a3^^3_|_c3 _3a6c become when a-\-h + c = 0 ?
16. Find the value of x*^ -y^'' -\-z^ -^-Zx-y^z'^ when x- —y- +z'^
= 0.
Formulas [7] , [8] and [9] . — Examples.
1. Simplify (2a;-8?/)3 + (47/- 5x)^ +{^x~ y)^ -
%2x-'3ij) {4.y-5x) {3x-y).
By [8] this is seen to be {(2x- 3«/) + (4y — 5x)-\-i'3x-y)}^ =
(0)3=0.
2. Prove that(a— i)3 + (i-6)3 + (c-fi)3 = 3(a-i) (b-c) (c-a).
In [8] substitute a — i for x, b — c for y, and c — a for 2; for
these values x-\-y+z = Q, and the identity appears at once.
3. Prove {a-^b + c)^ —{b -\- c — aY — {a-\-c—b)^ —{a-^b — c)'^ =
24.abc.
In [7] leta; = 6+c — a, y = c-\-a-b, z = a+b—c, and therefore «
4-// = 2c, '//-|-2 = 2a, z+x='lb, and this identity at once appears.
FtTNDAMKNTAXi FORMULAS. 19
Exercise x.
1. Cube the following: l—x-\-x-, a — b — c, 1 — '2x-\-r,x- —4z^.
2. Simplify (a-= +2z-l)^ + {9x-l)(x" + 2a;-2) —
8. *Prove that (x-\-y){y+z)(z+x) +x?/z = (x + y-i-z)(xj/ + i/z+zx)
4. Prove that {ax — by)"^ -f- a^y'^' — h^x^ -\-^ahxy {ax ~ by) =
{a^ - h^) (z^ +y^).
5. Simplify {x-27jy+{y-2zy^ + {z-2a;)^+S{x-y-2z)x
(y—z-2x) {z—x — 2y) + {x+y+z)^.
6. SimpHfy (2a;-' - Sy^ +4z-)- + {27j-^ - Sz= + ix^)^ +
{2z- -3x--\-iy-)^.
7. Simplify {2ax-by)3+{2by-cz)^ + {2rz—axy-\-
B{2ax-\-by — cz) {2by + rz — ax) {2cz+ax — by) .
8. Prove {x^ -^Qx-y — y^)^ + {'^xy{x + y)]^ = {{x- y)'^ -\-^x^y]
X [x^+xy+y-]^.
9. Prove ^x^+y^+z^) -{x->rV+z)^ ^{'^x-^ 4y-^ z) {x ~ y)^ +
(iy + iz+x) {y-z)-+{iz-\-4:X+y) {z — x)~.
10. Ti x-{-y-\-z = 0, shew that a:3-(-?/3_|_23 _ga.^2_
11. Ifa; = 2!/ + Sz shew that sc^ — St/^ — 27z^ — lSxyz = 0.
12. Shew that {x"" ^xy + y"^)^ + {x'^ ~xy + y-)^+^z^ -
Cz"- {x^-'rx^-y''+y^) = 0,i{x^+y^+z^'=0. '
13. Prove that 8{a-\-b + r)^ - {a^ b)^ - (6 +c)3 _ (c+a)^ =
S{2a + b+c) {a + 2b+c) {a-[-b+2c).
Prove the following :
14 {ax — hy)^ -\-b^y^ =a^i:^ ■\-?jahxy{by — ax).
•Note that the right-hand member is formed from the left-hand one by changing
additions into multiplications, and multiplications into additions; hence in (x+y+
t).{x.y+y.z+z.x; the si<;ns -f- and . maybe inteichauged throughout without alter-
ing the value of the expressioo.
20 FUNDAMENTAL FORMULAS.
15. a^+b^-tc^-3abc = ^{{a-b)^ + {b-c)^+(c-a)^} x
(a+b + c).
16. (a + b + c) {(a + b-c) [b +c-a) + {b + c-a) {c + a-b) +
{c + a-b) {u + b — c)} = {a. + b — c) {b + c-a) (c+a — b) + 8abc.
17. a^ + b^+c3+2labc={a + b + c)^ -3{a{b-cy+b{c-a)^ +
e{a-b)^}.
18. {a+b+lc){a-b)2 + {b + c+-la){b-c)^+{c + a+7b){c-a)^
'=2{a + b + c)^ -5iabc.
19. (a+b + c) {(2a-i) (2b-c) + (2b-c) (2c-a) + (2c-a)x
(2a-b)} = (2a-b) (2b-c) (2c-a) + (2a+b-c) (2b + c-a)x
(2c + a-b).
20. li x'^{ij + z) = a^,y^(z+z) = b^, z^(x + y) = c^, ariidixyz = abc,
shew that a^ +b^ +c^ +2abc=(x+y) (y-\-z) [z+x]
Expansion of Binomials.
We have from formula [5]
(a+b)^ =a^ +3a'^b + 3ab^ +b^ ; multiplying by a + b we get
(a+-h)^ = a^ + '^i^b + Qa^b''' +Aab^ + h^ ; multiplying this by
a+bviQ get
(a+b)^=a^ + 5a^b + lOa'ib^ + \Q>a^b^ + bab"^ + bf^ .
From these examples we derive the following law for the form- .
ation of the terms in the expansion of a+b to any requked
power : —
(1). ThQinclex of a, in the^?-sf term, is that of the given power,
and decreases by unity in each succeeding term ; the index of 6
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 anv 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 f^om the
extremes of the expansion, are equal.
MULTIPLICATION AND DIVISION, 21
Exercise xi,
1. Expand (j;+ 3/) «, {x + y)', {x+y)\ (x+t/)".
2. "What will be the law of siifus if — y be •written for y in fl) '?
3. Expand (a- 6)', (rt-2MS (26-a)4.
4. Expand (!+?«)«, (7« + l)^ (2ot+1)6.
5. What is the coellicient of the 4th term in {a—b)^° ?
6. Expand (x3_y)4^ (a-262)s, (a^-^h^)^.
7. In the expansion of (a — b)^^, the third term is 66^^ "/y^, find
the 5th and 6th terms.
8. Shew that {x+y)'^—x'^—y'^ = oxfj{x-{-ij)(z^+xy+y^).
9. From (8) shew that 2{{a - b)^ + {b - c)^ + (c - a)"} =^_
5{a-b) (b-c) (c-a) {(a- 6)3 + (6-c)3 + (c- a)2}.
Section III. — Horner's Methods of Multipucation and
Division.
Examples.
1. Find the product of kx^-{-lx^+mx+n and ax^+bx + c.
Write the multiplier in a column to the left of the multipHcand,
placing each term in the same horizontal line with the partial
product it gives :
kx^ +Ix^ +mx -|-w ;...Q
-^hx
+ r
akx'' +alx^ -\-amx^ -{-anx^ p^
-\-hkx* -i-blx^ +hmx^ -\-bnx p^
-{-ckx^ -\-clx^ •\-cmx+cn p^
akx^ -f {al-\-bk)x*' + (aw + bl+ck)x'^ + («« + bm + cl)x- +
{bn-\-cm)x-\-cn P.
Art. VI. The above example ha^ 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 multipher be unity, the coefficients of the multiplicand
will be the pame a? th^se of the -first partial product, and may be
used for them, thus saving the repetition of a line.
22
MULTIPLICATION AND DIVISION.
2. Multiply 3x4 -2x^-2x + 3 by a;3 +3.-?; - 2.
+3
— 2
-2 +0 -2
+9 -G +0
-6 +4
+ 3
■6 +9
0 +4
I Sx^ + lx^ - 12.c4 + 2.^3 _ 3,,.3 + 13a; _ 6.
3. Find the product of (x-3) (a; + 4) (x-2) (a; -5).
+ 4
1
-3
+4
-12
4-24
-2
1
^1
-2
-12
- 2
-5
1
-1
-5
-14
+ 5
+ 24
+ 70
-120
X4
- 6x3
-9x3
+ 94x
-120.
4, Multiply x3 - 4x2 + 2x - 3 by 2x^-3
1
-4
+ 2
-3
2
0
0
3
2
-8
0
+ 4
0
0
-6
0 0
') 0
-3 +12
0
-6 + 9
2x«-
-8x-'
+ 4x'*
-9x3+12a;2-
-6x +9
[x^ X X"*' =X*]
I lu this example the missing terms of the multiplier are supplied
iby zeros ; but instead of writing the zei'os as in the example, we
|may, as in ordinary arithmetical muitiplicatioa, " skip a line "
for every missing term.
5. Multiply x* -2x3+1 \v x* -x3 + 3.
1
-1
+ 3
1 +0 -2 +0+1 , ^ 4_ ,,
_1 _o +2 -0 -1 1-^ ^^* ~^ J
+3 +0 -6 +0 +3
dx^
+ 6a-* -7a;-
+ 3
MUT.TIPLICATION AND DIVISION.
29
e. Fin(3th«valueof(a;+2)(a-f8)(ar-f4j^a; + o)-9(x + 2)(x+3}
X (.^ + 4) + 3(j, +2)(.r-h 3) + 77(a;-i- 2) - 85.
+ 5
-9
+ 4
-♦-3
+ 2 i
1 -4
+ 4
-IG
+ 3
+77
1 +0.
+3
-13
+ 0
1 +3
+ 2
-13
+ 6
4-38
-26
+ 78
-85
9
7. Find the coefficient of aj* in the product of x —ax^ + bx^^
cx-T-d and x^-\-}jx-\-q.
1 —a +6 -c
— ap
+ ? ■
+ d
Exercise xii.
Find the product of
1. (l+a; + a;2+a;3+x4)(l-a+a;3_a:7+a.8_^.i24-;i-i3).
2. (l+xs)(l -a;5 +x^){l+x+x^ +x^ -{-x^).
3. (x-5) (x+Q) ix-l) {x+S); (2x^-x^-\.l) {x^-x+2.)
4. {x= + 5x3 _ iGa; - 1) {x^ - 5x^ - 16x + 1).
6. (6x«-2=+2x*-2a;3+2a;2 + 19a;+6) (3a;3+4a;+l).
Obtain the coefficients of x^ and lower powers in
6. (l + ^yx-ix^ + ^^X^-^^^X^) (1 - ),X
7. Multiply tlx' -x^+2x-A by 3a;5-2a;2 -x-l.
13.3 _ 1 ^.3 _ r. -,.4\
524 MULTIPLICATION AND DIVtSION.
Simplify the following :
8. (x+l) (a;+2) (a;+3) + 3(a;+ 1) (a; + 2) - 10(x+l)-f9.
9. x{x+l) {x+2) {x+B)-dx{x+l) {x+2)-'2x{x + l) + 2x,
10. x{x-l){x-2){x-S)+dx{x-l){x-2)-2x{x-l)-2x.
11. (x-l) {x+l) {x+S) {x + 5)-14:(x~l) (x+l)-rl.
12. 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 the «im
of the second pair.
+x^ +a;i*
13. Given that the sum of the three following factors is equal
to —1, find their product.
(1) x +z'^ +x^ +a;'3
(2) x'^ +x^ +a;io +x^^
(3) a:4 +x^ +X'' +x^.
Art. VII. "Were it required to divide the product P in the
first of the above examples by ax'^ + hx+c, it is evident that could
we find and subtract from P the partial products p^, p^, (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 in
each term oij)^, p^, respecUvely.
(1)
X -hx^
(2)
x^ +a;8
(3)
x^ +.r*
(4)
X« +X''
ITDLTIPLICATION AND DIVISION.
25
Oi ^
+
I I
4-
+
+ 1 1
+
+"^
+
H
+
+
+
I I
2
m
.2
>»
CO
rS
2
-
rw
0
<D
0
a
©
^
CO
fcjD
-♦J
*cn
0
0
P
rO
:^
a
5
tp
'>-
tp
>
"Ejd
■^
S-i
a
qa
0
-f3
0
a
0
o
o
_'g
3
'?
"-3
1
G
0
a
^
0
-4^
a3
CO
-+3
a
©
'53
>
CO
0
f^
■©
^
.2
a"
o
a
j:<
^
_>
-73
<D
"3
^
-§
o
.2
?^
•g
.a
?c
"0
a
_a
1=1
o
%
2
'03
>>
0
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CO
Pi
'3
a
0
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a
to
-1^
«
&i
fl
r^
a
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CD
s
a
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0
2
CQ
0
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g)
1
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%
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0
to
p
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a
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0
O)
w
a
03
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a
1
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©
-1-3
a
X
^
^
CO
rO
(-;
a
0
CO
'>
1 ,
o
-1-3
i=!
0
■40
CO
to
?c
-1-3
fcC
1
a
-1-3
.2
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0
a
o
^
><
c
rs
a
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CD
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-4-3
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c3
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a
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'3^
0
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o
to
1-5
n
m
■0
-4J
"5
2
'-1-3
a
0
2
a
s
-t3
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c3
"o
0
0
2
to
0
to
"^3
<fl
3
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to
'S
"^
0
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CO
-(-3
'Si)
.3
0
oq"
-1-3
0
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0"
"Eb
'>
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0
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-ts"
">
2
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eg
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0
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a
a)
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c3
^:
m
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to
a
c3
.2
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-1-3
ca
©
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0
-1-3
CO
1
>>
OQ
a
s
a
a
0
i=H
©
a
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0) ^
2
0
0
-5
CO
2
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a
0
a
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a>
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1
=0
CO
<D
.a s
'S 'S
0 0
0
CQ
5 J
1 en
26 MTTLTIPLICATTON AND DIVISION.
2. Divide Bx^ +lx^ - 12a;* -+-2a;» - Bx^ + 13a:- 6 by a;' 4- 3a? - 2.
j 3 +7 -12 .+ 2 -8 +13 -6 , o^^.^^^i
_3 _9 ^. 6 -0 -f6 - 9 \^ ^ -^ f
+ 2
-4+0-4 +6
3x4-2x34. 0 -2z+3
Compare this example with the second example of Horner'a
Multiplication, performing a step in multiplication, then the cor-
responding step in division ; then another step in multipHcation
and the second (corresponding) step in division, and so on.
3. Divide a; • ~ 3a;« + Ax^ + ISx^ _ 7V4 12 by xJ - Bx^ + 3a:- 1.
+ 3
-3
+ 1
1 _3 +0 -4 +18 +0 -7 +12
+ 3 +0 -9 -86 -27
-3 -0 +9 +38 +27 [a;7-a;3=x4].
+ 1 +0 -3 -12 -9
I xi +0 -3a;2-12a;- 9; 6a;2+ 8a; +3
The quotient is therefore x^ —ox'^ - 12a;— 9, and the remainder
6a;2+8a; + 3.
4. Divide a;» - 3a;' -Ca;"' +2a;4 + 5a;3 + 4a;2 + 1 by x^ + 2a;- 1.
The zero coefiicient in the divisor may be inserted, or it may be
omitted and allowance made for it in the 2a; — line. See examples
4 and 6 in multiplication.
— 2
+ 1
1
_3 +0 -5 +2 +6 +4 +0 +1
„2 +6 +4 -4 -6 +2
1 -3 -2 +2 +3 ^1
1
_3 _.2 +2+3-1; 0 +5 +0
[x^ -r a;3 =x^] . The quotient is therefore x'' ~Bx*
+ 3a;— 1, and the remainder 5x.
2a;3-4.2a;2
5. Divide lOa;" -lla;^-3a;4 + 20a;3 + 10a;3 + 2 by bx^-Bx^A-
2x-2.
MXILTIPLICTl ION AND D^V^SION.
27
Arranging as in
, 10 -
+3
-2
the ordinary method, we Jiave
-11 -8 -f20 -t-10 -1-0 -f2
6-3-6 +12
-4 + 2 +4-8
+ 4 -2-4+8
5
2
-1-2 + 4
24-12-flO
24x3-..12x+10
5a;3_3a;3+2x-2
We first draw a -vertical line with as many vertical columns to
the right as are lesg by unity than the number of terms in the
divisor. This will mark the jjoint at which the remainder begins
to be formed. We then divide 5 into 10, and thus obtain the
first coeflBcient 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 the nest 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 woric would be embarrassed with fractions. Yv'^e may
then proceed as in the following examples :
6. Divide z^-dx^+x^+Bx''-x + 3 by 2a;3+a:3 -8x + l.
Let 2x = y, or x = — .
^ 2
Substitute ~^ for x in the dividend and divisor, and we have
¥1
2^
24
+
y
+ 1^_
-1- 23
+ 3 -
%3
23 ' 23 2 ' '" ■ 23 ' 23
_ y5 — 2x3?/4+227/3 + 23x8j/2-24?/+25x3
+ — ^ + 1
2^
y3+y8_2x3y+22
p ~~~
_ y^ - 6<At+ 4</3 + 24y3 - 16?/+96
■y3 +2/2 -6.7 + 4 A.
28
MULTIPLICATION AND DIVISION.
Dividing y* -G?/^ + 4?/3 +24?/3 - 16?/+96 by yS 4.^9 _ 67/+4 by
the ordinary metliod, and the quotient by 2^ we have
y^-7y+17
23
J_ 39^^2 - 1147/ - 28
p- "73^^/2_62/-+.4 •
Substituting for y its value 2a;, and simplifying we get
2
7.7; 17
1 39x2 -57a; -7
8' 2a;3+a;2_3a.+i*
5.
By comparing the dividend of ^ 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 the same
numbers. Dividing then by Horner's division we get the 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, 23. Hence
the work will stand as follows : —
a.5_3a;4+a;:
12 4
5+ 3x2- a;+ 3-2a;3+x2_3s5+l
8 16 32 12 4
-1
+ 6
-4
1 -6 +4
-1 +7
+ 6
+ 24 - 16+96
-17
- 42 +102
- 4 + 28-68
1-6+4
1-7 +17
a;2
* Quotient = -p —
-39 +114+28
Ix 17 1 39x2-
114.7; 28
^^ 4
x^
2
^ ^ 2x3 +x
7x 17 1 39x2
4 8 ~ 8' 2x3 +
2-3X + 1
-57x-7
x2 - 3x+"l*
*It v/iU, in general, be as convenient to multiply the dividend by such a. nnia-
her as will make its first term exactly divisible by the ftrst term of the divisor, and
afterwards divide the quotient by this multiplier.
MULTIPLICATION AND DIVISION,
7. Divide 5x'+2 by 3x^-2x+B.
29
5x^
1
0 0
3 9
0
27
0 +2
81 243
+ 2
-9
5
0 0
10 +20
-45
0
-50
-90
0 + 486
-280
+225 +1260
5
+ 10 -25-
-140
- 55 +1746
+ 2 H- 8a;2-2a;+3
Coeffs. of Quotient = 1 —
3 ^ 32 ys 34
-2 +9
1 65 --"^M
Quoncnt= 4. — -
3^9
'J.^)X
27
Exercise xiii.
140
84 3-2+3
55a; -582
81 3a;2-2a;+3"
1. Divide Gx-^ + 5x^-llx^-Cix-+10x-2 by 2a:2+3a;-l.
2. (5x« + Gx^ + l)r(x2 + 2^ + l).
8. {a^-Ga + rj)-r{a^-%>-{-l).
4. (x-5 -4a;3^2 _ 8a;2^3 _ 17^.^4 _ i2f/S)-^(a;3 _2xv_3//2).
5. (a6 - Sa'^x^+du^x^-x'') ^ {a^ -'Sa^x + Sax^ -x^}.
Divide
6. 4a;4+3x2-3a; + lbya;2-2x + 3.
7. lOx^+oa;'^ -90x3 -44x2 + lOx'+l byx2-9.
8. x^ —x^y-\-x^ 11^ —x^ 11^ ■\-xy^ —y^ \>y x^ —y^.
9. Multiply x^ — -ix^'i + 6x2r(2 -4x^3 + a* by x^ + 2xa + a-,
and divide the product by x^ — 2x3(/ + 2xa3 — «*.
Divide
10. x'>-ax4 + /-x3-fex2+rtx-l by x-1.
11. 6x''+7x4+7x3+6x2 + 6x+5 by 2x2+x + l.
12. 60(x-'+^4) + 91x//(x3_2/2) by 12x2 - 13x(/ + 5y».
so
JVITTL.TIPIJCATTr'N ANT> DIVTSTON.
13. Gx'> -iSlz'' -hldx-*^ + 81x^ -8-ix- +8ex- iSi by*- 80.
14. Crx"^ -x^ +2x'^ -2x^ +2x^ + 19x + Q by Bx^+4x + l.
15. a(«+25)3-6(2a + 6)3 by (a-b)^.
16. {x+i/)^^-B{x+y)H + 3(x-fy)z'--^z-'hy{x+y)^ -\-
2(x+ !,)z+z"-.
17. 10xi" + 10a;« + 10a;»-200 by x' 4-:o=' -.c + 1.
J 8. 6?/?x^ 4- {lm-{-cm)x^ +cnx^ -\-abx+ac by hx+c.
19. Multiply 1+ 2^a;-18.'c3 by 1 - L3a;3 4. a^c^ and divide the
product by l+yrc — Sx^.
Find the remainders in the following cases :
'20. (x'- H- 3:c2 + 4x + 5) -:- (x - 2).
21. (x^-3x'-^+x-d)^{x-l).
22. (:c'i-f4a;3-f6a; + 8)-(a;-h2)
23. ^27a:i-v/4)-^(Sa;-2^).
24. {3x^ +ox'^ -dx^-hlx^ -ox-\-S)-^{x" -2x).
25. (5x4 + 90a;3 -1-80x3 -100a;+500)^(a;+ 17).
Art. viii. The following are examples of an important use of
Horner's Division :
1. Arrange x^ —Gx^-\-lx—5 in powers of a; -2,
II -6 7—5
2 2 —8 —2
1
-4
-1;
-7
2
2
-4
1
-2;
— 5
2
-1-3
1:
0
Hence, a;3-6x"+7x-5 = (a;—2)3-5(3;-2)-7, or as it is gen-
erally expressed, x^ - Gx'^ -f-7x— 5 = ?/5 — 5y — 7 if y = «— 2.
MULTIPLICATION AND DIVISION.
81
S. Express x*-hl2x-+-ilx-^ + 66x+28 in powers of. x-^^6.
-3
1
12
-3
47
-27
-GO
2S
-18
-3
1
9
-3
20
-18
6i
-G
10
-8
1
6
-3
2-
-9
0
-3
1
3;
-3
—7
1;
0.
Hence x'^ + 12z^+^lx^ +6Gx+28 = y/4 - 7?/- +10 if y=x+'6.
After a few solutions have been written out in full, as in the
above examples, the writing may be lessened by omitting the
Hnes opposite the increments (—2 in Ex. 1, and 3 in Ex. 2), the
multiplication and addition being performed mentally. The last
example written in this way would appear as foUows :
j 1 12 47 G6 28 .
-3 11 9 20 6 (10)
1 6 2 (0)
1 3 (-7)
1 (0)
Exercise xiv.
1 Exp
2.
3.
4.
5.
6.
7.
8
y.
ress x^ — 5a;- +3x — 8 in powers of x — 1.
x^-\-dx''+Gx+9 " x+1.
x^ -8a;3 +24x2 - 32a;+97 in powers of x- 2.
a;4 + 12a;3 + 5a;2_7
3.c5-a;3 + 4a;2 + 5.T-8
a;4-7a;3 + llx2-7a;+10
a;3_2x--4.c+9
a;»-9a;27/+6x/y2-82/»
x^ — 5x^1/ + 5x1/^ — y*
x+2.
x-2.
x-1^.
x— §.
x-2y.
x-y.
12
SYMMETRY.
If).
'• 8x^-{-11x^y+10xjj^+8?j^
<«
2a; + y.
11.
- x3-^x^+^x-^^4
((
i^-TV
12.
" a;* + 8a;3-15a;-10
<t
x+2.
CHAPTEE H
Section I. — The Principle of Symmetry.
Art. ix. An expression is said to be symmetrical with respect
to two of its letters when these can be interchanged without
altering the exj)ression :
Thus if in a^ + a^x + ax^ + x^, 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 x^ +b^x-l-ha-]-a-x. if Vfe interchange a and 5, there
results x^ + a -a; + ai + i^a; which is identical with the given ex-
pression ; but it will be seen that the expression is not symmetrica]
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 a^ +i3 4-c3 — 3abc remains unaltered by changing a into b,
b into c, and c into a, and is therefore symmetrical with respect
to these letters. So, a^b-{-b^a-\-a^c+c^a+b2c-{-bc^, and (a-b)^
^ (6_c)3 -j- (c — a)3, are each symmetrical with respect to a, b, c.
Again (x-a) (a — b)^ + {x—b) (b-c)^ + (x-c) (e - a)^ 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 b, b into c, c into d h into k, and k into a, without
altering the expression.
SYMMETRY. 88
A symmetric function of several letters is frequently represented
by writing each type-term once, preceded by the letter 2 ; thus for
a-|-6+e+ . . . -\-l we write 2a, and for ah-\-ac-\-ad-\- ....
•\-br-\-hd-\- . . . (?*. e. the sum of the products of every pair of
the letters considered) we write 'Lab.
Exercise xv.
Write the following in full :
1. 2«26, 2(« — ^)^, 2«(/'— c), 2«'^(x — r), y,aH^c, Z{a + b)
X{c—a){c—b), 2 [(a + c)--b^}, and y,a{h-[-c)^, each for a, b, c.
2. :SaAe, y,a^b, 2a'bc, 2{a — ^), and i;a2(a-i), each with
respect to a, 6, c, d.
Shew that the following are symmetrical :
8. (x + a) {((+b) {h-{-x)-\-abx, with respect to a and b.
4. (rt+i)^ + (rt — 6)3 with respect to a and b, and also with
respect to a aud — />.
5. (rt&— xy)^ — ("+'j— 03 — y) {«6(a; + y/)— a:iy(a + /-')} with respect
to a aud A, aud also with respect to x aud y.
6. a'-{b—c) — h^{(i — c) — c^{b — a) with respect to a, 6, c.
7. {ac-]-bd)- -\-{hc — ady with respect to a^ and b-, and also
with respect to c^ aid r/-.
8. x^ +^*^+3x//(a;2+x// + //2) with respect to x and ?/,
9. {x^—y^+'Bxy{2x-\-y)}^ + {y^-x^+3xy{2y-^x)}^ with res-
pect to X and y.
10. a(a-}-26)3-4-/>(6 + 2a)3 with respect to a and 6, and also
with respect to a and — b.
n. ab[{{a + c){b + c) + 2c{a-hh)}^ - (a-cy (h-c)^] with
respect to a, b, c.
12. a^h"^ +b'^c^ +c'^a^ +2abc{a + b+c) with respect to ab, be, ca
With respect to what letters are the following symmetrical ?
18. xyz + 5xy+2{x''+y'^).
14. 2{a^x'^ +h^y3)-2ab{xy+yz+zx).
1.5. {P-k^)--\-iii^{ f+h\^->-(9j-h-2g-'y.
B4 SYMMETRY.
16. (x+y) (x—z) {y — z) — xyz.
17. a^b'^-\-h^c^-{-G'^a'^-'labc{a + h-c).
18. aj6-//«+2«-3(x-2-2/'-')(;?/2-z2) (2-^40-*).
19. (a.-H/,)3+(a + r;)2 + (7;-c)4.
20. (a + i)4 + ((»-c)4 + (i+c)4 + (fl + r)4.
21. (a4-^>)4 + (a-c)4 + (/;+c)4 + (,<4-c)4 + (c- /,)4.
Select the type-terms in :
22. «3+2ai + />2+2k + c2+2m
23. a(&2_c3)4-i(c3-a3)-fc(a2-5-')-}-(f^-f/>) (^ + ^') (^ + «)-
24. «(64-c)2+Z^(c-ha)3-fc(a-fi)2-12aic.
Write down the type-terms in :
25. (.x-+?/)5, (aj _y)^(a5+Z/)'' --^^ -?/■'•
26. {x^yy + {x-y)\ (x+yY -{x-yy .
27. (a;+?/+2)4, {x—y-zy.
28. (a+6+c+t?)4, (aO+&2_^c-'+ci2)2,
29. (a-fZ*)3a.(/;_|_o)3 + (c-fa)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-\-h-\-c+d-\-e-\-&o.Y
This expression is symmetrical with respect to a, b, c, &e. ;
hence the expansion also must be symmetrical, and as it is a pro-
duct of ^(X'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
a 2 and nb are type-terms.
Now (a + b)- =a- -\-2ab-{-b^ ; and the addition of terms involv-
ing a, b, c, &.G., will not alter the terms a^ -f 2«6, but will merely
give additional terms of the same type. Hence from symmetry
we get
SYJIMETKY. ^5
(a + ^-f ,:4-i+f+&c.)- =a- +2fr6+2ac+2rtrf + 2flt'+
+ ^2 +2i:"+2W+2k' +
a.c2 +2a/+2t;e-|-
+ (/2 -\-2de +
+^2 +....,.
This may be compactly written
(2(t)2 = £0.24-22^6.
2. Expaucl (u + h)^.
This has been foimd by actual multiiilicatioii — 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 (« + 6)3 =«3_|_^3 ^n[a~b-{-b-a), where n is numerical.
To find the value of n, puta = ft = l, and we have (1+1)3 =
l + l + ?i(l + l); .-. 7i = 3.
b. Expand {x+y+z)^.
This is of three dimensions, and is symmetrical with respect
to a:, y, z. AVe have
(x+i/+2)3 = {{:x+,j)-¥z] 3 = (:c+v/)a +&C.
= x^'\-^x^y+k(i., Avhich are type-terms, the only other possible
type-term being xyz.
Now, since the expression contains 'dx'-y, it must also contain
3x-2, that is, it must contain 3x-(?/4-z)- Hence
(x + y^-z)-^ = x^-^dx^-[y+z)
+2/3 + 32/2(z+a;)
+z^ + dz^'{y\-x)
+ n{xyz), where n is numerical, and
may be found by putting x = y = z=l in the last result, giving
(1 + ] +1)3 = 1 + 1 + 1.-1-8(1 + 1)4-3(1 + 1) + 3(1 +!)+»;
.•. n = G.
36 SYMMETRY.
4. Similavly we may shew that
{a + b-[-c+d)3= a^-^3a-{b+c + d) + 6bcd
+ b^-\-'3b^{c + d+a:) + Gcda
+ c^ + Bc^{d+a+b)-[-6dab
-\- d^ +Bd^{a + b-\-c) + Gabc.
5. Expand (a + b + c-^Szc.)^.
The type-terms are a^, a^b, abc.
Expanding (a + b + c)^, we get a^ + oa^b-\-6abc-{-&0»
Hence by symmetry we have
6. Simplify {a-{-h-2cy+{h + c-2a)^+{c + a-2b)^.
This expression is symmetrical, involving terms of the types
a^ and ab. Now a^ occurs with 1 as a coGfficient 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 Ga^ 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 thiid square, and hence — bah is the second
type-term in the result: hence the total result is G (a2-|-i2_^c8
•—ab — bc—ca).
7. Simplify {x-{-i/+z)^-^{x-y -z)^ + {y~z-x)^ -\-{z — x-y)^.
This is symmetrical with respect to x, y, z; and the type-tenns
are x^, Bx^y, Gi'yz :
(1) x^ occm-s in each of the first two cubes, and —x^ in each
of the second two cubes, :. there are no terms of the type x^ in
the result,
(2) Sx^y occurs in the ,^rsf and ihird cubes, and —Sx^y in the
second and fourth, .'. there are no terms of this type in the
result.
(3) G.vyz occurs in each of the four cubes, /. 24x^2 is the total
result.
8. Fi-ove (a^ + b^+c^+d^) (rv^ +x^ -\-y^+z-)-
{alV^^^bx-\-cy+dz)- = {'ix -6w.')2 -f-(ay — [cu,')2 + {az — dwy -\-
{by-cx)^+{bz-dx)^+{cz-dy)^.
SYMMETKY. 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 interclianged witliout
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', h^x-,
— a^w^, —b^x"^, — 2rtiiia;, and are the terms oi {ax — hw)^ -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
(a;2_y.)3 + (^2_2;g)3 ^(z^-^x>jy-?.(x^-yz) (y^-zx) (z^--xy) is
a complete square.
The expression will remain symmetrical if (x^—yz) (y^—zx)
(z^ —a;?/), instead of being multiplied by -3, be subtracted fi-om
each of the preceding terms, thus giving
(x^ -yz) {{x^-yzY- {y^ - xz) {z'^—xy))
4- (?/2 - 2x) { (j/2 - za;) 2 - (z3 - xy) {x^ yz) }
■^[z--xy) {{z'^—xyY-{x--iiz) (y^ -zx)}
= (x^ —yz)x{x^-{-y'^-^z^ — 2xyz)
-f-ifec.
+ &C.
= (a; 3 -|-?/ 3 _j_^ 3 _ pjxjiz) (a; 3 +.V ^ +s ^ — Sxyz) .
Exercise xvi.
Simplify the following :
1. (a+64-c)2 + (a+6-c)3-f(5+c-«)2+(c-f«-?;)3
2. (a-6-c)2 + (i-a,-c)2 + (c-a-6)2.
3. (a-t-Z^H-c-d)2 + (i+c+(Z-a)2-{-(c-|-(Z+rt-i)2-|-
{d+a-\-b-cY.
4. {a+b+cy - a{b-^c -a) -h{a-\-c -b) -c{a-\-b - c).
5. {x-\-y-\-z+ny -\r{x-y - z-[-nf +{jc-y-{-z -n)- -\-
{x-'ry—z — n)^.
6. (a+6+c)3 + (a+fe-c)3-f(6+c-a)3-f(c+a-6)3.
38 SYMMETRY.
7. {x - 2y - 82) 2 + (7/ - 2^ - dx) 3 H- (2 - 2x - 877)2.
8. (??ia+?i/*-l-rc)3 — (ma-\-nb — rc)'^ — (jii+j-c — ??/a)3 —
(rc-f- WW — «^) ^ .
9. «{6-}-r)(i2-fca-r/2)4.&(c+a)(c24-a2-62)_[_
c(« + 6)((<2_^i2_c;-').
10. (fl/y + 6c+crt)2 _ 2abc{a-}-h+c).
Prove the following :
11. {ax+h]/+cz}- -{-{bx-\-cy-\-az)^ +{cx-{-ay-{-hz)^ +
{ax-'rcy-\-bz)^ ^{cx-\-by-\-az)^ -{-{hx-\-ay-{-cz)^
= 2(a2 + (;2-|-c2j(a;2-{,2/2-|-22)-[-i(ai+ic+ca)(.ryH- ?/z +;?a:;.
12. {a + h+c)^+{b+c-a)^ + (c+a-by-\-{a-\-b-c)*
18. (^a + b + c)^ = Za'^ + ilaH + eia2b^ + V2Za2bc.
14. (£a)4=s«^ + 4Sa3ft + 62rt2/;2 4-i2sa2ic + 242rt6c^.
15. (a2+/j2+c3)3 + 2(a&+6c + ca)3-3(«2 + i2+c;2)x
(^,?;4.ic + C«)2 = (a34.i3_|_c3_.3aic)2.
IG. (a-/.)2(6-c)2 + (i-f)2(c-a)2 + (c-a)2(a-&)2 =
(a2+/>2^.e2_a6-r/c-6c)2.
17. (2rt-?;-c)-'(2&-c-«)2 + (26-c-a)2(2c-a-6)2 +
(2c-a-&)2(2a-fc-c)2 = 9(«2+fc2 + c2_afe_6c-oa)2
18. (rtr2 + 2?9rs+Ps2){aa;2 + 2te// + fv/2)_
-( a?-a; + b {;nj 4 sa;) -j-csij}^ = {ac-b^) {ry - sx) 3 .
19. {u^ +ab + b''){c^ + cd-\-d'^) = {aG + ad + MY~ -\-
{ac-\-ad+bd){bc-a(l)-\-{bc-ad)^.
20. Sbew that there are two ways in wliich the given product
in the last example can be expressed in the form p- +p(j + g-, and
two ways in which it can he expressed in the form p^ -pg+g^-
21. 6{iv^ ^x^+ij^ +z^)^ ={w + x)'^ +{w-x)^ + {iv-\-yy +
{w-y)^ + {w + z)^ + (w-z)^ + {x+yy-j-{x-yy + {x+zy +
ix-z)^ + {y+zV-\-{y-zy'
22. |{(« + />+6')5 + (a-i-c)6 + (5-c-a)6+(c-a-i)'>} =
x\\a-\-b + c)^-]-{a-b-cy + {b-c-a)^-t{c-a-by}x
ij(a+6+c)2 + (a-5-c)2-|-(i-c-a)2 + ((;-«-i)2}.
THEORY OF Dr%'ISORS. 89
Section II. — Theory of Divipors.
Any expression which cau be reduced to the form /7y"-f-^'e""^ +
«c"""*+ • . • . + ■ • ■ +hx + k, in which n is a positive
integer and a, h, c, . . . . h, k are independent of x, is called
a PoLTNOME in x of degree n.
The expressions /(a;)", F{x)", pix)"", are used as general syniL . ;
for polynomes ; the index n. m. indicates the degree of the poly-
nome.
Theorem I, If the polynome/(x)" be divided hyx—a, the
remainder wiU be/(rt)".
Cor. 1. f{xY — f{aY is always exactly divisible by x — a.
(Particular case: k" — a"is always exactly divisible bya* — «).
Cor. 2. If /(a)" = 0, /(a;)" is exactly divisible by x — a, i.e.,f{x)"
is an algebraic multiple of x — a.
Cor. 3. if the polynome /(a;)" on division by the polynome
^{x)'" leave a remainder independent of x, such remainder will be
the value of /(a;)" when ^{x)'" = 0.
Examples.— Theorem 1. .
1. Find the remainder when x^ —'lx^-\-13x^ — IGx^ -\- Ox — 12 is
divided by a; — 5.
The remainder wiU be the value of the given polynome when 5
is substituted for x. (See Art. III.).
1
-7
5
+ 13
■ 10
16 +9
15 -5
-12
20
1
-2
3
-1 ' 4;
8
Hence the remainder is 8.
2. Find the remainder when (x — a)^ + (x — h)^ -j- (a-\-h)3 is
divided by ar+a.
For X substitute - a, then {-2a)^ +(-n-h)^-\. {a + />) 3 = _ Sa^.
8. Find the remainder Vfhen x^+a^ + l>^ + {x-{-a){x+b){a-{'b)
is divided by x-i-a-\-b.
40 THEORY OF DTVISOKS.
For X substitute —(a + l) and ^ve get
-(a + b)^+a^ + h^+ab{a + b) = -2ah{a+bl See Formula [6].
4. Find the remainder when {x^ + 2ax — 2a^)^{x^—2ax—^a'^)
+ B2{x - «)4(a;+a)4 is divided by x^ - 2«3. '
x^ — 2a^ may be struck out wherever it appears.
Tliis reduces the dividend to
(2ax)^(-2ax)+B2{x-a)^[x + a)^ = -IGoAx^ -^d2(x^ -a^)*.
In this subafcitute 2a^ for x^ and it becomes
-64«« + 32rt.8 = _32a8,
which is the required remainder.
Exercise xvii.
1. Find the remainder when Sx'^-{-(iOx^ + o4x^ — 60x-\-58 is
divided by a; + 19.
2. Find the remainder when px^ — dqx--{-'drx — s is divided by
z — a.
3. What number added to 4;c' 4- 34aj4 + 5Sx^-h21x^ - 123a;- 41
wiU give a sum exactly divisible by 2x-+13 ?
4. What number taken from lOaj^'^ - 20a;8 -lOa;^ - -SOa;*-
8*9a;24-20 will leave a remainder exactly divisible by 10a;- — 11 ?
• Find the remainders from the following divisions :
g. (a;4-l)5_a;5 ^^^. + 1, aud (a;+rt+3)3 - (a;+«+l)3 ^ x+2.
6. x"+y''^ x-y; x^''-{-y-'^ ^ x+y; x^"+'^+y'^''^-^^ ^ x-\-y.
7. (a; + l)3+a;3 + (a;-l)3-f-a;-2.
8. (a;-a)3(a;+rt)3 + (a;2-262)3 -^a;3+.';^
9. {x^ +ax+a^\{x^ -c(x+a^)-{x^ -'dax-^2a^)(x^ -\-Bf(x + 2a-')
-- x"- -\-2n^ .
10. (9a3+6rtZ» + 4A3)(9a3-6a/;+4Z/2)(81r/4_30rt3i3 + IGi*) -i-
f3a-26)2.
11. a'{x-a)^-\'h-{x-b)^ -^x — a-b.
12. {ax + by)^-{-a^y^-\-b^'x^ — 8abxy{ax + by) -r- (a-i-b){x4-y).
18. x'^ + ci'^ -{• b'^ — oabx -i- x — a + b; also -4- a; + a— 6 aiso*
x~ a — b.
THEORY OF fciVlSuSS. 41
14. Any polynome divided by x — 1 gives for reruaindev the
sam of the coefficients of the terms.
Examples. — Cor. 1.
1. x'^-i-y'^ is exactly divisible by x+y.
In " x^ - a^ is exacth' divisible by a; — «," substitute ~y for a.
2. m.r^ — px^ ^qx-{-}n -r p +q is exactly divisible by a;+l.
This may be written
{)> x" - px- -\-(jx} — {;«( — 1)3- p(_i)2_|.^^_l)i. ig exactly divi-
sible by a;- (-1).
3. {x- -{-Qxii-{-hj^Y -\-{x" -\-~xy-\-Ay^)'^ is exactly divisible by
(;>-+2?/)3. For (x3+G.r?/-f 47/3)5 -(_a;2_2a;y -42/2)5 ig exactly
divisible by {x'^+C)xy-\-Ay^) — {—x^ — '2.xy—^y^), which is
2(a;2-f 4a;iy+47/3) = 2(a;+2?/)2.
Exercise xviii.
Prove that the following are cases of exact division :
1. a;'2n+i_f.j/2n+i H- a; -f-y ; x-" - y^n ^ a:-f ?/,
2. a-134.,/13 -^a;4_j_^/4 ; a;3o^y3o _i.a.(;+yC. also -=- a!i«>4-2/"'
also -^- X' +y^.
8. (fla;+6?/)5 + (te+fn/)-' -^ (a + i)(a; + ?/).
4. («a; + i?/-fc.;)3 — (6x+f?/H-fl2)3 h- (a — i).« + (i— c)^-i-(c — a)2.
5. (2?/-cK)''-(2a;-?/)«-r3(7/-a;).
6. (2i/-a;)2'»+i+(2a;-?/)2"+i--i/+a;.
7. {my — nxY — {mx — ny)^ -^ (m+n) (y — x).
8. (a:-|-2/)« + (a:-i/)«^2fx2+7/2).
9. (a;2-fx^+y2)3 + (a;3_a;^ + 2/2)3H-2(a;3 + //2}.
10. (, + i)9_(a-i)9 -^2/>(3^/ 2+^/2),
11. (•j'^ + 5ft.i-+i2)7_|_(a;2_Z,a,-+i2)T ^2{x+b)».
12. (a + 6)*«+2+(«-6)4«+2-=- 2(a3 + i2).
13. {x^ + Sxy{x-rj)- y^\^ + {x^ -Qxy{x-y)-y^}^-^2{x-yY
14. 3x-3-.5a;2 + 4x-2-r-x-l.
42 THEORY OP DmSOBS.
15. Any polynome in x is divisible by «— 1 when the stun of
the coefficients of the terms is zero.
16, Any polynome in x is divisible by ic+l, when the sum
of the coefficients of the even powers of x is equal to the sum of
the ('oefficients of the odd powers. (The constant term is in-
cluded among the coefficients of the even powers).
Examples. — Cor. 2.
1. Show that a{a-\-1h)^ — h{1a-\-hy is exactly divisible by a +6 .
By Cor. 2, the substitution of — b for a must cause the polynome
to vanish.
Substituting ; a{a- 2a) 3 -f a{2a - a) ^ = - a* +a* = 0.
2. Show that (ab — xij)^ — {a + b — x-y){ab(x-ry) — xy{a-\-h)} u
exactly divisible by {z — a){y—a), also by {x — b){y-b).
For X substitute a and the expression becomes
{ab-ayy - {h - ij){ab[a + y) — ay{a-{-b)} =»
aHb-y)^-{b-y){a^{b-y)}=0.
The expression is, therefore, exactly divisible by ic — a. But it
is symmetrical with respect to x and y, hence it is divisible by
y — a, and a,s x — a and?/ — « 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
hy(x-b){y-h).
3. Show that 6{a'^-\-b^+c^) - Bia^+h^ +c^){a» + h^ + c») is
exactly divisible by a+i+c.
For a substitute — (6+c) and the result which would be the
remainder were the division actually performed, must vanish.
^6{- {b-{-c)^ +b- +c^} - 5{-{b + c)^ +0^ +cS} {{b-\-c)^ + b^ + c^}
= 6{ -(6+c)5 + 6^ +c^}-j-dObc{b+c){b-^ +bc-\-c^). See [1] and [6] .
The expansion being of the 5th degree, and symmetrical in b
and c, it will be sufficient to show that the coefficients ofb^, b*Cf
63c> vanish, she coefficientr, of b^c^, be*, c^ being the coefficients
THEORY OF DI^^S0R3. 48
of the former terms in reverse order. Calculating the coefficients
of these type-terms we get
6{ -564c -1063c8-...}+30(64c + 263^2 + ...),
which evidently vanishes. Hence the truth of the proposition.
4. Ifa +6 + c = 0, ^(a5 +b'' +c-=) = ^[a- +b^ +c3)-^(a3 + 63 +c^).
In the last example it has heeu proved that the difference of the
quantities here declared to he equal, is a multiple of a + 6-fc, 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 — by)^ + {bx-ay)^ — {u^+b^){x'^-y''^)-7-a,b,x, y, a+\
x-y.
2. ax^ - (rt2 +6)a;2 +62 -f- ax-b. (Substitute ax for b.)
c J {ax+bTj)^-{a-b){x+z){ax+by) + {a-b)^xz-^x+y.
\ {ax-by)^ -{a + b){x+z){ax-by)-^{a + b)^.iz -^ x + 'y.
4. da^x^'-iax^ —10axy-^Ba^xy + 2x^y-{-oy^ -r-2ax — y,
6. l-''2a*x—16-'32a^x--^4:-8a^x^ + -Qax'^ — x^ -i- 'Gax—2x^.
6. x^ +x^y'+X'y-^y^ -i- x^-ry.
7. {c-d)a^-\-G{bc-bd)a + d{b^c-b2d) ^ a + 3b.
8. 3r{x-^y)^+y{^\x-y)'^ -^x-y.
9. a{a + 2h)^-b{b + 2a)^ -i-a-h, also H- a+b.
10. a^-^2a^b + a^b^+a2x^-2abx^+b^x^ -i- {a-h){x-ha).
11. a(ft-c)3 + 6(c-a)3+c(a-6)3 ^ {a-b), (b-c), (c-a).
12. a3{b-c)-\-h3{c-a) + c^{a-b) -J- (a-Z>), (b-c), (c-a).
13. a4(6-c) + 64(c-a) + c4(a-6) -4- (a-b), (b-c), (c-a).
14. (a-6)2(c-J)2 + (6-c)2(ci-a)3_(,Z_6)s^a^.)s ^ ^a-b),
{b-c), {c-d), {d-a).
15. {(a-i)2+(6-c)--^ + (c-a)2}{(a-5)2^2 + (&-c)2a2 +
(c_a)263|_|(a_6jS^.4-(6-c)2a + (<:-a)2Z>}2 -^ (a-i), (b-c),
(e-a).
16. (a;+!/)(!/+~')(2+a;) +a;.V2 -r a.- + J/+2.
44 TBT'.OKY OF DIVISORS.
17. ab{a' — b^)+oc(b- -c")-^ca{c^ ~a^) -^a + b-^e.
18. {ab-bc-ca)l-an^-h^c^-c^(i^ -^a-k-b-c.
19. (rt + 26)34.(26-3c)3-(8c-a)3+rt.3 4.863 _27c» +
?+26-3c.
20. aH^+b^c^+c^a^ -Za^h^c^-~ab^bc-^ca.
ExAilPLES. CORS. 3 AKD 2.
1. Find the value oiAx^ -\-2x^ - ox'^ + 23.C+6 when 2^2 = 3a5 ^ 4,
Since 2a;2 — 3a;H-4 = 0, we have simply to find the remainder on
division by 2a;2 — 3x+4, and if it is independent of sc, it is the
yaiue sought, Cor. 3.
14 0 9 -5 28 6
3 ■ 6 9 15-3
~4 -8 -12 -20 4
~2~| "2 3 5 - 1; 0 10
Hence the required value is 10.
2. What value of c will make x^ — 5x" +7x — c exactly divisible
by a;- 2.
If 2 be substituted for x, the remainder must vanish. Cor, 2.
j 1. -5 7 -c
2 2 -6 2
!l -3 l;2-c
Hence 2 — c=0, or c = 2.
3. What value of c will make 6x^ — 5x^+cx^-20x^+19z-5
vanish when '2.x^=dx—l ?
By Cor. 3, the remainder must vanish when the given poly-
nome is divide by 2a;2 — Sx+l. 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 first express cx^ in lower terms in x, and
then divide and find the requii-ed value of c from the remainder.
1st. Method, (see page 28),
g _10 4c -160 304 -160
3
-2
18 24 12C+36 36c -420
-12 -16 -8c-24 -24cH-280
8 4c-f-12 12c -140; 28c- 140 -24c+120
THEOKY OF DIVISORS. 45
Hence 28f; = 140 and 2 It- = 120. Both of these are satisfied by
2nd Method, x^ = ix{3x-l) = ^x^- ^x=i{3x-l)-ix =
2\x-^ — ix— lix — i ; .-. cx^ = l^cx — ic.
Substituting for cx^ in the given polynome it becomes
6.^5_5a;4-20x3 + (13c + 19)x-|c-5.
Divide and ajjply Cor. 3.
6 -10 0 -160 28c + 304 -24c- 160
3 18 24 36 -420
—2 -12 -16 - 24 280
"6 8 12 -140; 28c -140 -2Tc + 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 clearly that 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* -Bx^ + ix^ — Bx+4:, given x^ =x — l.
2. a:^-2a;4-4a;3+13a;^-lla;-10, given (a;-l)2 = 2.
3. 2x-^- 7*4 4-12x3 -11x2 + 2a; -5 given (x-l)3 + 2=:0.
4. 3x6 + llx= + 10x3+7x2+2x + 3 given x' + Sx^ -2x-t-5 = 0.
5. 6x^ +9x« - IGx* - 5x3 - 12x3 - 6x + 60 given 8xA +x-4 = 0.
What values of c will make the following polynomea vanish
under the given conditions.
6. x4 + 13x3 + 26x2+52x+ 8c, given x + 11-0.
7. x4 -2x3—9x2 +2cx- 14, given 3j-+7 = 0.
8. X* — 4x3— x2 + lGx + 6c, given x2 =x+ 6.
9. 2x4 - 10x2 +4cx+6, given x3 + 3 = 3x.
10. 2x4+x3-7cx2 + llx+10, given 2x = 5.
4:6 THEORY OF DIVISORS.
11. 4a-4+ra-2 + 110ic-105, given 2a- 2 - 5a;4-15 = 0.
12. 3x^-~16x'^ + cx3-5x^-lUx-{-200, given x^ = Sx-A.
13. What values of ;j and g vfiW ma.'ke x'^-\-2x^ —lOx^ —2^^+^l
vanish, given x^ = 3{x — l) ?
14. What values oi p and q will make a^ ^ _ g^ 1 0 _^ 10^ k _ 15^16
-}-29a* —pa^+g vanish, given (a^ —2)2 =«" — 3 '?
Theorem II. If the polyuome/(x)" vanish on substituting
for a; each of the n (different) values a-y, a^, a^ . . . a„
f{xy = A{x-ay){x-a2){x-('^) .... {x-a„)
in which A is independent of x and consequently '6 the coefScient
of af in /(a;)".
Cor. If /(a;)" and ^(a;)'" both vanish for the same ni different
values of a;, /(a;)" is algebraically divisible by f (a:)™.
Examples.
1. x^-i-ax^+bx + c will vanish if 2, or 3, or —4 be substituted
lor X, determine a, b, c.
The coefficient of the highest power of a; is 1 ;
.-. x^ +ax2 + bx + c= {x-2){x-3){x + 4:) =x^ -a;2 - 14a;4-24.
.-. a= -1; b= -14: c = 24.
2. x^+hx^-\-cx-{-d will vanish if —3 or 2, or 5 be substituted
for a;,*detennine its value if 3 be substituted for x.
The given polynome =(x-^-3)(x — 2){x — 5);
.: the required value is (3 + 3)(3-2)(3- 5) = -12.
3. aa;3 4. 3/;^2_|_3ca;+c? will vanish if for a; be substituted —3,
or 1-, 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 a;
.-. ax^ + 3b.v^-\-3cx + d=^a{x+2,){x-\){x — H)
:. a(3 + 3)(3-i)(3-li) = 45; .-. a = 2.
:. 'lx^ + 3hx^ + 3cx+d = <2,{x+3){x-\){x-\\)
.-. 6 = f, c-= -'6k, d= H
THEORY OF DIVISORS. 47
4. If x^ +px^-{-qx+r vanish for x — a or b, or c, determine p, q,
and rin terms of a, b, c.
x^ +px^ +qx-[-r = (x - a){x - b){x - c)
= x^ ~{a + h-\- c)x^ + {ah + bc+ca)x — abc
.'■ p= —{a-^b-\-c) or — 2rt.
q= ab + bc+ca or Sr//;
»• = — abc or — 2 abc.
0. If x^+jJX' -i-']x-\-r vanish for x = a, or b, or c, determine the
polynome that will vanish for x = b + c, or c+a, ©r a + h.
Since x^-\-px^ +qx+r vanishes for x = a or b or c,
a; 3 _2)a-2_|_,^^._^ -^j^ vanish for a;= — a or —6 or — c,
and — ^i-— a+^+c;
But the required polynome will vanish for
x= —p —a, or —p—b, or —p — c;
that is, for x+p= -a, or —6, or —c.
Hence it is {x-^-j')^ —p{x+p)^+q{x+p)—r =
x^-\r'lpx^ + {p^-\-q)x-\-pq —r.
The following is the calculation in the last reduction. (See
page 31).
\ -p q -r
1 0 q ; pg-
1 p; p^ + q
1; 2p
1 ■
6. In any triangle, the square of the area expressed in terms of
the lengths of the sides, is a polynome of four dimensions ; and
the area of the triangle, the \engths 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 13 divisible hy a-{-b — r. Similarly it is divisible by b-{-c-a
and bv r + a — b.
48 THEORY OF DI^'lbORS.
2nfl. The area vani'^he'! if the three sides vanish tn^rether,
hence if a-\-h-{-r = 0, .4 = 0, aud consequently A is divisible by
We have thns found four linear factors, but A is of only four
dimensions.
.-. A = vi(a + b + c){h -^-c - a){c+a-b){a+h - c),
in which m is a numerical constant.
But 63 or 3G = m(3 + l + 5)(i + 5-3)(6 + 3-4)(3 + 4-5)
=f51Gm ; .-. 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 « = 0, 6 = c, and .-. a-^b = c, which is
included in 1st. ; if two sides vanish simultaneously, the three
must vanish).
EXAJIPLES ON THE CoROLLARY.
7. Prove that (z+l)^ ^ -x^ ^ -1x - 1 is divisible by
Factoring the latter expression we find it vanishes for ar = 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-\-y 4-z)^ -x^ - y^ —z^ is divisible by
{x-\-y + z)^ - x^ — y^—z^'.
The latter expression vanishes il j;= —y, so also does the former.
By symmetry they both vanish if ^/=r — z and ii z= —x. Hence
they are both divisible by {x-{- y){i/ +z'(z+x). But this expres-
Bion 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 {{<( + by -}-{c + dy}{a-b){c-d) +
{{b+c)'' + {a + dy}{b-c)[a - d) + {{b + d)'' + {ci-ay}{b-d){c-a)
is algebraically divisible by {a — b){c — d){b — c){a-d){b — d)(c — a)
y^^a + b+c + d), and find the quotient.
Let a = b and the former polynome reduces to
{(^a+cy+{a + dy}{a-c){a-d)-}-\{a-\-dy-{-{c+ay}{a-d){c-a)
THEORY OF DIVISORS. 49
which vanishes, the second comiilex term differing from the first
only in the sign of one factor, having (c — a) instead of (« - c).
Hence the former polynome is divisible by a — b, and by sym-
metry it is also divisible by a — c, by a — d, by b — c,hyb-d,hyc — d.
Again, (a + b)^ +(c-\-d)^ is divisible by {a-\-b)-}-(c -rd); for, on
putting a + 6= — (c -f-c/), it becomes {—{c-\-d)}''-{-{c + d)^ which
= 0.
Similarly the other terms of the former of the given polnomes
are each divisible by a-\-h-\-c-[-d, and consequently the whole is
so divisible.
Now all these factors are different from each other, hence the
former of the given polynomes 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.
Putrt = 2, b = l, c = 0, d=-l, and divide. The quotient will
be found to be — 5.
... {(a + by+{c-\-dy}{a-h)(c-d) + {{b + cy+(a + dY} X
(b-c){a-d)+ {{b+d)^ +{c + a)^}{b-d){c - a) = -5{a-b){c-d)
x{b-c){a-d){b-d){c-a){a-\-b+c + d).
' N.B. — It is not always necessary to find the factors
of the divisor, as the following examples show.
10. Prove that x2+x+l is a factor o^ x^'^+r,^ + 1.
.-^2 4-a;+l will be a factor oi x^'^+x'' +1 provided
a;i4+a;7 + l=:0 if x2+a;+l = U.
Ifa;2+a: + l =0
.-. x^+x-+x =0
.-. x^+x^--\-x+l-=l
.-. X3 =1
, . ^*=landa;i2 = l
x'' =x and x^^ = x^
..x^'^+x'+l =x^+x + l = 0
.. ic2+a;+l is a factor of ic^^+x^yfl.
60 THEORY Ol-^ DIVISOKS.
Art. XII. Two other methods of proving this proposition
are worthy of Botice.
1st. x^+x+1 will be a factor of x^^+x'' +1 provided it is a
factor of {(.-t;i4+a;7 + l) + a multiple of (x^+x + 1)}.
a;^^+.<;' +1 differs by a multiple of x^ i-x+l from
x^^+x''^{x^+x+l)+x^{x^+x-^l)+x'+x'^{x^+x + l) +
x{x^+x+l) + l
= x^^{x^-{-x + l)+x^{x^-^x+l)+x^(x^-^x+l)-ix\x---rx-rl) +
(x^+x+l)
= {x' 3 ^^-i +a;6 _|.,^3_j_i)(^3 +a; + 1).
Hence x^ +x+l is a factor of x^ ^ -\-x'^ + 1.
,-, -, x^^+x'+l a;3i-l x-1
2nd. — „ —- = - . — — . -^ — - =
x^-\-x-t1 X' —1 .c-^ — 1
{x^^-l){{x^^-l)-x{x^^-l)]
{x' - l)(a;3 - 1) -
(a;2 1_l)(a;>5_l) ^ x{x^^-l){x^*-l)
{x'-l){x^-l) {x^-l){x'-l)
But we see at once that on reduction both of these fractions
give an integral quotient, hence {x'^'^-^x~ -i-1) -i-x^ +x + l gives
an integral quotient.
11. x^+x+1 is a factor of (x + iy -x'' -1.
If a;-+ic+l = 0, (a^+l)^ -ic^— 1 will vanish also, for in such
case a;+l= —x^.
... ^r^j^xy ^x' -\ = {-x'^y -x-" -X^-x^^-x'' -\,
■which by the last example vanishes if x^ +x+ 1 = 0 ;
.-. je-+^ + l is a factor of {x-\-\y —x"^ 1.
For X substitute — and multiply by y^ and \f respectively,
V
and this example becomes
a;2_j_a;,,y_j_^2 is a factor of {x-\-yy —x'' —y''.
THROUY OF DIVISORS. 51
Exercise xxi.
Determine the values of a, b, c, d, e, in the lollowing cases : —
1. z^ + Sbx" + 'dcx-\-d YiimHhes lor x = 2, or 3, or 4.
2. x^+cx^+dz + e " " a;=l|or -3 or 4^.
3. z'^+bx'-\-cx+24: " " z = 2ov-d.
4:. ax^-{-bx'^+cx+dO. " " x-=3or-5or2.
C). nx^+cx^ -dOx + e. " " «= 1^ or -4, or 2^.
6. 81x'i + 6rx2+4f?a;+« " " a;= ij or -3} or 1^.
7. ax'i'+bx^-hcx' -81 " " ,« = | or f or 3.
8. ax'^+cx^+dx+e " " ,f = 2 or 1^ or -1 and be-
comes 14 for a;=l.
9. ax^-{-cx-\-d vanishes for x=i.\, or 23-, and becomes 49 for
x = 3, determine its value for a; = — 3.
Given that x^ - px^ +qx — r vanishes for x = a, or b, or c, deter-
termine the polynome that vanishes for
10. x = c.-\-l, ori + 1, or c + 1.
11. a; = a — 1, or 6 — 1, or c — 1.
.1 1 1
12. x=l -—, or 1-—, or 1-—.
a 0 c
13. x = (tb, or be, or ca.
14. x = a^, or b^, or c^.
f r ^
15. x = a{b + c),oi- b(c + a), of c(a-{-b). ^a(J)+c)=q 1.
I ^* i
16. x= or or - — \ — '—=- l.,i-
c a 0 [ c c )
Prove that the follovfing are cases of exact division :
17. {X -- 1) 1 3 _a;G _L(a;2 _3. J. 1)2 ^ ^s _2x^ +2x- 1.
18. {x-l)^^-x^-\-{x'^-x+l)^-^x^ — 2x^ + 2x-l.
19. (a;-2)io(2a;— 5)i»-«io + 2iO(a;3-4.r+5)5^
a.3_0a;2 + i3a._io.
20. (x2+4:X+3y^-x'^-x''-5x-S^x^+6x^+8x-\-B.
21. (9a;-4)2i(a;-l)2i-.a;3i-(9a;2-14a;+4)3i-=-(a;-l)x
(9a;-4)(9x2_l4a;+4).
22. {6(a;-l)}i3_(2a;3 + 3a;-4)>3_[.(2a;3_3a;-f2)i3_^
{2;c^ +Bx-i){2x^ -8x + 2){x--l).
52
THEORY OF DIVISOKS.
23. {2{z+l)(x-2)]-^-' -\-(x^ -Sx+3)^' -idx^ -5x-iy^
ix-]-l){x-2){x^ -Sx-\-3){3x^ -5x-l).
24. {6{x- !)}!«- (2a;3 +8a;-4) i ^ - (2a;=— 3a;+2)i ^ +
2(2x2+3a;-4)8(2a;3-8a; + 2)4^(a;-l)(2a:2+3a;-4)(2a;2-3a; + 2)
. 25. {2(x+l)(a; -2^,[2o_fa;2_3a.+3)3o_(3a.3_5a:_i)2o_^
2(a;--3a; + 3)9(3a;3-5a;-l)ii^(.T4-l)(x-2)(a;3-3ic+3) x
(3a;2-5a;-i).
26. l+a;4+a;8 -^ l-frc'4-a;3.
27. a;io+a;^?/^+^yio -^ a;3-|->'^^•// + ?/^•
28. l-^x^+x^+x^+x'^^ -j- l + x+a;2+ic3+a;*.
29. l+x^+x^-\-x^^ + x^^ H- l + a:+a:3+a;3+a;4.
30. a:i^+a!i°^^+a;^?yi*'4-//'^ -^ x^+x-y+zy'^-\-y^.
31. a;!' +a;-i+a;3+x--J-l -=- a;4+^--+.c3+x+l.
32. l+x+x--{-x^+x^+x^+x^^ ^
1 +a;+a;2 +a;3 +a;4 +a;* +x6.
Find the quotient of the Ibllowing divisions in which B denotes
the product
{b-c){c-a){a-b){a-d){b-d){c-d) ;
33. {h^c^ +a^d2){b-c){a~d)-i-{c2a^ +b^d^){c-a){b-d) +
{a^b''+c^d'^){a-b){c-d) ^ D.
34. (bc+ad){b^ -c^){a^ -d^) + {ca + bd){c^ -a^){b^—d^) +
{ah+cd){a^-b^){c2-d^) -r- D.
35. {b + c){a + d)(b^-c--^){a^-d^)-{-the two similar terms h- D.
36. {b^+c^){a2+d^){b-c)(a-d)+ " " -^ i).
37. {6c(6 + c)3+a£i(a + tZ)3}(6-c)(a-f/)4- " -^ D.
38. {6c(i + c) + a%^+rf)}(&3_c2)(a2_,/2,^. » _^ j».
39. {6c(63+c3)+af/(a3+cZ3)l(ft_c)(a-(i)+ " -=- D.
40. (6+c-a-rf)4(6-c)(a-(;)+ " -^ 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.
THEORY OF DIVTS0E3. 53
42. In any trapezium the square of the area expressed in terms
of the lengths of the parallel sides and the diagonals, is a poly-
nome of four 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.
{z-a){x — h) {x—b){x-c) {x-c)(:>-—a)
^' {c^a){c^^b) + (a-b){a-c) "^ (b'^){b':^) -^ = 0, if a,
b, and c are unequal ; for this is a polynome of two dimensions in
X, but it vanishes for x = a, and, therefore, by symmetry for x = b,
and for a; = c, that is, for three different values of x, hence it
vanishes identically.
2. {(a + 5)2-f(c+c/)2}(rt-ft)(c-J) + {(6-f6)3 + (5 + c03}
{b-c){a-d) + {{c + a)^ + {b + dy}{c-a){b-d} = 0.
Substitute b for a and the expression becomes
{(6+c)2+(6 + rf)2}(6-c)(6-rf) + {(c + &)3 + (5+^/)2t(c_6)(6_rf)
which vanishes, hence the given expression is divisible by a~b,
and consequently by symmetry it is divisible by (a—b), (b - c),
(c — d), {a-c), (b — d), and (a-d), But the given expression is of
only /o?tr dimensions, while it appears to have six linear factors,
hence it vanishes identically.
Exercise xxii.
Verify the following :
X^r-Z^ , (x--b--)(y^-b2)(z^-b^) (X^'-C^){y2_c!i)(z2_c2)
z=x^+y-+z^-b^-c^.
54 THEORY OF T>TVTS''Pg,
1 1 1
{x-\-<(){a-b){ci-c) [z+b){b-c){b-a) {x+c){c-a){c-b)
1
(x + a){x-\-b){x+cy
5. bc{b^—c-) + ca{c" — a-)-{-nb(n^ — b^> =
{a + b + c){a^{b-c)^b'^c-a)-\-c^[a-b)}.
,, a-\-x a+y/ ^ a+z ^ a
x{.c-y)[x—z) "*" y{v-x){y-z) z{z-x){z-y) ~ xtjz
"^2 (5 _ cy+F 2 (c - a ) + c'^ (« - ^)
(«,3 + ?,3)(,C+,p)(^3+J2).
(rt -i)(6 — c)(c — rt)
10, {-x+y+z){x — y-\-z){x+y—z)-{-x(x — y+z){x-hy-z) +
y{x + y-z){-x + y+z)+z{-x+y+z){x-y-\-z) = Axyz.
(^3-^2)3 4-(&3-c2)3^(g3_^3)3
(a+l;)(6+c)(c+ttj
(a-6)3 + (Z'-c)3+(c-a)3.
2(x// + 7/2+2a;)2.
Theorem IV. If the polynomes J\x)". (pix)"^ {n not less than
w) 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.
THEORY OF DIVISORS. 55
(This is called the Principle of Indeterminate Coefficients. The
full use of it cannot l)e exhibited till the'student is able to work
simultaneous equations.)
Examples.
{a-bXa-c){a-d) + [b-<i){b-cXb- d) +
':^ rfs
{c-a){c-L){c~d) + {d-a){d-b){d-c) " ^•
Assume
ix-a){z-b){x-c){x-d) "
^i B C D
m which A, B, C, B are independent of x.
Multiply by {z-a){z-h){x-c){x-d).
:. x^ = {A ■\-B+ C'+ Z))j;3 +terms in lower powers of x.
Now this equahty holds for more than three values of x, hold-
ing in fact for all finite values of x.
:. A^B-^C+D = Q
Again multiply both sif es of (a-) by x^ci
{x-b)ix-c){x-d) - ^ l^^rr + -^~;r + -^-TJ^-^-'*)'
Put x = a
m
a-
= A.
{a-b)(a-c){a-d)
^' ^^"""'"-^ (.-.)(/-c)(6-J) = B. *o.
Adding
(<^-'^)i"-<^)(«-^) "^ (i-a)(6-c)(7>-rf) + {c-a){c-b)(c-d)
(a-b){a-c) "*■ (6-c)(Z;-a) + ~(7^a)(V=6)"
= (a-r6 + c)2.
66 THEORY OF DIVISOES.
Assume x^—i)x^-^qx—r = (^x — aj(x — b)[x — c). (<^).
.". x^+j)x^+gx+r—ix+a)[x+b]{x + c). ((3).
x^+px^ + gx^-\-rx A , B , C
- 3 —^ =x+^p+ \ J + ■ (y).
x-' -px^+qx — r ^ x — a x-b x — c ^'
Multij)ly by x^ —px'^+qx — r and equate the coefficients of the
terms in x-. {In multiplying the fractions in the right-hand
member of {q'), use the factor side of («).}
q = q-2p-+A + B-{-C
:. A + B + C=2p^.
Multiply both members of (y) by ic — a
x(x + a)(x+b)(x-i-c) ( ^ B C ) ,
— 7 7 -/ r^~ = A-\-]x-\-2p+ J + (x - a).
Put x = a,
2a^{a + h){a + c)
[a — h){a— c)
By symmetry
= A.
2fe2(64-r)(6 + a) _ ^c^{c + a){c+b)
^b-c){b-a) -^^-^^ {c-<(){c 6) -^'
a^a+b){a + c) _^ bJ{b-}-c){b+a) ^ c^(c+a){c-^b)
{a — b){a — c) \b — c)(b~a) {c — a){c—b)
= ^(A+B + C)=p^.
= (a + b + c)^.
3. Extract the square root of 1+x-j-x^ -}-x^ -\-x^ -\-&c.
Assume the square root to be l+ax+^x^-\-cx^-\-(lx'^-{-&c.
.-. l-^-x-^x^ +x^-\-x^-{-&c. = {l + ax-\-bx^-{-cx^ +dx^+ &c.y
= l-\-2ax + {a^ +'Ib)x^-\-2{ab+c)x^ + ^2d-{-2ac + b^)x^ + &c.
.♦. 2a = 1 .•. « = ^
2h^a^ = l .: 6 = i(l-i) =1-
2{c+ab) = l ... c = i-(^x|)=A
2d + 2ac-^b^ = l .-. d^h{l-r%-^\} - x%V
.-. ^/{1+x+x^ -{-S^G.) = l-\-lx + ^lx- +f\x^ + j\\x^ + &C.
(Note. — As it is frequently necessary to determine the coeffi-
cient of a particular power of x, a few ipreliminary exercises are
given on this subject.)
THEORY OF DIVISORS. 57
Exercise xxiii.
Determine the coefficient of
1. x"^ m {l-\-ax^^-^(l + bxy+{l-cxy.
2. x^ in {l+x-\-2x- +^x^)il-x+3x- +x^ -5x^\
3. X* in (l+.c+2.>-3+3a;3+4a;*+&c)(l-a;+x3-a;3+a;*-&c).
4. jc3 in ^(a;-6)(a;-c)(a; — rf) + i:>'(a;-a)(a;— c)(a;— d) +
C{x — a)(x — b){x — d) + D(x — a)(x — b)(x — c).
5. a;* in (l-aa;)3(l + a5:)«.
6. a;4 in (H-«a;)3(l -ix-)^.
7. In the product
(l+ax+bx'^+cx^+&c){l-ax+bx^-cx^-}-&<i.)
prove that the coeffi.cients of the odd i)owers of x must be all
zeros.
Determine the value of the following expressions :
,.1 1
{a-b}{a-c){a-d) ^ (b-a){h-c){b-d)
1 1
{c-a){c-b){c-d) "^ {d-a){d-b){d-c)'
q ^ I *
\a-b){a-c){a-d) ^ {b-a){b-c){b-d) "^ *""
It)- 1 Tw w jr -h three similar terms.
(a — o)[a—c){a — d)
11. ^ + " "
(a — b){a— c) {a — d)
l-^- 7 TTT \i 1\ + three similar terms.
(a — b){a — c){a — a)
13. 4- «« "
{a-b)(a-c){a-(I) ^
a{a+b){a+c)
{a-b){a-c) ^ '^'^
jg^ a3(a+fe)(«+c) _^ ,^
(a — 6(a — c)
• (a_i)(a-c) "^
68 THEOKY OF DIVISOKS.
17 a{a+b){a+c){a+d) --14.
'• ' • 7 TVT TT T\ + three similar terms.
{a — b){a—c){a — d)
cr-(a + b)(a + c){a+d) _^
^^- {a-b){a-c){a-d)~^
(a— 6)(a — c)(«. — (Z)
20. , ^rv> . 4-two similar terms.
[a—o)[a—c) '
[For numerator use x^.-{-2px^-{-{i}^-^q)x + (pq - r).}
{2a+b\2a + c)
21. -7 TTT T" -f two similar terms.
(a — b){a — c)
[For numerator use x^ — 2j9a;- + (/>' i-?)*; — (f 9 —■'■)•]
a(& + c) . ,
22. PT/ "\ + two similar terms.
{a — b){a—c)
[For numerator use x{x+p).']
b+c+d ^^ . „
23. -, TTT r? 3^ + three similar terms.
{a — b){a — c){a —a)
a^(bc + cd+db)
^^- {a-b){a-c){a-d)^
bc-i^-cd + db
OK ,J 4_ <« " ««
-^^^ (a-5)(a-c)(a-^) ^
Extract the square-root of (to 4 terms) :
26. 1+x. I 27. 1-a;. | 28. l + 2.x-}-3a;* +4x3-1. &c.
29. 1 - 4a; + 10a;3 - 20a;3 + 35a;* - 56a;' + 84a;« .
30. Extract the cube-root of 1+x. (To 4 terms).
Art. XI. 1. Find the condition that pz^+2qz-\-r and p'x^
+ '2q'x+r' shall have a common factor.
Multii^ly 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 z.
p'px^ + 2p 'qx 4-_p V I ;;)• 'x^ + 2qr'x + rj- '
pp'x- + 2pq'x+pr' I p'rx^-\-2q'rx+r'r
2( pq ' —p 'q)x+{pr' ~p '/•) ! {pr'-p'r)x + 2{qr' -r'q).
Multiply the former of these remainders by (pr'-p'r) and the
latter by 2{pq'—p'q), and the difference of the products is
{pr' —p'r)^ —A.{pq' - p'q){qr' — r'q).
TBEORY OF DIVISORS.
59
But if the given polynomes have a hnear factor this remainder
must vanish, or
(pr' —p'r)^ =A(pq'—p'q){qr' — r'q).
U the given polynomes have a quadratic factor, the linear re-
mainders must vanish identically, or (Th. III.)
pq'—p'q — O, pr'—p'r = 0, and qr'—r'q = 0,
p q r
or, _ = -i- = —
p' q' r'
2. Find the condition that px^ + Sqx"^ + 3rx+s shall have a
square factor.
Assume the square factor to be (x — a)^. On division, the
remainder must bo 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 remainder.
a
P
3q
pa
3r
pa- +3qa
»
a
P
pa + 3q
pa.
pa^ + 3qa+3r ;
2pa^ +3qa
R
V
2va4-3o :
3{ija^4-2aa + r)
:. pa^-\-2qa-{-r = 0:
.". px^ + 2qx+r is divisible by x—a (Th. I. Cor. 2),
or, px^ +3qx^ -^3rx+s axid jJx^ +2qx+r have a common divi-
sor. Multiply the latter polynome by x and subtract the product
from the former, and the proposition reduces to
If px^-{-3qx^ +3rx-'rs have a square factor, ;ja;^-t-2(7a;-}-r and
ya;2_|.2ra;+s -will have the square-root of that factor for a com-
mon divisor.
.3. li px^-\-3qx'^ + 3rx-{-s yanishfor x = a, or b, or c, find in
terms of x, p, q, r the value of
x — a X — I) x — c
Beduce to a common denominator and add the numerators
Bx^-2{a + h + c)x + (nb-{-hc + ca)
{x — a){x — b)(x — c)
60 THEORY OF DTVISOKS.
Multiply both numerator and denominator hjp and reduce by
Tb. n., and Ex. 4 of Th. 11.
8 (pa; 2 ^2qx-T-r)
pz^+3qx'^ -i-'drx-i-s
r^m+i ^.m+1 ajTO+i S(23x"^+^ + 2(/x'"+^-}-rx'''-+^)
' ' X — a X ~ b X ~ c px^ +3ja;3 -fSric+s
4. If 2)x^ + 'dqx^ -{-Brx+s vanish for x = a, or b, or c, express in
terms ofp, g, r, s, the following, a + 6 + c, a^+b^+c^, a.3-hi^+c'
, a"' + 6™+(;'".
Divide a;^+^ by x — a.
1
a r?3 a^ a"* aw^+l
1 a flS a3 d'" ; a"«+i
Similarly divide a:'"+^ by x—b and also by x~c.
add together the quotients
;^w+l 2;'"'+l x'"^'^^
i 7 + =Sx'^ + ia + b+c)x'^~^ + (a- + b^ ^. c2W»->
x~ a X— h X - c ^ ' ^ '
Hence, by the last example, the required expressions are the
coefficients taken in order, beginning with the second, of the
terms in the quotient of 3(|;a;"^+3 -]-2qx'"'+^+rx'''+^) ^ {px^+Sqx^
+ Srx-{-s). These may now be found by Horner's Division.
0. Writing Sj for a + 5+c, Sg for a^-\-b^-{-c^, &c., express
(a— 6)* + (6 — c)4 + (c — f/)* in terms of s^, s^, .s-g, s^.
By actual expansion
{x--ay+{x-b)'^ + (x-cy =
SxA-A{a-{-b + c)x^-^G{a^+b^-i-c2)x^-4.{a^+b^ ^c^i)x+
a^ + b^ + c'^ = Sx^-4s^x^-i-6s^x^~4:S^x + s^.
Put z = a, —b, = c in succession.
(a_6)4 ^(c_a)4 =3a4_4sja34-6s2rt3-4s3a+S4
(a_&)4_(.(5_c)4 =r3ft4-4s^63 + 6s2/'2 -4.<?3/) + S4
(i — c)^ +(c-a)4 =3c4 — 4sic3^_ esgcS — 4.';3r+S4
... 2{(a-6)4 + i6-r)4 +(c-a)4}=3s4-4sic^3 + 6s„^-4.93.9,+3s.t
.-. (a-6)4 + (/;-c)4+(c-a)4 =s,S4-4.s-i.S3 + 3s'
in which s^ is written for 3 or 1 + 1 + 1, i.e., a'^ + ^^+c".
TBTEORY OF DIVISORS.
61
Exercise xxiii. (a).
1. Determine the condition necessary in order that x^ +'px-\-q
and x^-'cp'x + q may have a common divisor.
2. The expression x^-^Za'^x^ ^'dbx^ +cx^-\-odx'+oe^x-\-P
will be a complete cube if
e d c — rt®
ah oa*
8. Prove that ax^ + hx^c ?aid a-^bx^ + cx^ will have a common
quadi-atic factor if
fc2c2 = (c2 _a2_}.J,2)((.2_c[2+a^>).
4. Prove that aa-^-\-hx"+c and a+bx^ + cx^ will have a com-
mon qaadiatic factor if
a262 = (((2 _ c2)(fl2 _c3 ^ic).
6. Prove that ax* + bx^+cx+d and a+hx-^cx^+dx* willhaA-e
a common quadratic factor if
[a-{-d) ^ = {b-c){bd-ac).
6. x^+px- +qx+r will be divisible by x-+ax + b if
a^-22m^-\-[p-+q)a+r-pq = 0, and b^ -qb^ + rpb-r^ = 0.
7. x*+px-^q will be divisible by a;- 4-ax-^b if
a^-^qa^=2J^ and {b^- + q){h^ - q^ = p^^ .
8. Determine the condition necessary in order that a;* + 'i;t»3'^
+Gya;2+4ra; + « m?.y have a square factor.
'U. x^-\-4:px^-\-Qqx'^ +4:rx+t vanish for x = a, or 6, or c, or c?.
find in terms of x, p, q, r, t, the value of
X" x" aj" x"
+ 1 H }--
* a; — a a; — 6 x —c x — d'
10. 2 a, S«-, 2a^, S«*, 2;«^, 2a®-
11. ^(a-h)^, 2(a-i)*-
12. Determine the valuesof the expressions in Ex. 9, 10, 11, for
the polynome a:* — lix^ -j-x — 38,
62 FACTORING,
CHAPTEE III.
Section I. — Direct Application of the Fundamental Formulas
Formulas [1] and [2]. (x±y)^ :=x^±2x!/-^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;4-80ic27y3_|_400?/4 = (2a;3~20?/2)(2a;3-20?/3)
8. (a—b)^ + (b-~c)^-h2{a--b){b-c). This equals (a- 6 4-^- c)
X (a — b + b — c) = {a — c){a — c).
4. x^ +y^ +z^ -\-2xy — 2xz ~2yz.
Here the three squares and the three double products suggest
that the expression is the square of a linear trinomial 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
1st. 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
x-\-y — z, or —x—y-{-z= — (x+y — z), and the factors are
{x+y-z)(x-hy-z).
Exercise xxiv.
1. 9h/3_|_12w?, + 4; c2'»_2t;™ + l.
2. ^«-2v/-'523+z« ; iex^y^ + lQxy^-{-4y^.
3. 9a2634-12a6c+4o^; 36a;>2-24a;i/3-|-4?/*
PACTORINa. 63
5. (rt + 6)2+c2-2c(a+6); 9x« -^x^y^+r^x^y^.
G.z^'+(x-yr--Ozfx-y);-r-] +- -2-
b I \ a
8. (a;2-x^)2_2(a;2-xy)(a;^/-2/2) + (a;?/-,y2^2.
10. (3;>;-4!/)2 + (2a;-3;/)2-2(3x-4?/)(2x-37/).
11. (x^ -xy+y^)^ + (x^ +zy+y^y + 2{x^+z^l/^-\-y^).
12. (oa;2 + 2x?/ + 7i/3)2 4.(4a;3 +6//2)3 - 2(4a,-2 +67/3) x
;5a-2-L-2a;i/ -1-72/2).
18- ItJ +(ir) -2ly) •
14. a2-h^3+c3-2<r6-26c+2rtc.
15. a* + fc4+t4-'2«26«-2a2c3 4-262^3.
16. (a-6)S + (/>-c)2 + (6--«)2 + 2(a-i)(&-c))-2(a-6)(c-a)
+ 2(6-c)(a-c).
17. 4a4-12a26 + 953+16«2c + 16c2-24?yc.
Formula [4]. x^ —y^ = (x+y)(x—y).
Art. XIII. In this case we have merely to take the square-
root of each of the squares, and connect the results with the sign
4- for one of the factors, and with the sign — for the other.
Examples.
1. {a + hy-{c-\-d)K
This={{a+b)+(e+d)}{(a+b)-(c-{.d)}
= [a+b + c-i-d){a-{-b — c—d).
2. Factor (z^+^ry+y^)^ -(x^ -xy-hy^)^.
Here we have
{{x^ + 5xy+y^) + {x^ -xy + y^-)}{{x^' + 5xy-[-y^)-(-'^ xy^y-')\
= 2{x- + -2xy+y-){Gxy} = 12xy(x+2j)K
3. a^-b^ -c-+2hc.
This = a2-(i-cj3 = (a + h -c)[a—b + c).
64
FACTORING.
This = 4rt-//2-(rt2+/,2_^2)2
= {'lab +a^ -h^ - C-' ,(-lab - a-^ -I- -{-r-2).
The former of these factors = {a + b)'^ — c^ = {a + b-\-c)(a-^b—i:)i
and the iatter = c- —(a — b)^) = (c+a — b){c — a+b).
.'. the given expression
= {a-{-b + c)[a + b — c)(c-{-a—b){c--a-\-b').
Exercise xxv.
1. 49a3-462.
2. 9«3_ii3.
3. 81a4-16/)4.
4. lOOx- - 3G.y2.
5. 5n2b-2i)hx-y^.
6. 9a;«-18i/'i.
7 9 /-S _ 1
8. 4.ij^-^x-z^.
9. 81a4_l,
10. a-i--1654.
11. a>6_ii6.
12. a2-/j2+2&c-c>.
13. (rt + 2i)■•^-(3a;-4J/)^
14. (.-K-' +^3 j3_ 4^-2^2/
15. (x-t-7/j^— 4;i2^
16. (3x + u)2-(5;rH.?)»
17. 4a;37/2 _(a;2 4.^/2 _;j2)2,
18. ix'^+xij — y'Y — ix'^—xtj—y-)^.
19. (x2_y2+22)2_4^223.
20. (a+&+c+c^)2-(a-6+c-rf)''».
21. (2+3x+4a;3)3_(2_3x+4x-')'.
22. {ci-'-{-b''+4.aby'-{a'^->rb^y^.
23. (fl.2 -Zy2 4.c2 _fZ3^2 _ (2ac— 2M)3.
24. (x3-^3_23)3_4^223.
25. (a6_a3i3 0-^0)3 _(,f6_5a3i3_^fcG)3.
26. ai2_?,i2-|..Ga363_6/^9a3 + 869a3_Sa«i«.
27. (x2+_y3+23 -a;^/— 2/2 — 2:x)2— (x?/ + ?/2 + 2X)*.
28. (a;- +^2 ^gS _ 2ccy + 2xz - 2?/^) -((/-f z)3.
2r3a2_«4
./■4
29. 2a362+262c-2+2c3a
80. a;4+i/4+z4-2iv;2*/2-27/3s3-2£2a;2
FACTORING. 65
Fos^ruLA A. {x~r){x-'rs)=X'-{-(r+s]z-rri.
SXAMP::,ES
1. x''-9x-^20 = (x-'-5\{x^^4:).
2. {x~y)2-^-x-?j-110^(x-y + n)(x-y-10).
3. (o^-ob + b^ |2 + 6b{a2-(,b-{-h2)- 4024. n^- ^
^(rt2 _ rti4_i3) + (2rt + 3i) [ {(^2 -ah-\-b2) - (2a -Sb]}.
4. (a;3_5a;)2-6(x2-ca;)-40 = (x3 -ox4-4)(a;2 _ J^-IO).
5. (ffx4-% + c)2 — {vi —n){a.r-\-by-{-c)—vtn
= (fla; + % + c — ]ii){ax + % + c4- //,j.
Art. XTV. It will be seen that the first (or covimon) tei-m of
the required factors, is obtained by extracting the square root of
the first term of the given expression, and that the other teims
are determined by observiuG; two conditions :
(1) Their product must equal the third term of the given ex-
pression.
(2) Their sum (ah/ebraic) miiltiplied 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 form x^ + bx±: c, we may proceed as follows :
1st. Write down every pair of factors whose product is c.
2Dd. If the sign before c is +, select the pair of factors whose
suvi is b, and write both factors x+, if the sign before bis -\- ; x — ,
if the sign before 6 is ~ .
3rd. But if the sign before c is — , select the pair of factors
whose difference is b, apd write before the laryer factor x-'r ox x— ,
and before the other factor as— or x-\-, according as the sign be-
fore 6 is + or — .
Examples.
1. a;2 +9a;^20. The factors of 20 in pairs are 1 and 20, 2 and
10, 4 and 5. The sign before 20 is +, hence select the factors
whose sum is 9. These are 4 and 5. The sign before 9 is +,
hence the required factors are (a;-f 4)(x4-5).
(56 FACTORING.
2. a;2 - 8.r-f 12. Paii-s of factors of 12 are 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 — , hence the factors
are (x — 2)(x—G).
3. a;2_21a;— 100. Pairs of factors of 100 are 1 and 100,
2 and 50, 4 and 25, 5 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;-f 4)(a3 — 25.)
4. aj^-f 12a;-108. Pairs of factors of 108 are 1 and 108,
2 and 54, 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 a;— before 6, the other factor; hence the
factors are (a; — 6)(a;-|-18).
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
{x-2) (x-6)
Ex. 3 above. Smce the sign of the third term is — , write the
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
fx+4) {x-25)
6. a;2-81r4-64.
Here we have the factors
1, 64
a'-2, a;-82
4, 16
and since the last term has the sign +, and the middle term has
the sign — , we write — over both columns.
FACTOEIJJO, grr
6. x"-^l2x-Gi.
- +
1, 64
2, 32
.r-4, a; + 10.
Here, since the last term has the sign — , we write the sign
1^ + ) of the middle term, over the column ot larger factors, and
the sign — over the other column.
7. x*-10x-2-144.
Here we have the pairs of factors :
+ -
1, 141
'A, 72
4, yo
a; + 8, a;- 18.
And since the sign of the third term is — , we write the sign oi
the second term (in this case — ) above the column of lai-yer
factors, and the other sign (of the pair +) above the other
column.
Exercise xxvi.
1. X2-5.C- 14; a;-'-9a;+14; x^+lx+12.
2. x2-8x+lo; x^-ldz+Sk; x- -7x-G0.
3. 4a;2-2a;-20; 9x2 -150a;4-600.
4. ^x2+Mx-3(i;25z'+i0x+lo; 9a;« -27^3 4.20.
5. ^\x^ + Hx-\-12; 16a;4-4a;2-20.
6. x'i-(a3 + Z/3)a;2 +6/2^3 ; 4(^._}.v/)a _4(^._|.^)_fj9,
7. (x^ +2/2 j2 - (a2-fe2)(a,-2 +^-^)-aHK
8". {u + h)^-2c{a+b)-dc^.
9. (x4-?/)2 + 2(x2 +y2)(^x + ;j) + {^^ - y'^y-
10. (a + 6)2-4a/y(a + 6)-(«2_62j2.
11. {x:-+x>j+y^Y+x^-y^-5xii-'2ij^-2x^.
12. «2_2rt(6_c)-3(6-c)2.
68 FACTORING.
13. (x'^+y^)2+2a^{x°-{-y'') + a^-b*.
U. {x^ -lOxy -4:(x^ -10x)—0Q.
15. (x2 - Ux + 40)2 - 25(^2 _ 14a:+40) - 150.
16. {x^ -xi/ + >j^)'^+2xy[x-'-x>ji-ij^)-Qx'i/K
17. 04-823 + 2; «4-2.c2-3; dx^ + Qx'i'y^ -lOy*.
18. c-'" + t'"-2; a;«-a:3 — 2; x'"'-2x"'y'' -8y^\
19. x-"'—{a — b}x"'y"~uhy-''.
Art. XV. Trinomials of the form ax^ +wa; + c' (n not a squa e)
may sometimes be easily factored fi'om the iollovving coiiBiileia-
tions : —
The product of two binomials consists of
1st. The product of the Jirst terms.
2nd. " " second "
3rd. The sum {alysbraic) of the products of the terms taken dia-
gonally.
These three conditions guiile us in the converse process of
resolving a trinomial into its binomial factors.
Examples.
1. Kesolve 6.e'^-10;c// + 6;/2.
Here the factors of the first term are x and 6a:, or 1x and 8x- ;
those of the third term are y and Qy, or 2y and 'dy. These
pairs of factors may be arranged
(1) (2) (3) (4)
X 2x y 2y
Gx 8x 6y By
Now, we may take (1) with (3) or (4), or (2) with (3) or (4) ;
but none of these combjnatious will satisfy the third C(nlition.
If, however, in (4) we interchange the coeflicients 2 and 3, then
(2) and (4) give
2x 3y. and
Sx 2y, where we can combine the " diagonal"
products to make 13, and the factors are
FACTOBING. 69
2a; — By, and
Sx - 2y.
The coefficients of (2), instead of those of (4), ruaght have been
interchanged, giving the same result.
2. 6x^-l5zi/+(Jy'^.
Here, comparing (2) and (3), Ex. 1, we see that their diagonal
products may he combined to give 15, and the factors are
2x—y, and Bx—6y.
3. 6a;--2U.c?/+6//2.
Here, again referring to Ex. 1, we see at once that it is usele^.-
to try both (2) and (4), since the diagonal products cannot be
combined in any way to give a higher result than ISxy. Birt com-
paring (1) and (4), we obtain by interchanging the coefficients
in (4) x—By, and
6x—2y, wliich satisfy the third condition.
Or, we might interchange the coefficients of (3), and take the
resulting terms with (2), getting 2x—6y, and
Sx— y.
4. Gx^+SSxy-GyK
Here the large coefficient of the middle term snows at once
that we must take (1) and (8) together. Interchanging the co-
efficients of (1) we have
Gx— y, and
a + (jy. The same result will be obtained by inter-
changing the cociiicients of (3).
Exercise xxvii.
1. 6x^-Blxy + 6y'.
2. 6x2-1-9x2/ -6?/ 2.
3. 56x^ - 7 6xy + 20}/^.
4. 56x2 -36x?/- 20^2,
5. 56x2-1121x^+20^2.
6. 6Gx2-68x?/ + 20i/2.
7. 66x2 -558xJ/- 207/3.
8. 56x3 +3ex/y- 207/2.
11. 6x2 - 16x// — 6?/2.
12. 6x2-f5x(/-6?/2.
13. 66x2+562x//-f20?/2.
14. 56x2-122x^-1-202/2.
15. 56x2-102x!/-20?/2.
16. 56x2 -229xi^-f- 207/3.
17. 56x3-94.r//-h20?/2.
18. 56x2 -276x//- 207/3.
9. 56x2-67xJ/-}-20?/2. j 19. 36x2— 33xi/-15?/2.
10. 56x2 + 3x(/-20i/2. I 20. 72x^ -19xy-iOy^.
70 FACTOKINO.
Art. XVI. More f/merally, trinomials of the form nx^-\-hx-}-f
{a not a square) may be resolved by Formula A, thus
Multiplying by a we get a^x^ -\-bax+ac. Writing z for ax this
becomes z^ + is-J-ac. Factor this trinomial, restore the value of
z and divide the result by a.
Examples.
1. 6x" +5x - 4. Multiplying by 6, we get (6.r) 2 + o(Qx) - 24 or
2 ■ -f. 52 _ 24. Factoring, we get {z - 3){z + 8), hence the required
factors are ^{6x-3)(dx+8) = {2x—l){3x+4:).
2. 6a;3 - 13:cy + 6//3. ^ Factoring z" - ISs^ + SGy^ we get {2 -Ay)
[z — dy), hence the required factors are ^(6u; — 4?/j(6.B — %) =
{3x-2y){2x--S7j).
3. 33-14x-40aj3. Factoring 1320 -14^-23 we get
(30-2)(44+z), hence the required factors are Jg(30- 40*) x
lu+i0x) = {3-ix){ll+lQx).
NoTB. — 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+4).
Ex. 2, above 1 3(>
2 18
3 12
|(6x-4)(6a;-9) = (3x-2)(2ar-3).
[Another method of factoring trinomials of the form ax'^ + bx-^o
is as follows :
Multiply by 4a, thus obtaining 4:a^x^ +4:abx+ iac. Add h'^ -b*,
jvhich will not change the value, Aa^x'^ + Aahx + b" —b^+iac ; by
[1] this may be written {2.ax-\-b)--^{b^ — Aac). Factor this by
[4] and divide the result by 4a.
FACTORING. 71
Ex. Factor 562-' + 137a; -27885. Multiply by 4x56 or
2x112, 1122a;^ + 2. 187. 112a;— 624G240. Add 1373 -1372, then
1122a;3 + 2.137.n2x+1372 - (1372 +624G240) = (112a;+137)3 -
6265009 = {(112a; -f 137) +2503} {(1 12a; + 137) - 2503} =
(112a;+2640){112a;-2366).
We multiplied by 4 x 56, we must, therefore, now divide by that
number. Doing so, we obtain as factors (7a; + 165)'(Sa;— IGO).}
Exercise xxviii.
1. 10x2+a;-21
2. 10.,-2 - 29a; - 21.
3. 10a;2+29a;-21.
^. 6a;2-37a;+55.
5. 12a3_5a_2.
6. 12a;3-37a;+21.
V. 12a;2 + 37a; + 21.
b. 15aG + 13rt362-2064.
9. 12a;-^-a;-l
10. 9a;37/3-3x?/i'-62/«.
11. 4a;3 + 8a;/y + 3?/3.
12. U^x^-lhz^-dx*.
13. 6a;4-a;3?/3-35.//4.
14. 2a;'i+a;3-4o.
15. 4a;4-37a;=2/2 + 9?/4s
16. 4(a; + 2)4 -37a;2(a;+2)3 -t-9a;*
17. 6(2x + 37/)3+5(6a;2 4-5a;^-6y2)_6(3a;-2(/)3.
18. 6(2a;+37/)4 + 5(6.c3-f oa;.y-6?/2)3 -G(3a;-2y)*,
19. Q{x^-\-xij-\-y^Y+Vd[x^+x-y-+y'^)-dSo{x''-z>i-\-y^)^.
20. 21(a;3 + 2xf/+2!/3)3_6(a;2-2a;?/-l-2?/3)3_5(a;4+4?/4).
Section II. — Extended Application of the Formulas.
Art. XVII. The methods of factoring just explained may be
appHed to find the rational factors, where such exist, of quadratic
multinomials.
Examples.
1. Eesolvel2a;3-a;y-20?/2+8x+4l7/-20.
In the first place we find the factors of the first three terms,
vrhich are
4x+5y, and
8a; -4r/.
Now, to find the remaininy terms of the required factors, we
must observe the following conditions :
79;
FACTORING.
1st. Their product must = — 20.
2nd. The sum [algebraic) of the products obtained ty multi-
plying them diagoually into the ?/'s, must =Aly-
3rd. The sum of the products obtained by multiplying them
diagonally into the ic's, must =Sx.
We see at once that —4 with the first pair already found, and
+ 5 with the second pair, satisfy the required conditions, and .'.
the factors are
Ax+^y -4, aud
3a;-4?/+5.
2. p2+275r-2(7'-*4-77r— 3r2 -{--pq.
Here the factors oi p^ + V9. —^q^, aro
p+2q, and
2^—q.
Now find two factors which will give - 3?--, and wliich mnlti-
plied diagonally into the ps and ^-'s respectively, will give 2pr,
and 7qr ; these aro tound to be — r taken with the Jirst pair, and
+8r taken with the second pair. Hence the required factors are
p + '2(j — r, knd
p — q-f- 3r.
Art. XVIII. But the following examples illustrate a surer
method.
3. x--\-xy-2.y^-\-2xz + lyz-Sz^.
Keject 1st the terms involving »,
2nd. " " y,
3rd. " " X.
and factor the expression that remains in each case.
1st. x^+xy-2y^ = {x~y){x+2y).
2nd. x^+2xz-2z^ = {x+dz)(x-z).
3rd. -2tj-'+7yz-Sz^ = {~y+Sz){2y -z).
Arrange these three pair of factors in two sets of three factors
each, by so selecting one factor from each pair that two of each
set of three may have the same coefficient of x, two may have the
rACTOKiNCj. 73
HiTiie coeScient of y, and two the same coefficient of s (coejficimi
including sign). In this example there are
X— y, aj + Pz, — y + ?z,
and x-\-'2y, x- z, 2r,- z.
From tlie first set select the common terms (including signs)
and form therewith a trinomial, x—y + cz.
Repeat with the second S3C, and we get x+2.y — z.
:.x^+xy-2y^ -f- 2xz+lyz-dz^ = {x-y+B;.}{x+2y-z).
4. Bx- -8xy-By^ -iriiOx+27.
1st. dx^-Sxy-3?/^ ={Sx + y){x—dy).
2nd. 8a,-3 +30x4-27 ={Qx+3){x+9).
8rd. -3^2 -fii7 =(2/+3)(-32/ + 9).
.•. the factors are ('Sx+y-\-d){x — By+0).
6. 6rt3_7a6-b2rtc-2063 + 64ftc-48r3.
1st. 6a3-7a&-20i3 = (2rf-5&)(3a+45).
2nd. 6a2-u 2ac-48c3 = (2/;(-f 6c-)(3a-8c).
3rd. -2U{y3+64ic-48c3 =(-5/>-i-t;c)(46-8c).
.'. the factors are (2a — 56-i-Gc)(3a + 4i — 8c).
Exercise xxix.
1. 7x^ -xy-Gy^ -6x-20y-lQ.
2. 20x3 -loary- 51/2 -68a; -42^-88.
3. 3x^+a;3«/3- 47/4 + 10x3 -17^/3 -13.
4. 20x2-20?/3+9xz/+28x+35jr.
6. 72x- - 8^/3 -I- 55a;y+i2y— 169x4-20.
6. x^ —xy— 12y'^ —5x—15y.
7. 8x2 + 18x//+9//3+2xz-s3.
8. Gx3-h6?/3-13xj/-8z3_2//c+Sxz.
9. 6x*- 102/4 + llx3//2- 25^3 j.i0y2^2Dr/2z3- 15x2 + 10x»z".
10. 15x4 -lG;/4 - 22x2(/2 4- loz* + 14^232 ^ 50x3;r2.
11. 4ft3 -1563- 4^6- 21c2-3G/>c—8rtc.
12. rt4 + />4+c4-2rt362_2;,3c2-2c3a3.
74 FACTORIXG.
Art. XIX. Trinoraials of the form ax*^ + Ix"^ +r can always be
broken up into real factors.
If a and c have different signs, the expression may be factore(3
by Art. XVI.
If a and c are of the same sign, three cases have to be consid-
ered : i. 6 = 2^/(ac), ii. 6>2v/'(ac), iii. 6<2y(ac)
Case I. 6 = 2]/(flc). 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.
EXAMPLSS.
1. 4a;4 + 5a;22/2 4-1/4.
Here we see that (f?/^)* will make, with the first two terms,
a perfect square, and we therefore add to the given expression
{^y'^Y —{%]j^Y . The expression then becomes^
-(2x2 +^2/^)^- Ay*-
= (2a;' +f2/' + |2/2)(2x2 +1 2/' - i2/2)
= (2a;2 +22/2)(2x* + -i7/^) = (a;2 +t/2 j(4a;2 +y2).
2. 3a;4 + 6a;2+2.
Here multiplying by 4x3, and completing the square as in
Ex. 1, we have
36a;4+72a;2 + 62 + 24-62 = (6x3 + 6)2_12
= (6a;2 + 6-T/12)(6x2+6 + |/12), which divided by 4x3 give
the required factors.
Proceeding as in Ex. 2 we have, by multiplying by 4a,
ax^j^hx'^ 4-c = {4a2a;*+4a6a;2 +63 - h^ 4-4ac} ^- 4rt
= {2aa:3 4-i + ^(i2_4«c)}{2a^3+/,_^(i8_4«c)}-4a.
FACTOEIN». 75
Exercise, xxx.
1. x^+7x--hl; 4a,-'5-Mlrc3+l.
2. a;4-f7a;2i/S+^4 ; Sx'^+5x^y^-\-y^.
3. Ax* + Wx-+S; S{x+)/)^ + 5z^{x + y)'+z^.
4. a;* + 7a;2_v-'+3-}-!/4; x'^ + 7x''y^ + Siy^.
6. 3x*+8x'^y^+i^\y^; 36x4+96a;2+56.
7. 5x*4-20a;3+2; 4rt* + 12«2 + l.
8. 4(x+2/)* + 12(x+!/)'22+24; 5.r*+20a:3y2+2y4.
9. 9x* + 14a;2+4; 2x*-{-12x'^{y + zy' + 15{y-^z)^.
10. 2a;4 + 12a;8 + 15; 7x4 + 40a;2 +45.
11. 8a;* + 36a:2y2+29?/*; 7ic4+20a;-r/=» -20^*.
'12. 7(rt-6)4+16(a-6)2c3+5c* ; |«4 + 3«2i3+54.
13. 3a;4+6x'.v'+2y4; 3(r^ + /;)* + 6(a3-63)3+2(a-i*)4.
14. 49a4-84a3i-'+2264 ; 25m'^ + 60m^n^ + 27n^.
15. 49(j??. + ?i)4-84(m^ -n2)2+22(?«-«)4.
Case III. b<2i/{ac). This case may be brought under
Art. XIII. The following examples illustrate the process of re
duction and resolution.
ExAliPLES.
1. a;*-7a;'+l.
We have to throw thig into the form a' — 6' :
a;*-7a;2 + l = (x2 + l)2-9x3 = (x2+l + 3.-c)(a;2 + l-8a:).
2. 9x* + 3z^y^+iy^ = {Sx^ + 2y^)^-9x-y^
= {3x^-+27j'^-3xy){dx^ + 2y^-^3xy).
3. x*+y^ = {x^ + y^y-2x^y*
= {x^+y'+xy,/2)(x^+y^-xyy^2).
= {x^+y^ +ixy){x-' +y" -^xy).
6. az* + bx^-^c^{^a. x^ ^-^/c)^ - {2 y{ac) -b}x^
= {ya. x^ + ^/c- y^{2y'^ -O^x} X
{j/a. x^+y'c+^/{2y^-^~b)xl
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 Jirst and
last terms (arranged as in Ex. 5) a perfect square, and to subtract
the same quantity. In Ex. 2, e. y., the square root of 9a;* = 3a;2,
the square root of 4j/* = 2?/ 2, .-. Sa;^ +2//*-^ is the binomial whose
square is required ; we need .". I'lx^y^ ; but the expression con-
tains 8x^y^ : .*. we have to add and subtract 12x^y^ — Bx^^y^ =
<dx^y-K
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 ; this 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. «*+T'ca;-//2-{-y*. Here i/x*=a;2, |/y4 = )/8, 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,
^^z^y^, we get l^aj^^s. the square roots of this are +lxy.
Hence the factors are x--\-y^ + ^^vy.
Note that since v/7/*= 4-//^, or --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'^ — llx^ y^-\-y^. Here, taking -\-y'', we have
x''-^y^-\-xys/Vd, anda:» + 7/2-xy/v/13.
But taking — ?/-, we have
x^ -y^+Sxy, &uA x^ -y^ — 3xy,
Sometimes both signs will give rational factors.
8. lQx*' — 17x^y^-{-y*. Here we have
(^Ax''^^y2^2xi/){ix^+y^-3xy, and also
{4^'^ -y^ +5xy){lx- -y^ - 5xy).
FACTOKING. rjrj
Exercise xxxi.
1. x4+2^2^2+9?/4, ^4_a.2^2_|_y4^ ^.4+^2_,^--;_|_,,4.
3. a;-*H-l, x-t+%-1, l-12//2-j-iG^-i.
5. y^-x'^ + Ux^!/'^, x^" +4:y», x^ + ix^ + 16.
7. ^^'" + 04/'", «^'"+4i/»'", ix4 + ^9gyi-5|x3.(/-'.
8. 4*4-8x2 + 1, 7*2^3 _x^.4_30//4, a;4 + aV'
9. 7»2^.4+,;-j^4_(2m/i4-^)x-2_,y2^ ^im ^ 2*'"--(/*'".
10. 16x4-25*2 + 9, 4*4 _ 1(5^2 +4^ 13*2^2 _y_^4 _ 4^*.
11. 4*4-12^l'*2^2+g^4^ *4+6*2+25.
12. </4+i4 + (a + 6)4, l+«44.(l+,,)4.
13. {x-^y)*-7z^x-i-yr--\-zK X
14. (rt-(.6)4 + 7e2(,/4.i)2+c*t.
15. 16</4 + 4(i_f)4_9,<2(/,_,.)2,
16. 4(« + //)4+9(a-i)4-21(«2_/;2)3.
17. {x^+y^-xy}'^-7{x^-i-y^y'+{x+yy.
18. (r72+ai + />2)4 + 7(rt3_^,3)2+(a_fc^4.
19. lGa4 + 4a2 + l, *4_4i^2+i6.
20. x4+81y8_63a:2,/4, l+24+2oz».
21. (a2 + l)4 + 4(a2 + l)2a2+16a4, (^+l)4 + 2(*2 _ 1)2^
n(a;-l)4.
Art. XXI. We cau apply [4] , Art. XIII., to factor expres-
sions of the form ax'*' -\-bx^ + rbx — r- a. This may be written
«(a;4 -ri) + i*(* -' + >•) = {a{x^-r)+hx]{x^-\-r).
ExAilPLES.
1. 6*4 + 4*3+i2x-54. This
= 6(x4 _ 9) + 4a;(x2+3) = (x2+3){6(x2 _ 3)+4x}
= (*2 + 3)^G*2+4^_i8).
78
FACTORING.
■4){ll(a;9 + 4)+10x}
2. Ila;4 + 10a;3-'i0a;-176. This
= 11 (.r* - 16) + 10a;(a;3 - 4) = (x""
= {z^-A){Ux^+lQx+U).
3. 40;«^ + 30:c3 +60.^-160. This
= 10(4ic4-16) + 15a;(2a;2+4) = (2ic2+4){10(2.r3-4)-f 15.r}
= (2a;^+4)(20a;'-*+15a;-40).
Note. — To determine r, take the ratio of the coefficient of x^
to the coefficient of x.
Exercise xxxii.
Resolve into factors
1. x'>^ + 2x^ + 6x-9.
2. 2a;4 + 2a;3+6x-18.
3. x^+3x^+12x-16.
4. 3a;*+a;3-4a;-48.
5. 5x^ + 4:x^ -V2x~4:5.
6. 10x4 + 5x3+ 30a; -360.
7. ia;4+20x3+4a;-T^^.
8. 25x'i-40x3+8x-l.
9. 37ix4-30.c3+48x-96.
10. 63x* - 39x3 +52a;- 112.
11. 810x4 + Va;3 + |x-2i.
12. 242x4 -33x2-3x-2.
lO. 4X -f-ipQX' ^5"^ ^Tj^'
14. 80x4- 32x3?/+ 64x?J-320«,'4.
15. 24x4-12x32/+30x.(/3-1502/'4.
16. 2x4+^s;3,^_8a;_y3_512,/'l.
17. 1 1x4 + 10x3 -12x-15fi
18. 40x4 + 30x3+60x-160.
19. 13x4-12x32/+72xi/3-468s^4.
20. 3x4 + 3x3^ + 12.r//3-482/4.
21. 5x4 +4x3^- 12x//3 -45^4.
22. 4x4- 14x3 //+28x//3-16y4.
23. x4+80x32/+16xy3_^i^y4.
24. 2x4-x3?/+6xy3_72y4.
Art. XXII. Formulas [1] and [4] may sometimes be ap-
plied to factor expressions of the form
ax^ -\- bx^ -{-ex- -\-rhx-{-r^ a.
This may be put under the form
a(x4+r2) + te(x2+»-) + rx3 = a(x3+r)2+Z-x(x2+r) +
(c — 2ar)x2, which can sometimes be factored.
Examples.
1. x44-6x3+27x2+162x+729.
We have x4+729+6x(.'c2+27)+27x3.
= (x2+-27)2+6x(x3+27) + 9x2-3Gx3
= {x24-27+3x}3-36x3, which gives the factora
a;3 _ 3.^_|_27, and x2+9x+27.
FACTORING. tjrg
2. a.4 4.4a;3_|_4a;0_|_20x+25. This
= (x3+5)2 + 4a;(a:3+5)-6a:2
= (a:2 + 5)2 + 4a;(a;3 4-5>+ Jx'2 - lOx^
= {a;2 + 5+2^-a,VlO} {a,-3 4.5-h2^;-t-aV10}.
Exercise xxxiii.
Eesolve into factors :
1. x4- 6x3 + 27x3 -162a;+729.
2. a;4 + 2a;3 + 3a;3+8x+16.
3. x'*+x3 4-3;2_|_a, + i,
5. ia;4-12x3_6a;2-12x-f4.
6. a:4 + 14x3 -25.1-3 -70X+25.
7. 16x*- 24x3- 16x3 +12x+4.
8. x* + 5x3-16x2 + 20x+16.
9. x4+6x3-llx2-12x+4.
10. x^+4x3//+x2(/3-+12x//3+9y<.
11. x4+6x3-9x2-6x + l.
12. x*+4x3/y-19x3!/2+4x?/3+y*.
13. 4x4 +4x3//- 65x3^3 -10x?/3+25j/'t.
14. X* +6x'7/ — 9x2?/3 _6x>/3+|/4,
15. X* + 6x3// + 10x2?/2+12x?/3 4-4^4.
16. 9x4 + 18x3?/- 52x2?/-^ -12x!/3+4?/4.
17. 11x4 + 10x3?/+39^Vt-*--2/'' +20x^/3 + 44?/*.
Section III. — Factoring by Parts.
Art. XXIII. To factor an expression which can be reduced
to the form a.F(x)+6/(x).
When the expression is tluis arranged, any factor common to
a and 6, 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 factoring.
, Examples.
1. Factor acx^ — ahx — hc^x + b^c.
Here we see that only one power of a occurs, and we therefore
group together the t^rms involving this letter, aud those not in
volving it, getting
ci{fx^ — hx) — hc"x-\-h^c
= ax{cx — h) — bc{cx — b) — {ax— hc){cx — h).
2. Factor m^x^ —mna'^x — mnx + n^a^.
Here we observe that a occurs in only one power (a^).
Therefore we have
— a^[mnx — n^) + m'^x^ —ninx
= —na^[mx — n) + vix{i)ix — n)
= (})ix — n)()nx — na-).
3. 2x^+'iax + Shx + 6ab.
Here we observe that the expression contains only one power
of both a and 6. W^ niay, therefore, collect the coefficients in
either of the following ways :
a{4:X + iib) + [2x'^+Shx),
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{Ax + 6b)-i-{2x^+Sbx)
= 2a(2a;+36) +x{2x+Sb) = {2x+db){x+2a).
4. abxy + b^ij^-\-acx — c^
— a(hxt/-\-cx) + b^y^ —c^
= ax{by-\-:;) + {by + c){by — c)= (by + c) (ax+by — c).
5. y^ - {2a + b)y'' + {2ab-\-a^)y - a^b
= _ ^y2 _ 2ay-\-a^)+7j^ - 2ay^ -\-a^y
==-b{y^-2ay + a2)+y(y2-%iy + a2)
^{y-b){y-a)^-
6. 2x^y-^2bx'^ - bx^y + Aabx^y -x~y^+iaxy- - 2abxy^~2ay^.
= b(2x^ — x^y+4:ux^y~ 2axy^) + 2x^y—x^y^ + iaxy^ — 2ay^
= bx{2x^ —x^y+iaxy ~2ay^)+y{2x^ —x-y-\-iaxy — 2ay'-)
= {y-\-bx){2x^—x^y-\-4:axy - 2ay'^).
FACTOKINO,
81
And 1x^—x^y-\-^oxy — 1a]i^
= a{Ax!i - "Iij- \+2x^—x^!/
= 2c,//(2^ - 1/) -\-x-{'2x -y) = {1ay+x^-){1x- tj).
7. x^ + {2a-b)x--{:2^ab-a^)x-a^b
= b{-x^-'lax-a^)-{-x^+'2ax-+a^x
= -h{x + aY+x{x + a)^ =(x-6)(x+«)3.
8. ]'X^-{l>-(l)x--\-{p-q)x+q
^qix"^ —x+l)-\-})X^—px'^-{-rx
= q{x^ -x+\)+px{x^ -x^-l) ={px+q){x^--x^\).
Exercise xxxiv.
1. x^y—x-z— 'y^+yz.
2. uhxy -\-b' y^ ■\-iicx — c^.
8. z^z^+ax^
4
6
6. x^ —b^x- —a-x-\-a^b^.
7. x^—a^x--b^x^ + a^b^.
8. 8x2 + Vlax+ Wlx ■+- 15a6.
9. a~^{ac-b^)x- -if-bcx^.
10. a3+(ac-63j^2_i,.^3.
2x3 — ax — 4/)x-+ 2a6.
x--\-'lbx-\-Zax + ^cib.
11. ^^^3 f (flc-6'/\r2-f^/'+r(rx-|-^//.
12. 7>.r3-(^ 4-7^-2 + (7' + 7)x-?.
13. a^-\-ab-\-1iic-^h^-\-lbc-'dc^.
14. a;3+(f,^ij^.2_,_^,/^i^_g^a^
15. iviix^-\-{^inq — np)x^ — {mr + nq)x-\-nr.
16. a;^ — (a + i + c)x* + {rt6 + &c+ac)x -abc.
17. x' + (a — 6 — r)x- — (a6 — be + ca)x+a6c.
18. x^^[a-^b — c)x^ — {bc—ca -al)x — ahc.
19. ^3^3 _y3j.2^_f,22;^_j_^3j^2 ^(,x^ yz-\-x^z — xyz-^-ay^z.
20. a^bx^-\-ab'xy -f- acdxy-\-bcdy^ — aefxz — btfyz.
21. a2_g3 _ a(^i_ciaj2_j.(.^^_^._^^(.2_
22. mx^ — 7ix^y-\-rx"z — mxy'^-\-ny^ —ry^z,
23. awx^ ^(^,)i(jy — nay-\-iv(z)x —nby^ —ncyz.
24. (am — bc)n)x--{-{a)H — bcn)x-^a7i + ncx.
25. a-b-c^—b-c-xy — a^c^yz + c^xy'-z — a^b'-zx + b-x^yz-\ a-z-^.'y
26. x' -m^x^-{n-7i-)x^ + {m^7i — 7n-n^x^—a{x^-{-n^—n).
27. l-(rt-l)x-(rt-/; + l)^-+(a + 6-c)x3-(6 + f)x4 + cx*.
28. a3x3-rt2(i_c4-,/)x2y_(a6c-aW+«cJ).r?/2 + 6v/?/3.
29. vi^npx^ — (n^p — vi-n^ —vi^pq)x^ —{n^-\-7ipq — in^nq)x — n^q.
82
PACTORTXa.
-{n^q-+n-^^x)y^.
Art. XXIV. Sometimes an expression which does not come
directly under the preceding form, may be resolved by first find-
ing the factors of its parts.
ExAiMPLES.
1. abx^ -\-aby^ —a-xy — h'^xy.
Here, taking ax out of the first and third terms, and hy out of
the second and fourth terms, we have
ax{bx — ay) — by(bx — ay), and hence
(ax — by) (bx — ay).
2. x'^-{a + b)x^ + {a2b + ab^)x-a-b2.
Here, taking the first and last terms together, and the two
middle terms together, we have
(x^+ab){x^-ab)-{a-]rb)x^+ab{a+b)x
= (x^-tab){x^-ab)-{a + b)x{x^ ab]
= {x^ — ab) {x" +ab — (a-\-b)x} ={x^ —ab)[x — a)(x —6),
3. ic3w_4a;m_|_3. This equals
X3m_^m _. 3(a.OT _ 1) = x"'{x-'"' - 1) - 3(ic'«— 1)
= x'n{x^+l){x'"'-l)-3{x'"-—l)
= {x'^-l){x"^{x'^+l) — S},
Exercise xxxv.
1. a^ —ab-\-ax — bx.
2. abx^ +b^xy — a'^xy -aby^.
3. x'^+ax^ -a^^x-a"^.
4. rt''^a; + 2a2x2-r2aa;3-f-a;*.
6. acx^ ■\-{ad — bc)x — bd.
6. 2&x*-5x3+a; -1.
7. a^ —b^ + (>x -ac — bx + bc.
8. a^ + {l + a)ab + b-'.
9. x'^+2xi/{x^—y-)—y^.
10. x^-y^+x'^+xy+y^.
xl. 2b + {b-^-A)x-2bx^.
12. x3 + 3a;2-4.
13. p3-jo2<7-2p(/3 + 273.
14. a3-fa2-2.
15. 3a264_2«62 1.
16. ^y»-3^ + 2.
17. 2a3-.7.2i _a?>2+2^/3.
18. i3m + J'2»i._2.
19. ySn_ %j1n^n _ 2 (/««2« 4- gSn^
20. a3_4rt52_f.3/;3.
21. a'2"i-3a^c" + 2c2i.
22. aa;3-(«2^.j)a;2 4.^,3.
23. 35.«2n_6a2a;"-9a4.
24. a2^2+2rt7jc2-a2c2-63c2.
25. am^ —ab^ -\-b^7n — m*.
26. |-6a2 4-27a4.
PACTORING. as
27. {x-7j)^ + (l-x+y)(x-y)z- zK
28. 2Am^ -28m^n + Gmn^ -In^.
29. a;w+"4.a;'!y«+a;'"y^+?/w+w.
80. x* + 2x^y-a^x^+X'7j^ -2axij^ -y^.
Section IV. — Application of the Theory of Divisors,
Art. XXV. By Theorem I. we prove that
^•" — a" is divisible by x— a always
'xf' — a'^ " " <■'■ x-\-a -when n is even
x'' + «" " " " a;+a when ?i is 0(/<i.
By actual division we find, in the above cases ; —
= a:''-^+ar''--a-f . . . a;a''-^+rt"-i ....(1).
X —a ^ '
aj" — a"
X 4-« ^
x' +«"
« +a
Examples
a;«-i_a;— iiff^ . . -a;a"-'+a"-^ (3).
1. Resolve into factors x^ —y^ ; here x—y is one factor and by
(1) the other is x^+xy+y^.
2. Resolve a^-^(b — c)^ ; here a + (b — c) is one factor; and by
(.3) the other is a^ -a(b-c) + ih- (-y.
3. Resolve a;'" 4-1 024?/io. This = (a;-)5 + {(2 7/)2}5, one factor
of which is a;^ + (2?/)3, andbv (3) the other factor is
= x^^ -ix'^y^-i-iax'-y'^ -Gix^y^ +256y^.
4. Resolve"(a;— 22/)^ + (2a; — ?/)3 into factors.
Here by (3) we have
(a;-27/)3 + (2x-v)^
\,-2y +2x-y = (^ - 2^)' - (^" '' 2^/^^-^ - 2/) + (2:. -y)'
;. the factors are
Bix-y){'lx^-13xij+7y^),
S4 FACTOPaNG.
By (1) we see that this = "^^^ = 'i-'+l/'K-'-v')
^ ^ X —y x — y
= {x + y){x^-xy + y-'){x''+xy + y^).
6. Eesolve x^ ^ — x^ ° a+x^ a- ~ x* a^ -\-x'' a*^ — x^a'^+x^a*
x
12_«12
-a;4<,7+T3a«-a-2a9-fa;a'"-rtii. This =
x + a
~ x+a ~ x + a
t= (x3 +«3)(^-4 -a;2a3 + a4)(x-«)(^2 J^xa + a^)i^x^ —xa+a"^).
Exercise xxxvi.
Factor tlie following : —
1. x^-y^, x^~l, ;c3+8, 8a3-27a;3, 8-}-a3a;».
2. a;5-aio, 27a3-Gi, a^S-is, a;io_32^5.
d. Find a factor wliicli, multiplied into
a^+(i^h-\-a'^b^ +ab^-\-b^, will give a^-h^.
4. By what factor must x^ — i.x'^ y+lQxy'^ — Qhj^ be multiplied
togivex'^-256//4 ?
6. F&cior x'' ^x^y+x'^y^ +x^y^+x^y^+x^ y^ -\-xy'^ f y'.
Find the factors of the followi-p'- :
6. (3//2- 2^-2)3 _ (3^3 _ 2^2)3^ „8_16^4.
8. b{x^ —a^)+ax[x'^ —a-) + a'^yx — a),
10. x^—y^+2xy{x^+x-y^-+y^).
11. (a3_5c)3 + 8/;3c3^ a;iw_a4»i.
12. x3-3«x2+3a2a;-a3 + /;3.
13. a;3 +8?/3 4-4a:?/(x-2 -2^// + -l?/2).
14. 8x3_Ga;y(2x+3//)+27»/3.
15. l-23:+4.c2-8x3.
16. a^ -\-a^bc-\-a^b^c^ -\-a^b^c-^ +ab-^c'^-\-b'^c'^.
FACTORING.
85
Art. XXVI. The principles illustrated in Section II., chap.
II., may be applied to factor various algebraic expressions, as in
the following cases ;
Examples.
1. Find the factors of
{a + b+c){ab + bc-{-ca) — (a + b){h+c){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 c are also factors. .-. abc
is a factor.
8rd. There can be no other literal factor, because the given
expression is of only three dimensions, and ahc is of
three dimensions.
4th. But there may be a numencal factor, m suppose, so that
"we have
. {a+b-\-c){ab-\-bc-{-ca)- {a+b)(b-\-c){c+a) = mabc.
To find m, put a = b-c=l in this equation, and «t = 1.
.'. the expression = rt&c.
2. Eesolve a^{b-c)+b^{c-a)+c2{a-b).
1st. For a = 0 this does not vanish. .-. a is not a factor,
and by symmetry neither is b nor c.
2nd. Try a binomial factor ; this will likely be of the form
b — c; put b — c = 0, i.e., b = c in the given expression,
and there results
a2 (c - f ) + c2 (c— a) + c2 (a - c), which = 0,
.". 6 — 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 isictoi; m (say), so that
^t^{b-c) + b^{c-a) + c^{a-b) = m{a-b){h-c){c-a).
To find the value of «t, give a, b, c, in this equation, any values
which will not reduce eitner side to zero; let«=l, b = 2, c = 0
86 FACTORING.
and we have 2 = r??( — 2), or ?»-= — 1 : so that the given expres-
sion = — (ii — i)(6 — c)(c — a), or {a — b)[b — c)(a — c).
3. Eesolve a3{b-i-c^)-\-bl{c + a^)-^c'^{a-{-b') + abc{abc+l).
Here we see at once that there is no monomial factor :
put h+c^ =0, i.e., h— —c^, and the expression becomes
a.3( -6'2+c2)-cG(c + rt2)+c3(a4-c4)_c3a(-c3a+l) wiiich = 0;
.'. b-{-c^ is a factor, and by symmetry c + a^ and a+b^ art; also
factors ; and proceeding as in former|examples we find m=l ; /.
the expression = (6 + c2)(f-fa2)(rt + i3),
4. Besolve into factors the exj)ression
(a-i)3+{6-c)3 + (c-a)3.
As before, we find that there are no monomial factors.
Let a — 6 = 0, or a = &, and substituting b for a the expression
becomes zero ; hence
« — 6 is a factor.
B}' symmetry b — c "
and c—a "
Hence the factors are
m,(a—b)(b—c)(c — a).
To find m let a=0, 6 = 1, c = 2, and we hav»
6 = 2m, or 7?z=3.
The factors are, therefore,
S(a-b){b-c){c-a).
5. Eesolve into factors
a3(6-c)4-63(c-a)+c3(a-6)„
As before, we find that there are no monomial factors.
Let a— 6 = 0, or a = b ; substituting b for a, the expression be-
comes zero ;
therefore a—b is a factor.
By symmetry b — c "
and c — a "
Now the product of these three factors is of three 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,
b-\-c and c-{-a would also be factors, which would give an
expression of six dimensions. It cannot be a trinomial factor,
unless a, h, and g are similarly involved. For instance, if a — b+c
were a factor, then, by symmetry, b—c-\-a and c — a-\-h would also
be factors, and the dimensions would be six instead of four. The
other factor must, therefore, be a+b + c. Hence,
a^{b -c) +b^{c- a)-{-c^{a—b) = m(a—b)(b — c){c - a){a + b + c).
To find m, put a = 0, b = l, and c = 2, and we have
— 6 = 6«i ;
:. m = — 1.
Hence the factors are
— {a — b)(b — c)(c — a)(a-\-b+c^,
or, {a — b)(a — c)(b — c)(a-'rb + c).
6. Prove that
a^^b^ + c^^S(a+b){b + c)(c + a)
is exactly divisible by a+b + c, and find all the factors.
Jjeta + b+c = 0, or a= —{b+c); substituting this value for a,
we have
-{b+c)^ + h3+c^ + Sbc{b+c), or
-(6 + c)3 + (6+c)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 tico dimensions, and cannot, therefore, be
a binomial ; for if a+b were a factor, by symmetry b+c, and c+a
must also be factors. The factors in that case would give a
quantity of four dimensions, while the expression itself is only
of 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 multi]ple of a^+b'+c^, instead of wliich we get
SS FACTOEING.
a2-f3(i-i_c)2, Nor can af ■}-{h-\-c)'-' be a factor, for symmetry
would requii-e two other factors, viz.: b--\-{c+a)", &ndc^ +{a-\-b)^,
thus giving a quautifcy of scffw dimensions.
The only factor admissible is, therefore, {a-\-h+cy.
Hence
a^ + i>-'-i9'+3{a-i-b){b+c){c-^a) = rn(a-\-b-^c){a + b+c)*
= vi[a-j-b-^c)^.
To find m, let a = l, 6 = 0, and c = 0, and we have l=m.
Hence the factors are
{a + b+c){a-l-h + c)(a-{-b+c).
7. Simplify
a{b-^c)^-i-b(a + c)^+c{a+b)2 — (a+b){a-c){b-c)
-(a-b){a-c){b + c) + {a — b){h-c)ya + c}.
Let a = 0, and the expression becomes
be '--^cb^-\- hc(b — c) — bc{b -\-c) - bc(b - c), which equals zero ; there-
fore a is a factor ; by S3'mmetry 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 fs-,ctQr.
Hence the expression =mabc.
To find m, let a=b = c = l, and we 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 x^+l is a factor of a;3" + 2jc-"-l-3.r"-|-2.
Let a;"-i-l = 0, or x"=—l, and substituting, the expression
vanii-Ves, therefore, x"-\-l is a factor.
9. Examine whether a'^ + b^ is a factor of
2a'^+aSb+2a^b'-i-^ab^.
Let a^ + b- =0, or a^ = —b-, substituting, we have
264_a6-''— 2Z-4+ai3 which = 0, and
therefore a^-j- 6- is a factor.
FACTORING. §9
10. Prove that, a^ +h^ is a factor of
JjQt a^-\-h^=0, or a^ = — h^; substituting, we have
-a^h^ -ab^-b^-^-a-b^+ab^-i-b^, \vliich = 0, aud
therefore a^-\-b^ is a factor.
Exercise 3cxxvii.
Eesolve into factors
2. hc{b - c)—ca{a -c ) -ah{b -a).
5. I^a+b)^-{b-^c)^ + {c-ay.
6. a(b-c)^+h{c-aY-\-c{a-bY.
7. {a->rb + c){ab + hc+ca)-ahc.
8. a^{c~b^) + b^{a-c-) + c^{b-a^) + abc{abe-l). -
9. a2(6 + c) + /j3(6- + a) + c2(a + />)4-2«6c.
10. {a-h){c-li){c-k)-ir{h-c){a-h){a-k) + {c-a){b-'i){h-li).
11. x^ij" -f x^y^+x'^z'^ +x-z'^ + ij'^z- +7j^z'*^ + 2x^y-z'^.
12. {a-by-^{b-c)^+{c-ay
13. aft(a+i) + /;c(/»4-c)+ca(cH-rt)+("^ + ^^+'-^).
14. a^{c-h^)+b^{a-c^)-\-rA{b-a^)+abc{a^b2c^-l).
15. a;4(.y3-z2)_|_.y4(23_^2) +24(^3 _,y2),
16. a;4^.y4_j_24_2a;2y3_2^223_2z2^2.
17. {b — c){x - b)(z — c) -\-(c — a){x — c){x — a) + (a — b)[x — a){x— b).
18. (a+fc)3 + (/,+c-)3 + (c + «)34.
19. Shew that a^ +<i'^b- —ah^ - b^ has a^ — b for a factor.
20. Shew that (x + ?/) " - a: ^ - ?/ ^ = 7xy{x+y) {x^ +xy+y^)^.
21. Examine whether x^ — 5x-\-Q is a factor of
a;3_9^2_{_26a;-24.
90 FACTOEING.
22. Skew that a — 6+c ig a factor of
a^{b+c)-b^{c+a) + c^fn+h)-^ahc.
23. Shew that a'^+3b is a factor of
and find tbe other factor .
24. Find the factors of a^{b-c)+b^{c-a)+c'>^{a-h).
Sectiosst v.. — Factoking a Polynome by Trial Divisoes.
Art. XXVII. To find, if possible, a rational linear factor of
the polynome.
Substitute successively for x every measure (both positive and
negative) of the term k, till one is found, say m, that makes the
polynome vanish, then x — m will be a factor of the polynome.
Examples.
1. Factor a;3-t-9x-+16j;+4.
The measures of 4 are +1, +2 and +4. Since every coeffi-
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
makes the polynome vanish ; thus
1 9 16 4
-2 -14 -4
-2
17 2; 0
Hence the factors are {x + 2){x^-{-7x-\-2).
The labour of substitution may often be lessened by arrang-
ing the polynome in ascending powers of a;, and using 1 —
(measure of k) instead of the measures of k. (This is really
substituting 1 -f measure of k, for 1-i-x). Should a fraction
occur during the course of the work, further trial oi that measure
of k will be needless.
FACTORING. gi
Examples.
2. Factor x^ - lOa-3 - 63a;+60.
The measures of 60 are +1, +2, ±3, ±4, +5, fee. Neither
4-1 nor - 1 will make the polynome vanish. Try 2 ; thus
: 60 -G3 -10 1
1 I 30
2 I 30 -161-
A fraction occurring we need go no further. — 2 will also give
& fraction, as may easily be seen. Next try 3 ; thus
60 -G3 -10 1
20
20 -14i
A fraction again occuring, we may stop. — 3 will also give a
fijaotion. Next try 4 ; tlius
I 60 -63 -10 1
1 i 15 -12
4 1 15 -12 - 5i
Next try —4.
I 60 -63 -10
-1 : -15
4 1 15 -19i
Ne^t ix^vaeg 5 we find it fails, then try — 5, thus
-1
60
-63
-12
-10
15
1
-1
5
12
-15
1;
0
The remainder vanishes as required ; the factors are, therefore,
(a;+5)(a;2-15a;+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 of x (including the constant
terra k) ; let the sum be called h^.
92 FACTORING.
From the sum of the coefficients of the even powers of x
(including k) take the sum of the coefficients of the odd powers of
x; let the remainder be called k^. (In the coefficients are in-
cluded the signs of the terms).
1st. If k^ vanish, x — 1 will he a factor of the polynome.
2ud. If k^ vanish, x-\-l will be a factor of the polynome.
3rd. If both k^ and k,^ vanish, x^ —1 will be a factor of tlie
polynome.
4th. If neither k^ nor k.^ vanish, (writing p for " a positive
measure of k greater than 1 ") ;
[a) We ijeed not try the substitution of p for a; unless^ — 1 be
a measure of k^, and j9+l a measure of k^.
(B) Nor need we try the substitution of —p for x unless /J-fl
be a measure of /tj, and p — 1 a measure of k^.
(In trying for measures, the signs of k, k^, and k^ may be
neglected.
Examples.
1 . Find the factors of a;^ - lOa;^ - 63a;+60. (See Ex. 2 above).
B.exek = QO; k^= 1 -10-63 + 60= -12,
k.^=-l- 10+63+60 = 112.
Tabulating the trial-measures we get
12
1,
2,
3,
4,
60
2,
3,
4,
5,
6,
10,
112
4,
7,
12
3,
4,
6,
60
2,
3,
4,
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.
98
In the upper table, 8 is the only measure of 60 that gives a
full column ; heuce of the positive measures of GO 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 .r4+12x3-40a;2-|.67a;-120.
A- =-120; /.•,=1 4-12-40+07-120= -80; »
A-2 = 1-12 -40- 07 -120 = -238.
80
1,
2,
4,
5,
120
2
3>
4,
5,
6,
8,
10,
12,
15, &c.
233
7,
80
4,
5,
16,
120
2
3,
4,
5,
c,
8,
10,
15, 20,
21,
'&c.
238
1,
2,
7,
14, 21,
The upper table gives us 6 as a trial-measure, and the lower
gives us —3 and —15.
Trying these
i we get
-120
07
-40
12
1
1
-20
6
- 20
n
-120
67
-40
12
1
-1
40
8
-40
35 1
i - 120
67
-40
12
1
-1
8
— 5
3
-1
15
- 8
5
- 3
1:
0
94
FACTOEINO.
Hence z+ 15 and x^ — Sa;^ + 5a; — 8 are the factors. The latter
cannot be resolved, for our tables above tell us we need try only
x—Q, a;+3, and ic+15. The first two have been found not to be
-factors, and 15 will not measure 8.
4. Factor a;* - 27a;2 -f 14a:+ 120.
A- = 120; /.-, = 1-27+14+120 = 108
A", = 1-27-14 + 120= 80.
108
1,
2,
3,
4,
9
120
2,
3,
4,
5,
6,
8,
10,
80
A
5,
108
v3.
4,
6,
9
120
2,
4,
5,
6.
8,
10,
80
1
2,
4,
5.
12,
15, &c.
16,
12, 15, &c.
The upper table gives ua 3 and 4, the lower table gives us - 2,
— 3, and -5. Using these in order we get
Hence « — 3 is a factor.
Hence a; — 4 is a factor.
Hence a:+2 is a factor,
and there remains a;+5, a factor.
Hence the factors are (x-3)(a;-4)(a;+2)(a; + 5).
5. Factor x^ - px^-\- {q - V)x^- +px - q.
k=-q; A-,=l-/> + (7-l)+2>-? = 0;
k.,=^l+p + {q-l)-p-q = 0.
Since both k^ and k^ vanish, the polynome is divisible by both
a; — 1 and a;+l.
120
14
-27
0
1
1
40
18
— o
-1
3
40
18
- 3
-1;
0
1
10
7
1
4
10
7
1;
0
-1
-5
-1
2
5
1;
0
-1
-p
1
-p+l
-1
q-1
-P + l
q-p
+P
P
q-p
1
FACTORING. 95
Hence the other factor is x- — px-\-q.
6. Factor x^ + 1ax^+{a^ +a)x--{-2n^x-^a^.
k = a^ ; /f^ = l+2a + (a2+a)-f2a^-f-a3 = («4-l)3;
A-2=l-2a4-(a3+r/)-2a2+a3 = a3_«2_„_i_
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 be rejected since k^ or (a-)-l)3 is not divisible by*either
a2 + l or a^±l. But ky is divisible by a+1, and k^ is divisible
by a — 1 ; thus we need only try the substitution of — a for x.
(See 4 0, page 92)
1
2a
a^+a
2a2
a»
— a
-a^
-a2
--a?
1
a
a
«2;
0
— a
0
-a3
10 ■ a; 0
Hence the factors are {x-\-a)-{x^-\-a).
7. Factor a;^ -[(,, + c)x^ +{b + nc)x-bc.
k = —be ;
kj = 1 — (a+c) + {b+ac) — bc= 1 —a+b—c + ac -be
kj= -l-(a+c)-{b-\-ac)'-bc= -(l+a-\-b+c + ac+bc).
The factors of k^, other than 1, are b and c. k^ is not divisible
by either 6 + 1 nor by c+1. However, A; j is divisible bye— 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 8G).
c
(b + ao) —be
— ac be
I 1 —a b ;
Hence the factors are (x — c){x^ — ax-{-b}.
96
rACTOEING.
Exercise xxxviii.
1. a^—9a^ + liJa-4:.
2. a;^-9a;3 4-2Ga;-24.
3. .r3-7a;2-j-i6a;-12.
4. a:3_i2:,.+ i6.
5. x^ + '3x^ + ox + S.
6. a;4_)_i^.3_|_i0x2+12a;+9.
7. a;3-8^+2.
8. a;4 + 2a;3-|-9.
9. m^ — Bm-n + Amn^ — 9,11^
10. a;3_^2a;3 + 2.
11.
5)ii^n-\-8i)i)}^ — An^.
12. 63+//2(._|.76,.2^ 39,-3.
13. ;/'4 -.4»/»3 4-3,;4.
14. a4_7,,3/^_j..28,(//3_iC64.
15. a:3-ll.T2-f39a;-45.
16. a;3H-5.T2 4-7x4-2.
17. «3_8a3_i93„_j.i95.
18. p^-Sp^-Gp-S.
19. «4 4-Grt3_3rt2_7a4_6.
20. rt«"-6a^"4-lla-"-6.
21. a4_41«2/,2_|_1664.
22. «4_^,2/,2_2((634-264.
23. p^-4p^-\-Gp-'4:.
24. a:-"4-4a;-''-5.
25. 2/^
26
27.
28. 2u'"-a-"-
,4 _ 5^34.8^2 ._ 8.
„4_2„3 + 3«2_2r,4-l.
«3+„2/,2_|_a/,2_3^3.
■a" 4-2.
29. a;4- 18x3 + 113.^3 _ 288x4-252.
30. x'^-dx^y-^-'ZOx^y^-ddxy^ ^18y*. '
Art XXTX. To find, if possible, a rational iioear factor of
the pulynoiBe
ox" 4- />x""^ + rx"~' -f- 4- ^'^ + ^•
First Method. Multiply the polynome by «"~^.
(aa:Y+b{axY-^+ac{ox)''-^ + -\-a"''h{ax)-\-a"-Vc;
or writing y for ax,
y^ 4_ /;_,y"-i .4. ^r ?/"-^ 4- -\-a''-'hy -{■ a^-^k.
Factor this polynoiie by the method of the last article, replace
y by ax, and divide the result by a"~^
Example.
Factor 3x4-|-5x3 -33x2 4-43x-20.
Multiply by 33 and express in terms of Bx.
(3x)4-h5(3x)3-99(8x)2 4-387(3x)-54&;
or, ?/4 -1-52/3- 99^2 _j_3872/_ 540.
FACTORING.
97
B.eveJc= -540; ^•,=1 + 5-90 + 387-540= -246;
Zo = 1 - 5 - 99 - 387 - 5i0 = - 1030.
82, 123, 240.
246
1,
2
3, G, 41,
540
9
B,
4,
1030
5,
246
3,
6,
41, &c.
540
2,
5,
1030
1,
(Trying by factors of 246
instead of by factors of 540,
for couvenienct).
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 subriti-
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 tliat 135 is not a multiple of 2. The other
factor is therefore y^ + O//^ -63// + 135.
Replacing ij by 3a; aud dividing by 27 ;
^V(3a;-4)(27x3-f81x2-189a;+135)
= {Sx-A){j:^ +3x3 -7x + 5),
which are the factors.
Art. XXX. Second Method. Writing »? for " a measure of
a," and p for a " measure of k, positive or negative ;"
For X substitute every value of p-m till one, s^j p'-^m' be
found which makes the polynome vanish; then in'x — p' will be
a factor. Should a fraction be met with in the course of substi-
tution, farther trial of that value ui p~in will be useless.
Should k have more factors than a, it will genei'ally be better
to arrange the polyuome in ascending powers of x and use values
of m 4- JO instead of p-i-iit, making jj positive and vi positive or
negative.
98 FACTORING.
To reduce the number of trial-measures, calculate b.^ and Jc^, as
directed on page 92, then 1, 2, 3 hold as on that page, but in 4
read ^? — w for p — 1 and p-{-m for p + 1.
Examples.
1. Factor 36*3 + l71a,2-22x+480.
k = 4.S0,k^= 36+171-22+480 = 085
^2= -36+171+22+480 = 637.
m may have any of the values +1, +2, +3, +4, +6, +9,
+ 12, +18, +86.
In forming the table write out the measures of ^^ ; 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., Jc^, keep the mea-
sure of 480, writing above it the value of m used. Should the
sum in either case not be a measure, another value of vi must be
tried ; when 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 y^i 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
-3
-9
-3
665
1,
5,
7
5
7
19
19
480
4,
6,
10
3
4
10
16
637
7,
rr
< .
13
1
1
1
13
Hence the only substitutions that need be tried are
4 ' 6 " lO' 3' 4' lO' 16' X '
Arrangement in ascending powers of x.
By actual trial, as below, we find ^| is the only one of these
giving a zero remainder.
FACTORING.
99
8
480
- 22
360
171
86
4
1
120
844
80
6
3
80
9|
144
10
-2
48
12-2
-320
228
-266
8
-3
160
-114
360
133;
-230
4
-9
120
- 95^
-432
10
-3
48
-45-4
- 90
21
-36
16
30
- 7
12;
0
(The coefScients are written only once, and understood for the
other lines of substitution.)
Hence the factors are 3a;+16 and 12x- - 7x-\-?>0.
The latter factor cannot be resolved, for 16 will not measure
30, and all the other factors left for trial by the tables above,
have been tried and have failed.
2. Factor lOx^ -x^(15i/-\-4:z) -z^{4:0y^ -6yz)-^
a;(60!/3 + 16?/2z)_247/3z.
Here m= ±1, +2, ±5. or +10. k= -2iyH.
^•l = 10 -(152/+4z)-(40y2-6yz) + (607/3 + 167/22) _24?/32
= 10 - 15y - 40y2-f-637y 3 - 2^(2 - 3v/ - 8?/2 + 1 2?/3)
= (5_22)(2-37/-8y/2+i22/3).
7,-3 = (5 + 2z)(2 + 32/-8y2_i2^3)^ as may easily be found
by making the calculation.
We get at a glance 2z a factor of 7c, 2z — 5 a factor of k^, and
2z+5 a factor of k^ ; hence taking m = 5, we are directed to try
the substitution — for x.
5
10 -(Uy + iz) -(407/2-67/2)
2? 4z — 6yz
(607/3 + 16,/=2)
-16?/-z
-247/3-
2%\
5 2 -By —87/2
127/3;
0
100
FACTORING.
Hence 5x — 2z is a factoi', tlie other being
The latter factor being homogeneous, the method of this article
may be applied to it.
771= +1 or
111 = 1,
±2
, /c =
2,
12,
1,
/.,
= 3,
-1
k, = 15.
3
12
15
1,
1,
3,
5,
3,
4,
5,
3
2
1
The other columns
are not full.
Hence the trial- substitutions (arrangement m ascending powers
of x) are i, f , i, =±.
12
-8
-3
2
1
6
- 1
-2
2
6
-1
-2;
0
2
4
2
3
2
1;
0
Final factor is 2y+Xr.
Hence the factors ai-e {x — 2//)(2x — dij){x+2)j), and these, with
the factor 5x — 2z already found, give the complete resolution of
the polyuome proposed.
(The factor ijx — 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 of ^).
Exercise xxxix
Factor
l.aa;3-20a:3+38a;-20; x^ -7x^y + lGx>/^ -12ij^.
2. 12x^-i-6x^yi-Z!j^ + 3y^ ; 8x^ -Ux+G.
8. 3x^-15ax+a^x-5a^; 2x^-{-9x''y-^7xy^ -Sj/^.
4. 254_7i3c_463c2_|.j(.3_4c4 ; l5a3-}-47rt8i + 13rt52_i263.
5. 4p^ + 8p^q+lp^q^^-hSpq^ + 3q^.
6. 150a;'i -125x^y+d'dlx^y^ +920zy^ - 1152?/*.
7. 36a;^— 6(9-7?/)a;3-7(9 + 14?/)x3//+3(49-40?/)a;?/2 + 180?/3.
8. lOx* -x3(] 5y+42) + j;3(40(/2 _|_Gy2) +x(G0(/3 -iG^^g) _ 24:y''z
DIVISION. 101
CHAPTEK 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 a2_2a64-62-c2 4-2«Z-(/3 hy a -\h-\-c—d. Here
we see at once that the dividend ={a — b)^-(c — d)^, and .•. quo-
tient = a-h — [c-d) = a — h — c-\-d.
2. Divide the product of a^-\-ax-^x^ and a'^-\-x^ by a^-\-a^x^
+x^. Here a^+x^ ^{a-{- x){a^ — ax-{-x^), and the divisor =
{a^-]-az-\-x^){a^ —ax+x^) :. the quotient is a -fiC.
3. Divide a^+a^b+a^c — abc-b^c-bc^ by a^-bc. The divi-
dend isa(«2_5c)-|-6(«2 _i(.)_|_e(a3 -he) .".the quotient =a+b->-c.
4. (a^+b^-c^ + Sabc)~{a-{-b-c).
Dividend =a^-{-b^-{-Sab{a-\-b) -c^ -Bnb{a+b)-\-Sabc = {(X.J^b)^
— c3 — dab(a -\-b — c) which is exactly divisible by a-f ft — c ; quotient
=a2-[-ft2_^(.2 _(ib-{-bc-\-ca.
5. Di\ide z^ —x'^y-{-x^y^ — x^y^ +xij'^ —y^ byx^—y^.
The dividend is (Ai't. XXV,) evidently (x^— ?/*') -^ (^3+;/), and
this divided by x^ —y^ = {^^+y^) -^ {x-\-y)=x'^ —xy-\-y-.
6. Divide b{x^ +a^)-\-ax{x^ - a'2) + a^{x+a) by {a-\-h){x-ira).
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-\-o{x^ — ax-^-o"^)
= {a-^b){x^ — a£-\-a^) .•. quotient =z^ —ax-\-a^.
7. DWidie apx^ •\-x^{aq + bp)-\-x'^{ar -\-bq+pc)+x{<ic-\-br) ->t- cr by
ax- -^bx-^"
102 DIVISION,
Factoring the dividend (Art. XXIII.) we have
/. the quotient = the latter factor.
S. Divide Q>x^ - ISaa;^ + l^a^x^ - l^a^x- ha^ hy 2a;' - ?>ax~ a?.
This can be done by Art. XVII. The divisor is 2a;3 -a^ - 3ax,
and we see at once that Sx- -\-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 -{-la^x^ ; but -{-ISa^x'^ is re-
quired. .'. -\-Qa-x^ is still required, and as this must come from
the third term multiphed into — Sax, that third term must be
— 2ax ; .". the quotient is 3x^-{-5a^ — 2ax.
j^OTE. By multiplying the terms - 2ax, —Sax, diagonally into
the x^'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 2a^-a^h-12a-b''-5ab^ + U^ hj a^-b^-2ab.
Here, as before, one factor is a'^ — h^—2ab; :. tivo terms of
the other factor are 2a^-Ab^. Multiplying, as in the last
example, we get -6a^b^ ; but -12a^b^ is required. .*. —6a^b^
is still needed, and +3a& is the third term of the required quo-
tient, which is therefore 2a^ — 4:b^-\-Bah.
Prove that
10. {l+x+x"'+ . . . . -Jr.v''-')(l-x+x^- .... +x''-^)
= l-f.x2+a;4+ .... +x--"-^
l-x" 1 -f a;"
Product = Y=^'- Y+^
^~''^ = 1+x^+x*-^ .... H-a;^"-*.
-x'
11. Divide (a^-bc)^-]-Sb^c^ by a^+he.
= (a2-&c)3 4-(26c)3 hy {a^-bc) + 2be
^ (a2 _/,c)2 -(^2 _5c) X 2ic+(2ic)«
= a^'-4.a^bc-\-lb^c^.
DIVtSON. 103
12. Divide l+2357947691a;^ by l-lla:4- 121^3
Dividend =l4-(lia;)»
= {1 -(llx)3.f (lla;)«}{l + (lla:)H
Divisor = { 1 + (11a;) 3 } -^ (l-fllx).
.-. quotient = U-(llx-)3+(llx)«}(l+llic)„
Exercise xi.
Finrl the quotients in the following cases :
1. 1 —x-\-x^ —x^ -^1—x.
2. l-2x^+x^ ^x^+2x^ + l,
3. x^^+a^x^+a^^ ^x*-a^x-+>i^,
4. 2;4H-4a;2i/3-32?/4-Hcc-27/.
6. l-4x3+12a;3-9a;*^l + 2a;-S.T'.
6. {a^ -2nx-\-x-){a^+'da'^x + Sax--\-x^) -^a'i -x^.
7. x^—i/^+z^-\-dx>/z^x—i/ + z.
8. 6a4_a36 + 2«262 4-13^63 + 454 ^ 2^»-3a64.4^».
9. 4:X^ — x^y" +Qxy^ —9i/* -r- 2x^+Sy^—xy.
10. rt*+i4_c4_2rt263-^«3_J2_c2,
11. 21a4-16rt3i-f.l6a2^3_5ai3 + 264 ^ Sa- -ab-{-b^.
12. 2ffl3_7a2_46rt-21 H- 2a2 4-7rt + 3.
13. {a^(b-c)+b^{c-a)-^c^{a-b)} ^ a~\~b-^c.
14. x^ — Sax'^-'r3a-x — a^-\-b^-—x—a+b.
15. a;*-?/4+0^-f-2a;322_2y2_i _^a.3_^2u-^s_i.
16. «•* — (a + c}x3 + (6 + ac)j;3 _;)fa; -^ x — c.
17. x3+a;3?/ +ic?/3+?/3 _^ x+y.
18. x'^ —x^y+x^y^ — x^y^-JrX^y^ -x^y^-'rxy^ —y' -^x*+y^.
19. a'i+i4-C*-2fl3fe2_2(;3_l -HVi2_/;2_c2_l.
20. a* - ai3_ac3-2y 36+264 + 26c3^3«3f_ 363c _3c4
■f a + 3c-26.
21. a36-6a;2-{-rj2a;-a:3 -^ (x + 6)(a-a;).
22. a(6-c)34.6i^c-a)3+f(a— 6j3 -^ ^^ -a6-ar+6c.
1*^4 MEASURES AND MULTIPLES.
23. a^b^ + 'lahc' - a^c^ - b^c^ H- ab+ac-i>o,
24. x^ + y^ + dxij-1 --^-x + y-l. \
25. x^-x"-^-^ x^~x+l.
26. a4-29a3-50a-21 -^a2_5«_7.
27. (2x-y)^a* - (x+y)2a^x^ +2{x+y)axi' - x^ -^
(2x~y)a^ -c {x-{-ij)ax-x^.
28. (a;3-l)a3_(a;3^^2_2)a2+(4x3+3a;+2)a-3(a4-l)
^ (a;-l)rt3-(a:-l)a + 3.
Art. XXXII. The Highest Common i? actor of two algebraic
quantities maVj in general, be readily found by factoring. The
H. C. F. is often discovered by taking the sum or difference (or
sum a7id difference) of the given expressions, or of some multiples
of them.
Examples.
1. Find the H. C, F. of {b-c)x^-i-{2ab-2ac)x+aH-a'^o, and
(ab — ac-{-b'''—6c)x-{-a^c-\-ab'^ -a'^b-abc.
Taking out the comm.on factor b—c we get {b — c){x'^ +'2ax+ab)
and {b~c){{a—b)x-a^+ab} ;
.-. b-cis the H. C. F. of the i^iven expressions.
2. Find the H. C. F. of
l—x + y+z — xy+yz — zx — xyz, and
l-x—y—z+xyi-yz+zx—xyz.
Their difference is 2y-]-'2z — '2xy — 2zx = 2{l — x){y-\-z).
Their sum is 2-2x-i-2yz-2xyz = 2{l-x){l+yz).
.-. theH. C. F. is (l-x).
3. Find the H. C. F. of x^+3x^ -&x^ -9x-3, and
a;5 _2.7;4_Ga;3+4a;2 + lBx+6.
The annexed method of finding the H. C. F. depends on the
principle, that if a quantity measures two other quantities, it will
measure any multiple of theii- sum or difference.
MEASUKES ANO MULTIPLKS.
1
1
+ 3
- 2
0-8
-6 + 4
-9-3 (a)
+ 13 + 6 (b)
5
+ 6 -12
-22 - 9 (c)
2
1
+ 6
- 2
0 -16
-6 + 4
-18-6
+ 13 + 6
3
+ 4
- 6 -12
- 5 (d)
15
16
+ 18 -36
+20 -30
- 2 - 6
-66 -27
-60 -25
-6-2
1 + 3
+ 3+1 Cf)
25 +30
27 +36
-60 -110 -45
-54 -108 -45
-2-6
-6-2
(a) X 2
(c)x3
(i)x5
(.)x5
{d) X 9
1 + 3 + 3 +~i {g)
E. C. F. = (a;+l)3.
The coefficients are written in two lines, (a) and {b). They
are then subtracted so as to cancel the first terms, (a) is next
multipUed by 2, and added to cancel the last terms. If (c) and
{d) had been the same their terms would have 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 (6)
do not contain the same number of terms it is more convenient
to find only (c), and then use this with the quantity containing the
same number of terms. The general rule is to operate on lines
containing the same, or nearly the sams number of terms.
4. Find the H. C. F. of 3a;3+2a;2-14^-+8, and
6a;3-lla;3 4-13x-12.
3 4- 2 -14 + 8 (rt)
6 -11 +13 -12 (6)
6 + 4 -28 -fl6 («) X 2
15 -41+28 (c) W-(«),
(5-7)(3-4)
H. C. F. = 3a;-4. {d)
If (a) and (i) have a common factor its first term must measure
8 and 6, and its last term must measure 8 and 12. (f) is not
106 MEASURES AND MULTIPLES.
therefore, the H. G. F. Eesolve (c) into factors. 5z — 7 is not a
factor of (a) and (b). If, therefore, (a) and (5) have a common
factor it is 3a; — 4. On trial Sa; — 4 is found to be a factor of (a)
and .-. it is the H. C. F. of (a) and (6).
5. li X- +px-\-q, and x^+j-x+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 = Q. or
s — o
{p~-r)a = s-q, .-. a = -^,, and
s —q a— s
x — a = x— -. —x+ '-_ — -•
p —r P—^
6. What value of a y^iW m?i,'ke a^x^+{a-\-'2)x-^l. and
^23.3 _j_rt2 —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 4a;"^ + l, and 4x" —1, which
have no C. F. Hence if the given quantities have a C. F., it
must be of the f@rma;+»i; dividing a2a;3_j.a3 _ 5 by x+m, we
have for remainder,
5-«^ 1 .. ox • ■>• ,
a^m^ + a^ -5 = 0, or m^=: ^„ ; .-. 'rn= — -/(t. -a-), m which
|/(5 — a^) must be possible and integral, .-. a^=4:, (a^^icrives
values to m which on trial fail) and a = + 2, of which the positive
value must he taken, a.nd .-. 2x+l is the C. F.
7. If the H. C. F. of a and b be c, the L. C. M. of
a6 - 66
(a+6)(as-/j3),and(«-6)(«3^&3)is ^— •
Let a = mc, h = nc, and .-. a^ = 7n^e^. b^=n^c^. Thus
(a +b ) = c (»« +n ); («- —b ) = c (m —n ), and
(a3_f.J3) = c3(m3+n3); (^3 -63) = c3(w,3 -nS).
,.. (^a-irb){aS-b^)=c'^{m-{-n){'m.^—n^), and
MEASURES AND MULTIPLES.
107
TheH. C. F. of tlie last expressions is c^(m*-n^), .'. the
L. C. M. = c4(„,,6_„G)= -_^_^ =—^2—
8. U (x-a)' measures x^+qx-\-r, find the relation between q
and r.
Let a; + 7?i be the other factor, then
x^+gx + r = i;c-ay{x^in)^x^ + {m - 2a)x%+{a^ - 2a«»)a;+OT« =
equating coefficients,
771 — 2a = 0, a- — 2rtw = *7, via^ =r
,-. J7t = 2a, and .•. a2_4rt3 _^^ 2^3 =7-, and
«^ = — -fr, 01 « '^ = - 07 ■» *^<^ ^^ = "^ °^' ^"^ = "4"
;-3 g3 J.2 ^3
.-. -=-^,or J-+^=0.
Or thus : —
Dividing x^ +qx+7- by (a;— a)' we find the remainder
{q + da^)x+r-2a^
and as this will be the same for all values of z, we have, by equat
ing coeffijcients,
and r — 2a^=0,
or q3 = -21a^
and r2 = 4a^ ;
^2 ^3
therefore "T" + 07 = 0, as before.
Exercise xli.
Fh3d the H. C. F. of the followmg :
1. 2x*+3x^+5x^+dx-S, Sx^ -2x^ + 10x^ -Cx+B.
2. x3 + (a + l)a;2-f(a + l)x-}-a, x"^ + {a-l)x^ -{a-l)x + a.
3. px^-{p-\-q)x^+{p-q)x + q, px^ -{p+q)x^ +(p + qyx-\-q.
4. ax^-{a-b)x^-{b-c)x~c, 2axJ^i-{-{a-{-2b)x^-\-{b + 2c)x+o.
5. l-3|a:-3Ja:2 + ia;3-x-i, l-l^^x-'dx^ + l^^^x^-i-x*.
6. ac^+bc"" + {a -|- 6)c"+*, a^c-'+a'^c" + c-"6' + b'c\
108 MEAST3ST',^ AND MUI/TIPLES.
7. a^x^ -\-a^ —2ahx^ +b^x'^ +a^h'^ -^a'^b, and
2a-^x^ - Sa^xS +3ffiC _ 2b^x^ + 5aH^x^ - 3aH^.
8. (aa;4-fc?/)3-(a-&)(:c+z)(rta; + %)+(a-i)2a;5;, and
(aa;-6z/)2-(a + fe)(ic+2)(«a;-%) + (« + 5)2a;z.
9. «(i2-c3) + ?y(c3-a2)+c(a2-63) and
10. ^s^+a-^ + a'^ + l, and a^t" _««»•+ «"• — !.
11. If a;3 4.fl^2_^j^^c^ and x^-\-a'x + h', have a common factor
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—b)'^+{b-c)^-i-{c-a)<^, and
(«2 - b^)-"' + {b^ -c2)^ + (c2 - a*)^.
13. Find the H. C. F. of
2(w3_2?/2_y+2)a;3+3(?/2-l)a;2 — (2i!/3_y2_ 2?/+l), and
8(?/3_4?/2-|-o?/-2)a;-+7(?/3_2?/ + l)x-(3^3_5_(^2 +.^ + l).
14. If cc24-j;a;+?, and a-- +??7j5+37. have a, common hnear factor,
shew that
{n—q)^-\- n{m—pY =m{m—p)(n — q).
15. Find the L. C. M. oi x^ -Sx^ + dx-1, x^ -x^ -x+l,
a;4 _ 2a;3 + 2a; - 1, and x* - 2a;3 + 2x- - 2x+l.
16. Find the L. C. M. of
a;3 + 6a;2 + lla;+6, x^ + 7x^ +Ux+S.
.^3 4.8a;2 + 19a;-f 12, and a;3+9a;3+26a:+24.
17. Find the value of y which will make
2(?/2+2/)a;2 + (ll2/-2)a;+4 and
2(?/3_i_^2)aj3_|_(ii2/2_2^)a;3+(2/2 + 52/)a; + 52/-1, 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
2:^3 _ ll;c2 + ITx - 6 ; find the other.
19. Ifa;+aand x-a are both measures of x^+px^+qx+r^
shew that pq = ^>
FRACTIONS.
109
20. If x^+qz'-j-r and x^ -tvix+n hare a oommon measure
(« — a)2, show that q^n^=m^r^.
21. If the H. C. F. ofa;3 +px + qa.nd o:^ + mx+n, be a-+a, their
L. C. M. is
22. If x2+5J5 + l, and x^+px^+qx + l, have a common factor
of the form x + a, shew that (;; — 1)3—5(^0 — l)-i- 1 = 0.
23. Jlx^+px'+q, and x^+mx+u, have a+a for then- H. C.
F., shew that their L. C. M. is
x^ +{m — a+p)x^ -\-p{m — a)x^ +a^ {a— p)x + u^ (a - p){m — a).
24. If X-+PZ + 1, and x^+px^ +qx-\-l, have x-a for a com-
mon factor, shew that a=
25. Find the H. C. F. of {a^ -b^y + {b' -c^y+{c^ -a^)^,
and a" (6 — c) + i^ (c — a) + c^ (« — Z>).
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 h, and S the H. C. F. of a, h. and e, then
the L. C. M. of a, 6, c, is -^. ^ •
27. If x-f c be the H. C. F. of x-3 +ax+.^, and x^ +a<x-\-h\ their
L. C. M. Will be x^ + (a+a'-c)x-+{aa' -c^)x+{a-c){a' -c)c.
28. Shew that the L. C. M. of the quantities in Ex. 2 (solved
above) will be a complete square ii x = y''^ -\-z^ —y^z^.
29. Find the H. C. F. of a;«+2a;«+3a;4 -2a.'2 + l, and
Qx''+ x' -{-llx'^ -Ix^ -"l.
Section II. — Feactioks.
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 FEAOTiO^b.
ExAMPLiES.
J ac-\-bij+ay-\-hc ^ c{a+b)+y{a+b) c-\-y
a/+'2bz-h2ax^bf ~ J\a+b)+'2x{a+b) ^ f+2x'
2 a^—ba^—a'-ib^ + ab^ a^a^+b^ -ab{a+b)}
a^—bw^ — ab^ + b^ ~ a{a'^ — b'^) — b{a^ — b'^)
_ a(a-\-b){a — b)' . a
{ci~b){a^—b^) ^ a^^b^'
g x^ +x^il-\-x^y^ -{-x-y^ -\-xy'^ +y^
x^—x^y+x^y^—x^y^^-zy^—y^'
Here the numerator is evidentlj {x*^ —y^) ^ (x-y), and the
denominator is ^ ~y , The result is .". ^^•^.
x+y x-y
4 {x+7/)'^ -x^ -y^ _ 5x^y + lQx^y^ + Wx^y^ + 5xtj^
(«-h/r*-f-'«M-/^ ~ (x+y)^-x^y^ + {x^ +y-t)^^x^y^
^^Uj a;3+y3-t--2.y>/(a;+y) }
(a;3 +^3 +0;?/) { (3;+2/) ^ +a;i/ +a;3 +2/2 - a;?/}
5xy{x+^j){x^+xy+y^) 5xy(x+y)
5.
2(a;2+5Cj/+v/3)2 2(a;2+a;!/+2/')
x2-12a;+35
a;3_ioa;2+3ia;_30
Here we see at once that the numerator = (a;— 51 (« — 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 c»uotient to be x^ —5x+Q.
x-1
. the result =
6.
x^ —?>x+Q
a;4 + 2a;^+9
J*— 4a;3 + Sa; — 21
The factors of the numerator are at once seen to be a;'+2a:+3,
and a;3 — 2x+3, of which the latter is one factor of the denomin-
ator, the other being (Horner's division) '•.'' — 'lx — 1: .'. the result
is ^±2Hi3
a;3-2a;-7*
FRACTIONS. ill
Exercise xlii
Reduce the following to their lowest terms :
x^-lx + 6 3xj/^-lSx!/+Ux
1. -
3 -2x2 -8a; -96 7//3- 17^2^6//
x*+a-x--\-a'*^ a-2-{-a;-12
x'^+ax^ —a^x — a^ x* — oa;^ +7a._ 3*
a;3_3a;+2 x'^+2x^+9
x^ + -ix'-5 x*-ix^-{-4x'^-9
2+hx x^ + 2x^-\-\2x
*• 2ft+(62^ja; - 2bx^' x\+4x
5a''+10a*x+5a^x^ ^Ox'^+x^-l
^- a^x+2a^x^+2ax^ -fx* ' 25^* + SajS^T^ITl*
.r' —x^y+x^y'^ — x^y^ •\-x^y^ —x'^y^ -\-xy^ —y^
x' -^•x'^y+x^y^4-x*y^ +x^y^-r-x-y^ -hxy^ + y^*
'a 6 ,
_ Sa''x*-2ax^-l ^^ T + T/ ^^+ ^'
4La^x^ —la^x^ — 'dax^-k-1 i a ■ o
a^{b-c) + b^{c - «.) +(-3 (a - fc)
aic(a — i)(6 — 6-)(6- - a)
Q (« + & + c)^
10. From Ex. 4 (solved above) show that
(a-&)*+(6-c)^ + (c-«i^ _ (r^6)3 + (6-c)3 + (c-«)a
(a-6)5 + (6-6-)*+(c-a)- " 5(a-'6)(6-c)(c-a) '
{x+yY-x^-y^
12. Shew that
112 FKACTIONS.
Art. XXXIV. In reduciug comT)lex fractions it is often
convenient to multiDh' both tex-ms of tiie comxJiex fractiOB by the
L. G. M. of ail the denominators involved.
Examples.
1. Simplify i(^+H)-l(l-H
Here the L. C. M. of all the denominators involved is 12;
.-. multiplying both terms of the oompiex fi-actiou by 12, and
removing brackets, we have
ex+S-8+6x _ 12a; _ 8a;
21 — 4a; — 17 " 4 — 4^ ~ l—x
„ a — h
l-\-ah
1 + ^^ —• Here multiplying both terms by 1 -^-ah, W3 get
l-{-ab
a{l^ab)-a-^h 6(a24-l)
Here multiplying both terms of the frac-
by 4 — a;, the given fraction becomes at
, and now multiplying both terms by 4, we
tion which follows a; — 1 by 4 — a;, the given fraction becomes at
1
I, 4 4
have = --'
Ax—4,-\-4: — x dx
It may be observed that when the fraction is reduced to tha
form — -^ — > we may strike out any factor common to the two
h d
denominators, and also any factor common to the two numerators ;
it is sometimes more convenient to do this than to multiply
directly by the L. C. M. of all the denominators.
REACTIONS.
a-b\
a+bJ ^
113
4. Simplify (— ^ + ^-^) ^ (^— ^^ - ^;^^pp! .
Here the numerator of the first fraction is (a4-6;' 4- (a — i)^
and tho denominator is a^ - h^ ; the numerator of second fraction
is {u^-{-b")^ — {a^ —b-}^y and the denominator is a*—b-^; the
former denominator cancels this to a--}-h''^, wiiich, oi' course, be-
comes a multiplier of the first nnmerator :
\a^+b^){{a + b)^ + (a-bf\ _ (^24^x3
•'• ^^ "^^^'^ {a^'+b'^ ) 2 _ (^3 - i ■■' ) 2 ~ 'laH-'"
Occasionally, vre at once discover a common complex factor,
strike this out, and simplify the result.
/I 1 \ > 1
here the den. = I — + "r — "v
1 1
— + — —
1
c
1 i i
0 •
/I 1 i\
= — f- -7- -f —
(^-
mon factor we have
1
1 1\ ^ .
— , and cancelling the eom-
o c I °
J^ , 2_ J_' ^"-^^ mif.fciiDlying by abc, this = ^^^
a ^ b ~ c ' bc+ca-ab
Exercise xliii.
Simplify the following :
1 - ^{1-1(1 -x)}
2. 1:
x—v
1+a x+p
114
FRACTIOWS.
«=-f &3
a-\-b •4--T-
x—1 ij — l t — 1
Sxt/z X y "^ &
yz- zx — xy 1 1 i
*• ?/ "*" z
_2 _2_ ^ ,^4 + 64 J, c4
^ «^ "^ ^^ "^ ci^ "^ a^b^c-
a 0 c
be ac at)
/a-f6 a'-\-h" \ la-b o^-b^\
\a-b ~^ a- -0^ ] "^ \a±h ~ a^^^j
1
1 1
— +
a h -j-c
2(1 -a:) (l-.'cV
„/a-M3 (a-oY /a~b\
FBACTIONS. 115
~b'
13.
15. /^-z:^' + -J-i^) ^ f J-^^^ Ir^n
16. find the value of
17. Find the ralue of ^/{l --,/(l -x)}
\l + bj Vl + 6/
18. Find value of
V(a+l>x) + -i/(a-hx) ^^^^ ^ ^ 2«g ^
l/(c-f ^a-') — |/i«-<^.c) i(l-fc-)'
Art. XXXV. When the sum of several fractions is to be
found, it is generally best, instead of reducini? at once all the
fractions to a common denominator, to take iwo (or more) of
them together, and combine the results.
Examples.
1. Find the sum of
x+y y — x x^ —y^
2x—2y iia;+% x"+y^
Here taking the first two together we hare
(x±y)^±^-y)^ = ^1±J^; now add this to _ ^^Zl"
2(a;2-2/2) x"—y" x^+y-
and we get {^^ +y^r- i^^_^J^^ ^ 4^/J ^
116
FRACTIONS.
2. Find the snm of
1+^ 4a; 837 l~x
r^ "^ 1+^3 + i+a;4 " r+^-*
Here, taking the first and the last together, we have
(l+.c-)2-(l-a;)2 _ Ax
l-a;3 "" l-a;3'
taking this result with the second fraction, we have
/ 1 1 \ 8^ .
[l + x^ + 1-x^-J - 1-x^'
now take this result with the remaining fraction and we get
''''[l-x'^ '^ 1 + x^l - 1-x^'
ar'" x"" 1 1
^- ¥:^1 - xr^ - ^-^ + ;^M^' T'"^^^S m pairs
those whose denominators are alike, we have
x'"-l a:-"-l
The work is often mr.de easier by completing the divisions repre-
sented by the fractions.
2ic+l 4:X+o
4. Find the sum of 1+ 0|773T\ — oTTo' By dividing num-
erators into denominators, this
3 _1 3_ 1
= '^-^'^+ 2^32 ~^~ 2x + 2 = 2^^ ~ 2x+-2
3a;-f3-a; + l _ x+2_
2a;2-2 ~ x^-l
X x—9 x-{-l x—8
5. t: 4- ^ — T - a '■ we have, by division
x~2 ' x—i x—1 x — Q •'
2 2 2 2
1+ ^32 +1- ^^7 -1- ^Ti -1+ ^36'^^
2 _2_ _2_ 2 2 (2a; -8) 2(2a;-8)
x-2 ^ ^^ ~ ^^ ~ a;-l "(^2)(a;-6) ~ (x~l)ix-l)
f 1 1 )
= (-ia;~16) |3.a_8a;+12 ~ a;3 -8x+lj
= {80-2C.x)-^{x^-16x^ + 8'dx^ - 152,«-f 84).
[denominator = {x^ - 8a;)2+19(x2 - Sa;) +84J.
FRACTIONS. 117
6. Find the value of
x-\-2a x+2h , 4ab
— TT + — Ki when X = -— v
z-2a ' X— 2b a + b
. -D ,. . . , ifi ib
Bydivision,l+^^3^-hl+^3:^^
/a b \
= 2 + 4 -jj^ + ~3;ji| ; but the quantity in -tbe brackets
(a+b)x — 4ah
.'. the value of the given expression is 2.
Exercise xliv.
Simplify the following :
1 X — a x^+ax+a^ x^ — a^
5 x + a x^—a'^
^ _rtM-i3 a^-Sa^b-j-Sab^-bs a{a~b)- b{a-h)
a^-ab + b'^ '^ a^ - b^ ~~;^2..f^b+b^ •
/I 1 2a s
3. ( + f- — , r| V
\a -\- x a — x a^+x'^J ^
t 1 1 2x
\a + x a—x a^+x'^j'
a b ab ah
a + b a — b ab—b^ a--\-ab'
^ Z+2x 2 - Qx 16x-a;3
5. — _ 4-
2-x 2 + X ^ a;2-4
g 1 _ 1 1
• 2 i3x-2//)/ 2 \3a;+2y/'
g x+1 a;-l l-3a; a; 1
* 2^^ ~ 2x+l ~ x(l-2x) "^ a;(4ic3 -1) + ^a--^^^
1 4 9 a;-l
2x+2 a;+2 '^' 2(x+3) (a:+2)(a;+3)"
118 FRACTIONS.
x — y x+y x^-\-y' x^—y*
+
( 1 1 1
\x -\- a X + b j
( o-\-x 4:ax 8a^x a — x
,^ 5a;-4 12s;4-2 10^+17
9 ^ lla;-8 18
a <i a2 2rt3_i3_a63
,^ 12a; + 10a , 117a + 28a;
Sx+a ^ 9a + 2a;
4x-17 8a;-30 l(^j^-^3 5a!-4
1^- "^irr ~ 2x-7 + ^2s"-"5' ~ a;-l *
a + 6+2c a + 6 + 2d
17. Find the value of -j^jz:^^ + a+6_2rf
4cc<
vlien a + h= ^'7^'
■I Q -^- _ _J_ •
(a-5)^" (a-?>r _ 1 ^ 1 _
1^- (a_5)»-l ~ («-&)" + ! (a-5)"-l "•" (a-6)'M-l•
^^^___ 1 1
1+.^ 1-^ 2 _ 2ce3
21- fZ^3 + i+:c3 ~ l-a;3 x^ + l'
FRACTIONS.
119
Art. XXXVI. The follo\ving are additional examples in
which a knowledge of factoring and oi the principle of symmetry
is of advantage.
Examples.
{x+z)-~^^ "^ {y+'xy^'-^' "^ {z + y)^-x
.2
Cancelling the common factor x — y+z in the two terms of the
first fraction, there results r- , hence hy symmetry, the
x-\-y-\-z
denominators of the other two fractions will be x-\-y+z, and the
numerators will be y-{-z—x, z-\-x — y; .'. sum of the three
numerators = a; +^+2, and the result =1.
ab be ca
2. Simphfy (,_^)(,._^) + (^rr]0(^T) + {b-c)ib-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—h)+bc{b — c)-\-ca{c — a)
we have
{a — b){b — c){c — a)
(a — b)(b — c)(a — c)
1.
~ {a — b){b — c){c — a)
2. Eeduce the following to a single fraction :
a b
{a-h){a-c)ix-a) "^ {b-a){b-c){x-b) ^ (c-a){c-b){x-cy
Here the L. C. ll.ia {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 —h{c—a){x—c)(x—a), and that of third is
— c{a — b){x~a)(x — b); and their sum is
— {a{b — c){x-b){x — c)-{-b(c-a){x — c){x — a) +
c(a — b)[x — a) (x — b)}.
This vanishes if a = 6, hence a — & is a factor, and .'. by sym-
metry b — c and o — a are also factors. Zl^ow the product of these
I'AU FRACTIONS.
is of the tliird de^-ee, while the whole expression rises only to
the fourth, hence «-. cannot he involved. The other factor must
therefore be of the form iix+n, in which m is a number.
To determine n put x = 0, and the expression becomes
aficja — 6 + 6 — c-j-c — a} =0; .'. w = 0, orthe other factor is vix.
' To determine w put « = 0. 6=1, c = — 1, and m will be found to
be 1. The numerator is .-. x{a - b)(b — c)[c — a), and the result is
X
{x — a)[x — b){x — c)
o o- Te a + h 6+c c-{-a
3. Simpliiv J. , _^ _^
[b-c){c-a) ^ {c-a){a~b) ^ (a-h){b-c)
L. C. M. of denominators is {a — b){b — c)(c—a) ;
.-. iirst numerator is a- —b-, and by symmetry
second " b'-<~,'And
third " c2-a3 ;
the sum of these = 0, which is the required result.
4. Eeduce
^-y ~^ y-z "^ z-x "*" {x-y){y-z){z-x)
Here the numerator becomes
2{y-z){z-x)+2{x-y){z-x) + 2{x-y){y-z) +
{x — y)^-i- {y — 2) - + (z — a;) ^ , which is evidently
{{■^^-y)+iy-^) + {'^-^)}'=0.
ff(3 + 263|3 (2a3+63)3
ff/3 + 263|3 (2a3+6^)
Observe that the denominators become the same by clianging
the sign between the fractions, and that the expression is sym-
metrical with respect to a and 6, The numerator of the first
fraction is a^~ + 6^963 -+-12fl^6^+8a36^, and by symmetry that
of the other is —b'^"—Gb^a^ — l'2b^a^—8b^ci.^. Their sum is .:
= (a«_6G)f^(G.|_iC+e,^3/,3_8a.3^,3| = (rtG_iG)(a3_^3^2
= («3_}./;3)^rt3 _i3)3^ and since the denominator of the given
expression is (a^ —6^)3 .-. the result is a^+i'.
FRACTTOVR. 121
Exercise xlv.
Simplify the following :
a-\-b b-\-c c-\-a
1 1 1
^- (a-i)(«-c) "^ {b-a){b-c) ''" (c-aj(r-/7)'
a — 6 5— >• c — a (a — h)(b — c)(c—o)
^- a+b "^ /7+"c ■*■ c+a "*" (a+/>)(6+o)(c + «)'
flS &2 r»
(a+b){a-}-c){x + a)'^{a-{-b){b-c){x+h) {a+c)(h - c){x+c)
a;3 y2 23
(x-y){x-z) ^ {y-x){y - z) {z-x}{z-y)
flZ J)3 (.3
^- (a - b)(a - c) "^ (b - a)[h - c) + (c - a){c - h)
" (W(R "(RJT^nFfe^)-
/a;5 - 2?/3\3 /2a;3-7/3\3
11. ___! + ^ +
(6+c-2rt)(c + a-26) ^ (c + a- 26) («+/»- 2-) ^
1
(ffl+6-2c)(i+c-2a)
,„ 62 -c2 c2-a2 a2-62
12. -— -^ + 7-— -TIT + /--^
(6 + c)3 "^ (c+a)2 ^ (a+6)2
13 «^ , ^ ,
(a-6)(a— c)(a;-fl) (6-«;(6- cj(a;- /A
C2
{c — a\{c — h)[x — c)
122
RATIOS.
14. %+g) J y{z+x) z{x+y)
(x~y){z-x) ' (y-z){x-y) "^ {z-x){y-z)'
15.
If5.
{a-i-b)^-\-{b-c)^+{a+cy-' , 2 2
0!
(a + 6)(^ — c)(a + c) a+c Z>-c ' a+6
1 1 1 .
+ „/, -w,. „x +
ici^a;-a)(a;-6) ^ a{^~a){x-a) ^ b{b-a){x—b)
Sectiox III. — Eatios.
Art. XXXVII. If -7- = -^ .-. ad = bc. Now,
-,. .• , h d
diviamg ad — he by ca we have — = —.... (1).
»• ad = bchjcd " — = —.... (2).
C d
d
ad = hchjah " -^ = 4" • • • • (S)-
Also :^^:^ = each of the given fractions . . . (4).
mh[-Y)+nd[-jj {mb+nd)^
(5).
ma-j- Jic ^.
mb-\-nd mb -j-nd ~ "' "mb'+ml"' ~ b d
A very im'portant case of this is w = 1, m = + 1, hence
a c a-{-c a — c
b ~ d ~ b-{-d ~ b — d
a—b c—d
Also —.-7 = —,—j (6).
For by (2) and (5)
a b a—b a-^b a — h c—d
c ~ d ~ c — d ~ c-\-d " a+b ~ c-\-d
- --1 4--1
Or tl • ^~ = -' = '^' - ^~^
' a-i-b a c ~ c+d
RATfOS. 123
Generally, toprove tbat if — = — , any fraction whose nu-
b d
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 .
h b ^ d'
and reduce the result.
By (2) the fractions may be formed of a and c, and b and d.
if — = — ■ = — , ■ LZ_ _ — or — or — (7)
b d f vib+nd+pf b d j
'"'{t) + "ii) + At)
ma-\-nc-{-pe
mb+nd+pf ~ mh-^-nd-^jij
a
{mb+nd->rpf)Y
a
vib + nd-^pf b
If^ = ^ and ^ = Z.
b d n q
ma+pc pa-T-mc ma pa
, ~ . = ^ ~ , = — or ^ or &c .... (8)
nb + gd qb±_nd no qb '
T-, ma pc via + pc , ,„,
For — = ^— = — -^- by (o)
nb qd nh + qd
pa mc pa^inc
qb ml qb±_nd
But ^ _ -P^^ hence the equalitv stated in (8).
nb qb
If ^ = A = ^ and !!L = A = ^,
b d f n q s
ma + pc + re pa-'-rc + tne via
— ,~ ~ r = , ■ ■ , — 7 = &c., = — = &c. . (9),
nb±qd + sf qb±sd + nf no ^ '
If an upper sign be taken in a numerator, the corresiDonding
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 t.Avios.
Examples,
a c oa — 4:b
1. If -r = — r, show that ,^^ — ^-y =
-Ji _ 4
Tlie given fraction = -»— ™— = »-»,■..-»«=»„ = _ — — — -
7-^ + 5 - ■
a c 2^(3 ^3^2^ 2c3+3c3(i
2- I^T = T'^^^*^^^ 3^il^463 = 3c3d^4^3'
Dividing the given fraction by 6^ we have
a G
and this becomes, on substituting for— its eqaal-^.
a 3 rt3
2p- + 8^
a2
c3 C
2 — ■\- 3
rf3 -r ^,^2 ^ 2c3_+3c3rf
8:7? - ^
a3 + i3 /«» \
3. If 3a = 26, find the vahie of ^^^3^. This = (p- + ll h-
(^ _ — [by dividing both numerator and denominator by
\b'^ b j
a 2
6»3] . But from the given relation — = ^r we have, by substi-
a
tuting for -r-'
c ^ a3 4-63 6 /«+ft\*
4. If -7- = -T- Prove that -,-",",v x -7 = — ,~i •
a b a-\-b
We have - = -j = -_^' Also
,3+^3 _ ^/^ \ /^ ^J ^ ^^.,,^d this multiplied
i3 - d'^ \63 ^ / U3 ^ j d^
C
KATI03.
12i
o- ii ^3 — :,^.. jj; ,— = --. , shew that a; = — -.
Multiplying botli terms of second fraction by x, it becoiaea
x^-{-ax^ — bx
^z~^^y.2Z£i)x' ^'^'^ ^^^^ °^ ^^ given fractionti =
diffei'ence of numerators
diflfereuce of
denomiuaiors ;
=
c
c
= 1
-•. X'-x-ax — h—x--
- ax-{-b
,
or 'lax =
2b
■ x
h
a
6. If
a
c
* ac + ce-^-ea
= -r, shew that , , , ,,. ,-.; =
/ bd + dJ-{-fb
A3+^S
For
ac
Td'
ce
ea ac-\-ce-\-ea
JU bd + df+fb'
By (7) making »i;
= 7i =
A.lso
a2
C3
«2 a.2-\-c^-\-e-'
~p~b^'-fd^+f--
By (7).
But
ae
Id
~6^
hence the required
equality.
Tiie problem is a particular case of (9), with all the signs +
and a for m, b for n, c for p^ &e.
(If the fractions yiven 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 c-fa
S{a — b) ~ 4(6 — c) "" 5(c — a)'
prove that 32a +35^4- 27c = 0.
Assume each of the riven frs.ccion3 = a;, so that a-^b = 3(a — bjx,
b+c = i{b — c)x, c-jra = iJ{c — a)£,
126 RATIOS.
or — ^— -4- — !— _i_ — ! — = a;(a — 6-f /;— c-fr — a) = 0.
o 4 5
/. adding these fractions we have 82a + 356+ 27c = 0.
This example might also be worked as a particular case of (7),
thus
a-\-h h-\-c c-\-a
^{a'-b) ^ 4(6 -c) ^ 5(7^
20(a4-6) + 15(&+c) + 12(c+rt) _ 32a+356+27c
"" 60(a- h) + 60(6 - c) + 60(7^ a) ~ 0 '"'
.-. 32a + 356+27c = 0 x -^^- = o
3(« — 6)
8. If— + '— =z — J— — — -I- ^l, prove that
63 ^ ,/2 fZ i 6 f^ ./' j
/ g + c+g \-
U + f/+// " 63+tZ3+y2
Transposing terms, &c., we have
t3
2rtC c3 ^3
2c« 6-3
-^ + ^ + y5 -
"df "^ rf^
= 0,
/
that is, the sum of two essentially positive quantities = 0 ;
.'. each of them must = 0 ; hence we have
a c ^ e c
- - -^ = 0, and -^ - - = 0;
O. C <; , . «3 f^2_|_c2^g3
Also
b d / 63 bz+d^-^p
a-\-c-\-e _ . ^^ /^+c+^\-.
/a + c+e\2 a^+c^+eP
XbA-d^'fl
b^d^fl 62+t/2-|./a
RATIOS. 127
Exercise xlvi.
_, ,„ <7 c a^ —ab+b^ c^ — cd + d"
1. If _ = -. prove ^,_^,, = ^Sri^-
n Tt a c «2_o3 /«-«v^ /«+c\^
2. H _ = _, prove __- = (^-_-^j = (— ).
y. Given the same, shew that each of these fractions
4. If 2x = dy, write down the value of
•2x^-x^y+y^ ^^^ ^^ x'^-3x^y + 2ij^
'x^>/+xy'' + 2>j^' {^x--y-'^)^
If — = — = — ^, shew that _- =
ma — 7}e—pe
b d f b mh — nd — yf
6. From the same relations prove that — = ( -, \
b^ \b-md—np
7. If '±^ = Af^-±^+^%then:«3^(i_«).(i+«).
1 — x a\l—x-\-x'^!
°- ^ ,, , -// - . = a, prove that « = .-, — ^•
^ Ti? wa;4-a+i mx—c—d b — c
9, If ! — — — — , prove;*; —
nx+a-l-c nx — h — d
10 If ^~^ '^""'^ c — a
ay-\-hx iz + cx cy + az ax + by+cz
then each of these fractions = , a + 64-c not being zero.
x-i-y+x
11. -^1'+^- = i±i^'= -^^±^,then8a + 96 + 56- = 0.
a-b 2(6 -c) B{c.-a}
\/o.-\- \/{a-x) 1 , , a — x 11 — a
12. If ^ .; r = — ' shew that = L-ri
-j/a — 1/ (a— aj) a a \l+a,
a;2— wz y^—ocz ^ , , ,
13. If — T r = -/5 T' and a;, y, z be lanequaJ, shew
x[\-yz) y^l-xz)
that each of these fractions is equal to x-^y-\-z.
«- h- c^
^ 18. If -^ = —, = , shew that rt2a.+62»+c2a
x/ —yz y- —zx z^ — xy ''
19. If -7^r — = ,-^' — = ~ — -,, thenwHl {a-h)x-\-
a-\-h — c h-\-c — a r+a — b ^ ' '
128 RATIOS.
1A Tr^'+-'^ + l y^+%1 + 1
14. it -—^ = -- — (^ r^-' shew that each of these
X' — Zx + o y^ — ly-f-o
fractions ={xy—l)-T (xy — S).
2ryx^-16 3(^3 _4) 4 3
15- If TOxTS = -2^34-' «bewthat^_^. = y
4&C iZ+SS y-h2c
16. If .V = ^^^ shew that •-_^^ + ,;^-^ =2.
1 /.S+Z^SX 1 //;2+,2N 1 /,2+«,2N
25a2 +2762 + 22^2=0.
^ 18. If -5 = -
x^ —yz y
= («2+62+c2)(a: + v/ + 2)
19. If -y-"^— =
ib-~c)y + [c-a)z = 0.
a C p. ,r,2 4_^2_!_^2\2 „4 + ,,.44.^4
^^- If T = T=^7 ^'^^^ (/^I'+TmT^' ^ /74+J4-474-
•,„ J>x4-oi/ cy + lz riz-\-cx
21. If r^' = ^^ = , shew that
(t —h h —c c — a
[a-\-h-\-c){x-{-y-\-z) = ax-r-hy -^cz.
^^ ^„ x^ — 5x^a — a^ + 6x'i" x-~a ■, r ,-,
22. It — 9—r-i — ; o i^^^~ = ~~"' shew that each of these
x^ +X''a-\-xa^ +(c^ x + n
expressions = ■* ,
1 in-h\ 1 (h-c\ 1 ir-a\
23. If TT -TT = — ;T^ = ^-uT/' ^^tl «' ''' ^ ^^
b \a-X-b> o\o+r/ i(jV''+f'-'
different, shew that 1 f)b + 1 1 6 + 1 5 = 0 .
^■ + 7/2.2 1 — 7/2
- ,' ) = ^~ ,- pioTethata-2+v2+22+2aTy3 = l.
2^5. If Jl— = JL- = ^1_, shewthata+?;+c = 0.
x — y ?/ - z 2 — «
26. If _ = — , prove that — - - -/^--r^-SV,-^-
b d a — h y V'c) -y{bd)
KATiOS. 129
27. It" — = — = — , tbeu each is equivalent to
b d f ^
- - ^ , hence siiow that
lb-{-iuU+n/
a b e -.
2z-\-^—y ~ ^x^%j-z ~ %j-r'lz — J
-c _ y 2
2a + 2i-c ~ '"Ib + 'Ac-a ~ 2c+2a-6*
28. L — = — , prove that _ ^ ( ^__ .
29. li' --^ = — L = -,-^, prove that
±{y-z, + -^{z-x) + I-{x-y)^Q.
30. If ^ "^ ^^ = — -^' -^ = ^ "" . . then vvill
±{i-x) -r -^(/n-^) + — («-z) = 0.
/x tiiy nz
31. Lz - ^ ^ ^ , and ?/ = ^-^ — . — /, shew that
// ■^
\/iaz- —ai I
a; = * -^
z
32 If ^'"1: = ^'"— = ''"-"^ = 1, shew that
a3 62 c2
33. If I:L = ^ = -. and 4 = ?^ . ^_ 1.
X y s w b^ c'
prove tnat —^ + -^ + -^ =3 . . ., v o •
a c
S4. Tf — = -r = -7- = &c., then
. a'^-c'''
_ a"c"^''-(a"-o"4-g")'
130 COMPLETE SQUARES.
a, a. fl, a„
65. If -r^ = T^ = ~ = . . . = 7^' then
»! Oo ^3 0^
^ + B+C .4 , i? , C
36. If 7 = f- IT + — '
abc a 0 c
&-n(i{A + B+C){a-\-h + c) = Aa + Bb+Cc,
A B ^ G _
then will ^;:^ + .-^^ + y:^ - 0-
and also + + ^
11 1
«+— b+-r c+—
a-h i/k zl x^ y^ a-^
37. If -5^ = 7T = -5-' and -^ = ^^ = -^ =1»
thenwill^ly + y + -j =j^ + ^ + 1?
Section IV. — Complete Squares, &c.
1. What quantity must be added to a; 2 ^2jx to make it a com-
plete square ?
Let r be the quantity.
Then x^+jjx+r = complete square - {x+ ^/7~')^
= x^ +2x yr-{-r.
Equating coefficients we have
_p2 / p
Or thus: Since {a-}-x)^ = a- -\- 2ax + x'^^ ; we observe, (See
Art. XII), thsit four times the product of the extrsmes is equal to the
squai'e of the mean,
4:X^r=j^"^' '
r = {-t;-] > as before.
COMPLETE SQUARES. 131
Or we may extract the square root and equate the remainder
to zero . thus
P
p
px-\-L-
4
2T'
Now, if the expression be a complete square, this remainder
must vanish ; hence we have
^ = T = (-2
2. Find the relation connecting a, h, c, if ax^+bx+c is a com-
plete square.
Assume ax- -{-bx-TC = {\/ n.x-{- \/ c )" =ax- +2-[/(ac).x + c.
Now, since this holds for all values of x, we have 2 ^ac = b, or
b- = iac, the relation required.
3. Determine the relation amongst a, b, c, in order that
a^x^+bx+bc-{-l>' may be a perfect square.
As in Ex. 1, we have Aa^x-{bc'\-h^) = h^x'^ ;
.-. i- - -f- = 1.
4u2 I
Or thus :
Assume a^x^ +hx-\-bc + b'^ = {rix+\/'oc + 1"^)^
= a'^x^+lai/'h^b^- + Z*c+63 .
Equating coefficients, we have b = 2aybc-^h^ ;
.". — _ — = 1, as before.
4a- b
The same result may also be obtained by extracting the square
root and equating the remainder to zero.
132 rOATPT.T'Tr, SOrAVTi;s.
4. Silow tlmt U x'^-ri^'x^-^hx" -^rx-~d be % 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^ +cx-\-d in the
usual manner, we have for the final remainder
, -x + d- —lb -
2 \ 4 / ) 4\
Now, if the expression be a complete square, this remainder
must vanish ; and, that it may vanish for general values of x, we
must have
a I a^\
*-y(^-t,'=^ W'
1 ; fl2
^^-Tr-T-'=^ (2);
Elimmating h - _, ^re have c- -«"(■/ = 0 . . . (3).
The coefficients must satisfy the equations (1) and (2), and
therefore either of these equations, together with the equation (3),
which results from them.
The same result may be obtained by assuming
a-* +ax^ + hx^ +cx^d = {x^ -f ^ax+ yd) ^
= x^+ax^ + 2x^yd
-f \a-x'^ -r nx s/d -\- d.
Equating coefficients, we have 2yd+\(i.^=b . . .' (1)
and ayd=c . . . (2).
From (2) we have c^ — a"d = Q, as before.
5. What must be the value of m. and n if
4x* — 12x'^ +25a;2 _4,»a:+8?r is a perfect square ?
Assume the expression = {(2.^2 _ 3a;+ n/(8».) } ^
= 4a;4-12.'c3-H4«2y(8«)+9a;--Cj-N/(8?0-!-Sn.
Equating coefficients, we have <6\/{Qn) = Am .... (1),
and4:-/(8?i) + 9 = 2.5 .... (2);
.-. %=2,
in = 6.
COMPLTSTE SQU.4IIES. 133
Or thus : ExtrR,ctIng the gqiiare root in the nrJinary ivay, the
remainder is found to be (— 4???-}- 24)3-4-8^—16 ; .■. we must
have 4m -I- 24 = 0, or m = 6,
and 8?? -16 = 0, or n—2.
6. If nr^-^bx'+ ex -'rd be a complete cube, shew that ffc^ _^js^
and b- =3ac.
Assume ax^-{-bx^+cx + d=(x+'J}i)- '
Equating coefficients,
b = Sa^d^ (1)
c = 3a^(i' (2);
dividing (l)by(2),l = ~;
c a.:
ac^=dh^.
Also, h^=9a*d^ (3);
dividing(3)by (2), ^ = 3^;
c
7. Find the relations subsisting between a, h, c, d, e, when
ax^ -\-hx^ ■\-rx'^ -^dxAr^ is a complete /owri/t power.
Assume ax^ -^-hx^ -\-cx'^ -^dx-\-e = {(i^x-\-e^)^
= ffx^ +4aVa;3 + 6aMa;2 + 4a¥a;-f-e.
Equating coofiacients, we have
h = 4rt^ e^,
c = Ga^e^,
d = 4caie^ ;
whence M = 16fl<'. n\
bc = 2iaie^ = QaAaiei = Qad. .... (2).
cd = 24:a^ei = 6eAa'e^ = &)e. .... (3).
8. Shew that x^+px^ + qx^+i-x+s can be so resolved into two
rational quadratic factors if s be a perfect square, negative, u.nd
equal to
134 COMPLETE SQUARES.
Since — s is a perfect square, let it be n^.
Assume x^ +px^ -^Q.^^ -\-'>'-''- "~ ■" ^
= {x^ + vix-{-n)(x'^ -^-m'x — n)
= re* + {m-\-m ')x^ -1- mm 'x^ — n (w — m ')x — n^ .
Equating coefficients, we have
m-\-7n' = p
mju' — q
r
m—m'= —
n
ni^ +2mm' ■\-m''^ =p^
4,mm' = 4.q ;
:. {in-
■m')^:
=p'
-4j
j-2
= n"
= -s.
'■ ^
- 4j0 2
Exercise xlvii.
1. What is the condition that {a — x){b — x) — c^, may he a per-
fect square.
2. Find the value of n vrhich will make 2a;- +8x4-", ^ perfect
square.
3. Find a vahie of x which will make x^ -f- Qx^ + 1 Ix" + 3x + 31,
a perfect square.
4. Extract the square root of
. 5. Find the values of m and ?i which will make
iz^—4:X^+5x^ —mx+n, a perfect square.
6. "What must be added to x* — n/(4x^ -lCx-+16)-4x3 in
order to make it a complete square ?
7. The expression x^ + x^ — 16x2 — 4x4-48, is resolvable into
two factors of tho form x-'+mx + G, and x^+nx-[-8; determine
the factors.
ex
' 8. Find the value^ of c which v/ill make 4x4 — cx^ + 5x^ + -^ + 1,
a complete square.
COMPLETE SQUARES, 135
9. Oulaiii tbe square root of
4{(a3 - b-^)cd-i-ab{c'i-d^)}2 -f {{a^ - h2)(c^-d^) - iahcd] a.
10. If {a-b):c- + {a + b)^x+{a^ - b"){a+b), is a complete
square, then a = Sb, or b = Sa.
11. Find the simplest quantity which, subtracted from
a^x^+iabx-r-Lacx+obc-l-b^c^, will give for remainder an exact
square.
12. a;* — 4rf3—a;2-f 16a; — 12 is resolvable into quadratic factors
of the form x^ +mx+p, and x- +77x+q^: find them.
13. Find the values of m which wili make x~ +max + a'^
afactor oi x'^ — ax^-j-a^x^ —a^x-{-a*^.
14. Shew that if x^ + ax^+bx'^+cx+d be a perfect square, the
coefficients satisfy the relations
8c =a(46-a2), and
64:d= (46-a2)2.
15. Investigate the relations between the coefficients in order
that ax- -^bij^ ^cz"^ +dxy-{-eyz-\-fxz may be a complete square.
16. If x^+ax^+bx-rc is exactly divisible by (x+d)-, shew that
- i(6 -d2)=-^=d{a-2d)
17. Determine the relations among a, b, c, d, when
az^ — ix^ + cx—d, is a complete cube.
18. The polynome ax^ -\-Zbx^ +ucx-\-d is exactly divisible
by (a-a;)3 ; shew that {ad-hc)^ = i.{ac-h^){hd-c^).
19. Find the relation between p and q, when 0:^+^3:2+5', is
exactly divisible by {x — a)^.
20. If x^+7iax + a^ is a factor of x^-\- ax^-{-a^x^ ^a^x^a'^,
shew that n^—n— 1 = 0.
21. li x^+ax^ + bx^ +cx + d,he the product of two complete
squares, shew that.
(46-a2)-'=64c/, (46-«3}a = Sc, a>/{Ba^ -2b) = Sh.
136 RELATION IN INVOLUTION,
22. Prove that »* -rpx^+q^^-^rx+s is a perfect square, if
p'^s—T, and q = —- -^ 2^/6-.
23. If a.c3+36^'-+3c.i; + c/ contain ax--\-1hx+c as a factor, the
former will be a complete cube, and the latter a complete square,
24. li m^x^ +px+ijq+q^ be a perfect square, fiudj:> in terms
of m, q, and x.
25. Find the relation between p and q in order that
x^+px^-{-qx-\-r may contain (a;-i-2)3 as a factor.
2G. If x'^+^j.c2_|_^^^_j_;. \^Q algebraically divisible by
'dx' + 'lijx-]rq, shew that the quotient is u; + —•
Eelatxon in Involuiion.
Art XXXVIII. If aa' = ^*6' = cc', prove that
1. {a + h'){h-\-c'){G+a') = {a'-[-b){h'+c){c'-\-a)
{a + b')xa' = aa'+b'a' = bb'+b'a'={b + a')xb'
{b+c')xb' = bb'+c'b' = cc'+c'b' = {c + b')xc'
{c-i-a')xc' = cc' + a'c' = aa^-ira'o' — [a+c') xa'
.-. {a+b'){h-\-c'){c-\-a')xa'b'c' =
{a'+b){b'+c){c'+a)xb'c'a'
:. {a+b'){h + c'){c + a') = {a'+b){b'+c){c'-\-a).
2. (^a + b){a+hl){a' -c){a' -c') = {a' +b){a' + b'){a- c){a- c").
{a + b)y^a' = aa' + a'b = hb' + a'b = (b'-i-a')xb
{a+b')xa' = aa' + a'b' = bb'+a'b' = {b+a')xb;
(a' — c) X a = aa' — ac = cc' —ac = (c' — a) xc
(a' — c') xa = aa' — ac' = cc' — ac' = {c~a) xc'
:. (a+b){a + b'){a' -c){a' - c') X (aa')^ =
{b'-i-a'){b+a'){c'-a){c-a)xbb'.cc'
B\xt bb'.cc'=(aa')^,
and (c' — a)(G-a) = (a — c)(a — c')
:. la-{-b){a-\- b'){u' -c){a' -c') = {u'+b){a'-^b'){a-ti){a -cf).
RELATION IN INVOLUTION. 137
Exercise xlviii.
If aa' = bb' = cc' prove that
1. {a-b')[b-c){c'-a') = {b-a'){a~c){c'-b').
2. {b-c'){c-a){a' - b') = (c -b')[b - a)[a'-c').
3. {c-a'){a-b){h' -c') = {a-c'){c-b){b'-a').
• 4. {a-b'){b-c'){c-a') = {a-c'){b-a'){c-b').
(a-b){a-b') _ (a—c){a-c')
^' (a'-fc)(a'-i') ~ (a'-c){a'-c'y
{b—c){b-c') _ {b-a){b-a')
^- {b'-c}{b'-c') ~ [b^^a^b'-a')'
{c-a){c-a') _ (c-b){c—b')
^* (e'-a)(c-a') ~ \c'-b){c'-b'T
8. Shew that the seven preceding relations may he derived
trom the single relation
\a-\-a'){bb' — cc')-^{v -{■b'){cc' — aa ') + {c-T<:'){c!,a'—bu') = 0.
138
SniPLE EQUATIONS.
CHAPTER V.
Simple Equations of one Unknown QtJANTixy.
Art. XXXIX. Preliminary Equations. Although the
following exercise belongs in theory to this chapter, in practice
the numerical examples should immediately follow Exercise I.,
and the hteral examples Exercise III. Like those exercises, this
one is merely a specimen of what the teacher should give till his
pupils have thoroughly mastered this prehminary work. But
few numerical examples are given, it being left to the teacher to
supply these.
Exercise xlix.
What values must x liave that the following equations may be
true?
1. a;-5 = 0. a;-3i = 0. x-a = 0. a:+3 = 0.
2. a;4-4i = 0. x + a=0. a: + 3 = 5. a;-4 = 6.
3. x — a~h. x-\-a = c. x-h= —c. 6 — a;=3.
4. 8-a;=10. 5 + :«=ll. 9+x' = 4. l-x=-5.
6. 84-iC-^-6. fl-a; = 36. 2rt = a;-f-36. Qa^oh-x.
6. 2a;-6 = 8. 3a;+8 = 20. ax = a^. mx=bm.
7. Bx = c. ax = o. ax = 0. [a-\-b)x = b-\-a.
8. {a — b)x = b-a. {a + bx) = {a+b)^. {a—b)x = a^-b^.
9. {a + b)x = b^-a^. {a^-ab + b^)x^a^+b^.
10. {a^-b^)x = a-b. {a^~b^)x = a + b. {a''- +b^)x=l.
11. {a-\-x — b) = {a-\-b). x — a-{-b = b~x-ra.
12. 'la—x — x — 2h. ax-\-bx = c. ax — b = cx.
13. ax — b = hx — c. ax — ab = ac.
14. ax — a^=bx — b^. ax — a^ = bx — b^.
SIMPLE EQUATIONS. 139
15. ax — a^=b^ — bx; ax-\-b-\-c = a + bx-\-cx.
16. a — bx — c = b — «x-{-cx; a-{-hx+cx" =ax — b^cx^.
17. bx — cx-+e = ex — b-cx"; 'dx = t; 4.x = ^.
18. 10a; = ^ — 1; ax= — ; ax= — .
c b
, „ . a ^7 ac^ ab"
19. abx = — -f — ; bcx= — -i_ — .
ha be
20. ia;=5; |a: = 8; -50; = 2; ■Sx=-Q(j.
21. •02x = 20; -32;= -2; -ix^-Q,
. . X ax
22. •18x=l-8; — = 6; -r = c
23.
T
0 a; ax .
— , =c; - = L
c a-\-b a~{-b
24.
a— 0
a a — b a-r-b
b ' a-hb ~'b — a'
25.
a
0— 't
a b -a a — h
a — 0 a + b b+a
26.
a+b
— a;
a-i-c
_ a-c .1 1.2
a + b' X -2, ' X
=
3
5
27.
1
X
1 . 1 a ^ a
ah ^^x h X
b ,
c
7
a;
= 1 + A
3 4
28.
1-
4 _ 33 1 . ^ j_
5a; 5a; 'd x
b
c
= 0.
29.
5
a;-7
=6 ' ; ' -7+
a;-7 3a;-4 4
9
-3a;'
30. (a;-4)-(a;-i-5) + a; = 3; 2a;- (a; -5) -(4 -3a;) = 5.
31. 2(3-a;) + 3(a;-3)=0; 2(3a;- 4)-3(3-4a;) + 9(2-x) = 10.
32. a(l-2a;)-(2a;-a) = l ; a;- 5(a-a;) = 6a;- 5«.
33. ma;(3a-4) + 3ma;-3rt + l = 0.
84. a(bx—c)+b[cx — a)+c(ax-b)=0.
85. a{ax — b) i-h(^cx — c)+c{cx — a) = 0.
^ SnTPLE FQUATION'S.
86. a{hx - a)-\-h[rx - h)-^r(ax -c)—-0.
38. 3(3{3(3a;-2)-2}-2)-2 = l.
39. 9(7{5(3a;-2)-4}-G)-8 = l.
41- i{ia{4(a^ + 2) + 4}4-6) + 8} = l.
43. t(l{|(3^:^-li)-H}-li)-H = 0.
44. ^{?T(^{|(fa;+4) + 8} + 12) + 20}-f 32 = 58.
45. l{|(t{^-(a:+7)-3}4-6)-l}=4.
46. r{g(;j{w(rHa;-a)-&}-c)-rf}-^ = 0.
47. (l + 6x)3 + (2 + 8a:)2 = (l + 10a:)=.
48. 9(2a:-7)2 +(4a:- 27)3 = 13(4x+ l^)(x+G).
49. (3-4a:)24.(4-4a;)2=2(5+4a;)2.
60. {9-4x)(9-5x)-{.4.{5-x){5-4x) = SG{2-x)9.
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. Thus, in order that (x-a)(x-b) may =0,
either a;-a must = 0, or x-b must =0, and it is sufficient that
one of them should do go.
Hence the single equation (x-o)(x-b)=0 is really equivalent
to the two disjunctive equations, either ar_a = 0 or a; - 6 = 0, for
either of these wHl fulfil the condition of the given equation, 'and
that is all that is required.
Similarly, were it required to find what -alues of a? would make
the product {x-a){x-h){x-c) vanish, they would be given by
a;-« = 0, ora;-7; = 0, or a'-c = 0 .-. x = aovborc.
Hence the single equation
{x—a)(x— b)(x —c) = 0
is equivalent to the three disjunctive equations
x — a = 0, or x — b = 0, or x—c = 0.
simple equations. 141
Examples.
1. Solve x'- a; -20 = 0.
The expression = (x-5){x+4:), -which will vanish if either of its
factors (Joes, that is, if a: - .5 = 0, or a-+4 = 0,
.'. x = 5, or a*— — ■^
2. SoUex*—.r^-x^+x = 0. •
This gives x'^{x-l) — x{x-V) = x[x—\\x^-V)
= x(a-— l)(a; + l)(x— 1), which vanishes for
a: = 0, a" = l, x-= —1.
3. Solve a;3 4-n2a;2 _crj; _ aS = 0.
This = a-(rc2 - a) -\-a'^{x'^ — a)
= {x+a^){x^ — a), which vanishes fa?
a;4-a2_o, anda;2— a = 0, or
x= — a", and X- = a.
4. Solve x^-{a-h)+a''-ih-x) + h^{x-o) = 0.
The factors of the expression are (Ex. 2, page 79)
« —a, x—h, a — h; hence the expression vanishes if
a:— (7 = 0, or a; — 6 = 0.
5. Solve 221ar2-oa;- 6 = 0.
Here we have the factors ITx — 3 and 13a:+2 ;
.•. the equation is satisfied by ITx — 3 = 0, or a; = fy,
and 13x+2 = 0, or a; = - ^V
6. Bolve 2a;4+2a:^ + Gx;-18 = 0.
In this case we have 2{x4 — 9) + 2x{x^ + 3)
= 2(a;2+3){x3-3+a:}, which vanishes for
x^+Z = 0, ora;2+.,._3 = 0.
7. Solve (x -a)^+ (o. - b) •> -;- (b -x)- = 0.
The expression is equal to B(x — a)[a — b)(b—x),
and therefore vanishes for a; — a = 0, or x = a;
and for x — b = 0, or z = b.
142 Simple equations.
Exercise 1.
1. If an equation in x has the factors 2a: — 4 and 2:c — 6, find
the corresponding vahies of x.
2. If an equation gives the factors 2a;- 1 and 3aj — 1, vrhat are
the corresponding values of a; ?
3. If an equation gives the factors dx^ —12 and Ax — 5, find the
corresponding values of x.
Find the values of x for which the following expressions will
vanish ;
4. x''-2x+l] Ax^-12x+d.
6. 9x^-4; x- — {a-\-h)2 ; x^-2ax + a^.
Q. x^-9x+20; 4a;2-18a; + 20.
7. x^+x-6: x^-x-U; 9a;3-9a;-28.
'8. 6a;3-J2a;+G; 6a;2-13a;+6; 6a;2-20x+6.
9. 6x^-5x-Q; 6a;3-37a;+6; Gx^-\-x-l2.
10. A certain equation of the fourth degree gives the factors
a;3 — a; — 2, and 4:X^ — 2a;— 2, find all the values of x.
Find values of x in the following cases :
11. x^-2bx^-Sb^x=-0.
12. x'^-ax^+a'^x-a^^O.
13. a;3-2.?;+l = 0; a;3-8a;+2-0.
14. a;*-2aa;3+2a3^— «4=0-
15. x^ + {b + c)x^ -bcx-b^c-bc" =0.
x — ax—b {a — bY _ x^ —a^
■ x — b X- a {x—a){x-b) {x—a){x — h)
17. x^-bx''-a^x-an = 0.
18. 3a;3-{-4flte2 -6a3/;2^_ 4(13^3 =,0.
19. x^{a-b) + a^{b-x)^-b^{x-a) = 0.,
(x-b){x-c) {x-c)(x-a) ^ ^
^"' {a-b){a-c} "^ {b-c){b-a)
■x-2a\ 3 /2a;- a\ 3
/a;-2a\
21. x\;:rr-7.i +«(ir+^; =-^'
SIMPLE EQUATIONS.
143
cfi feo- ax _ 1
^^' {l>-a){x-a) "^ (^^{a-b) "^ (^a-b){b-x) ~ a^b
24. Form the polynome which \\ill A'anish for x equal 5, or
-6, or 7.
25. Form the polynome which will vanish for x = a, or 4a, or
3(1, or — 4fl.
26. Form the equation whose roots are 0, 1, -2, and 4.
27. Form the equation whose roots are 1 + s/-, 1— \/2, 1 - ^^'3,
and 1 + y 3.
Art. XLI. In solving fractional equations, the principlea
illustrated in the section on fractions may frequently he appHe«!
with advantage, as in the following cases.
When an equation involves several fractions, we may take two
or more of them together.
EXAJIPLES.
1 o 1 8a; -t- 5 7a; — 3 4a;+6
1. Solve 4- = — -^ —
14 ^ 6a;+2 7
Here, instead of multiplying through by the L. C. M. of the
denominators, we combine the first fraction with the last, getting
at once
7a;- 3 7 1
6xT2 = 14 " T •■• 7x-8 = 3a;+l, anda; = l.
2 2a>f8| _ 13a;- 2 ^ _ Z^ _ =^+16
~9~~ 17a; - 32 ■*■ 3 ~ 12 "36
In this case, taking together all the fractions having only
numerical denominators, we get
8a;+34+12a;-21a;4-a;+16 _ 13a; -2
36 17a; - 32 '
25 13a;- 2
or
lb 17a; -32'
.-. 425a;— 800 = 234a; -36, hence x = 4.
144 SIMPLE EliUATlONS.
It is often advantageous to complete the divisions represented
by the fractions.
4x-17 _ 3|_-22u; _ A / i^
9 33 y; \ C>4
Here, completing the divisions, we have
4x 17 1 2x 6 rr,
lOa; X () 6
—TT— -2 = a;+-7- — — .-. — 2 = — — , or.e = 3.
V d X X
ax-\-h cx-\-d
4. + = «-fc
X — III x — n
ani + b i-n-ird
a -\ + 0 + — «4-c
X — III x — n
{am. + h){x— n) -\-{cn-\- d) (^x — vi) — 0
{aiiL-\-b + cii-\-d)x- («4-c-)//m4-i/<+(/nt.
5. Similarly may be solved
ax + h ex + d ex^ +/x — g
x — m x — n {x — m)[x — n)
ain+b cn+ d {e{iii + n)-^-/}x--emn — g_
" X —m x — n {x — vi)(x-n) ~
{am.-{-b){x — ny\-{cn-\-d){x — m)-{- {e{m + n)-{-f}x — emn- f/ = 0.
{{a+€)in-\-b + {(■-\-e)a-\-d+f}x — {a -tb-\-c)in.n -\- hn+dm-j-;i.
3x+l ~ x-1
43 13
••• 4^ - 3:h^1 + ^ + JT-i = 52, or
13 43
:, - -s — ^-^', •'• 39x- + 13 = 43a; — 43, and jc— 14.
x — 1 3x-fl
_ 25— Xx lGx+4:l . 23
7. !L_ . = i = 0-1- -.
ic+1 ^ 3.i;+2 x-tl
Taking the last fraction with the first, and multiplying the rC'
suiting ei^uatioo hy 15, we have
SIMPLE EQUATtONS. 145
240r-J-63 _ r-r , 5.r— 30 .
.-. SO - J^ = 75 + 5 - -^, or
3^ + 2 x + 1
o.r-f-2 a'-f-l
X- a x — h X — c ,;
o-irf a + e h-\-c
.-. ^^^ - 1 +--I1^ - 1 +:■--: - 1 =. 0;
''■ff ffl + c "^ h+a ~ '
which is satisfied bj^ a; — (r/. 4- ft + c) = 0 ; .-. a; = a +/)-{- p.
m + 71
9. -^ +
' X —0 x — c
x + n x—h x — c
m(.r — r) n[x — r) _
x — c. x — b
which may be solved as in Ex. 1.
10,
nx-Iro 4a::4-9 _ 15r4-_7 IS^r^-l?
x+l ~ 2x+l ~ 3.r+'l' "" ^oTfT *
I- -J^- - 2 - —±— = 5 + _^_ _ 4 - - or
^+1 2a;+4 3,r+l 3a: + 4
2 12 1
x+l 2x+4: Pjx + 1 3x+4'
3a: 4- 7 _ 3:r + 7
2a;2 + 6.i-+4 ~ 9a;2 + 15a-+4'
This can be divided by 3ar+7, givini? 3ar-f7 = 0, or x= ~^.
The result of the division is
1 1
= , or
2x3 + G.f-f4 9a:2 + 15a: + 4
9a-- +loa;+4 = 2x2 + Ga- + 4, or 7x^ = —9x, which we can divide
by X, :. x = 0 ; the result of the division is 7a: = —9, or »:= — 5..
146 SIMPLE EQUATIONS.
Exercise li.
lOx+n 12x+ 2 5a;-4
1.
13.
15.
16.
18 18.C-16 9
6x+13 9a;+15 2x+ 16
15 5a;- 25 ^ 5
Tx+l _ ^ X ^-^
a-1 ~ 9 a;+2
3. -t: — T = — ^ :7T-o + 3^
4a;- 7 2-14x ' H+x _ 10 - 3fa; _ 19
^- 2^^ + 7 + 14 ~ ^ 2 21*
2a; + a' Sx-a ^ ^
^- 3(a;-a) "^ 2(j;+«.) ~'^^'
a;
-4 3x— 13 1
6- 6a; + 5 "*" 18a; -6 3
3a; + 1 _ a;-ll _ a;-9 ^5 ^ ^
"^^ 2a;-15 ~ 2a;- 10" ' a;-5 "^ a;-8~""
^ a;-12 a;- 4 ^ 7 . 3a;- 19 3a;-ll
^- x3^ + ^^2 =2+ x37' -^^^13 + -^-7 = ^-
a;- 2 a;-l 5^. a;+l a: + 4 ^ ^
9- 2:^+1 + 3(^8) ~ G ' 4(a;+2) + 5x+ 13 ~ £0'
6(2a;2+3) 5-7a; ^ ^ 3 14
!«• -Vfi + 2:^='-^-^' ^7 + x-^ = X— 8
7a;+55 3a; _ Sa;^ + 8 . 17 15 32
2x+ 5 ~ 2 ~ 2x -4 a;-16 ^ a;- 18 ~ a;- 17
3-2ia; _ 28- 5x- _ IC
" 14(a;-l) ~ 3"^
2+2ia;3-^a;3 1 6
l-25a; 3-2ia; _ 28- 5x- _ 10a; - 11 x_
12. —^ 14(a;-l) ~ 3"^ 30 +3
6-5a;+a;3 '2 x-5
^__ 60+8^ _ 4^ ^ ^^
~ ' ■ ^ "^ a;+3 ~ a;+l
5j;2+a,._3 7a;2_3^._9
4
7a; -10
a;- 9
a; + l a;-8
~ a;-i "^ a;-6'
SIMPLE EQUATIONS. 147
17 x^-3x-9 x^-lx-ll a;3-6x-15
x-6 ^ x-d .. ^
18.
19.
20.
21.
22.
23.
24.
25.
26.
ixA-1^ 4.g+9 _ 4a; + 6
ix-\-Q "^ 4a:+7 "" l.c4-~4 "^
2^-3 2x-4 _ 2.t;-7
2^-^ '" 2^-5 " 2a:- 8
7a: + 6 2.r + 4|- x_
28 ~ 23a;-6 "^ T ~
a;2_5 a;3-ll a;3-7
a;-8
4a: +10
4.r + 8
2a,- -8
2a: -9
11a: a;-3
21 42
a:2-9
+ "2 iA-
a;3-6 ^ a;--2-12 ~ a;3-8
X - 1|4 2-Ga: 5.<;-^(10-3a:)
2 ~T3~ " ^ " 39
1— 2a: 1+x 1
3(a:2-a; + l) "^ 2(a;2 + l) "^ 6(^^-+l) ~ 9(a;3+l)
2.^2 +a;- 30 x-+4.x-A _ x^ - 17 2a;2+7x-13.
2a;-7 "^ x^^O ~ a:-4 "' 2x-3
a;-a a:-?> {a-hY _ 2(a-a;)
a; — 6 a;- a (a; — a)(a: — i) ~" a-\-x
12a:+10a j. 28a: + 117^/. ^ ^g
~3a:+^ ' ~2^+9^
^^ 13|x-5 13.^a;-ll _ 13;a:-7 13^a;-9
"'■ ISJ^^T^ "^ 13Sa;-12 ~ l^^^ "^ 134a;- 10 '
<,Q 1 1 X 16a;
2(a:-l)2 "*" 2(a:-l) 2(.'c3+i) " (^_i)(a;2 + i)
29. i(|x+4)- Zi
3
OQ 3» _ 81a;2-9 _ q _ i_ ^a;^-! _ 57 -3a;
* T (3a;-l)(a;+3) ~ '"^ 2 ' a:+3 2 '
81 1 + ^^L i?+i_ _ ^L+^'t? _
2(x-l) ~ 2(^+1) ~ a:2_2a:.fl
148 BI^JPLE EQUATIONS.
7;c-30 5x-l 2 -21a;
?.2
10|- ^x--3 21
42x-171 _ . 2a:-'J 1
7
6-3- - 10 + 03- iT. - -(* - ■
18^-22 l-t-Hx 101-Gi.>;
4-9x 5-12.C __^^*l~^_
8a;4-25 16^ + 93 _ IQx+m 6x+ 26
^^' 2;^^-^ "*" ^2x+Tl ~ "■2x + '"9 ■*" 2.^+ 7*
1 1 1 1 ^
gg I . ! _}_ — 0.
x-\-a-rii x—a-\-b x-r-a—b x — a—b
Art. XLII. The results deduced in Section III., Chapter
lY., may ofien be ax:)X5hed with advantage.
1.
Examples.
ax -\-h in
ex + d n
(ax + b)d-(cx+d)h
rnd—oih
(cu;+'i)i;t— («x+ Ij)c
~ 'fuo — me
ind — nh
na — mc
ax^-\-bx-{-c a
inx''^-^7ix-hp vi
<i.age 123).
(ax'+bx+c)j-ax^ ^ n_ , ^^^^ ^^2^
{mx'^ +nx+2j) — ■nix'^ m ^^^ ■ '■
hx + c a
nx-\-p lib
3x+7 _ 3x- 13
33+4 " X— 4
SIlfPLE EQUATIONS. 149
By (5) each of these fractions =
difference of numerators 20 3,r4-7 5
difference of denominators' " ~S ~ x-r^~ x+V
or -^ = — -, -. ^-=0.
vix-\-a + h mx-{-a-\-G
nx — c — d ~ nx — b— d*
vix-\-a + b nx — c — d
•■• ^+r+c = nx-b- d ' °- ^y 4' ^""S^ 122,
i)ix + a + b nx — c — d
I = — ; ; or ()i — m)x
b—c o—c ^ '
= a-\-h-\-c-\-d, .'. a; = &c.
^/{a■\■x) — ^{a — x) ~
Here by (6), page 122, we have
2y(a + x) rt+1
2 /(a — x\ ~ ~a^ ' °^'' ^^^'^^^""o ^^ 2 in left hand mem-
ber, and squaring,
^^x ^ (^zryiF' whence, again by (G),
2^ _ (a+l)--(a-l)2 _ 4a _ 2a
26t ~ (a + l)34-(«_l)2 - 2(a2+l) ~ a^^V
2a3_
■•* ^= asqpi'
6
^/(x — a + 6) — ■j/(a; + a — 6) _ a-b
l/(j;-a + fc) + v'^('C+« — ^) ~ 'iVi*
. i/ix-a-^-h) _ a .
y^Xx^a^h) " T'
squaring and again applying (6),
'■• ~ 2(^^6) ^ a3~6"3"' and u; = ^-^
150 SIMPLE EQUATIONS.
Exercise lii.
1 Lt?
9 0-+X fi{b-\-x^ ,
b-'r2x ' a -X ~ ' a — x
3. 'L±^ =
a — x
^ a-j-hx
a-\-b ~. f + c/-
g 2^3 _ 5a; -1-6 _ u;2-7a; + 5
2a;2-7a;+3 ~" a;3-9a;i-2*
9.
10.
ax~h + c (6 + c)2
i^. i/(«^+?/)+i/(x-.v) ^ ^^ ^i^g^ ^1^^^. ^:^
-\/{x+y)-^{x-y) y x-y
1x-l _ x+1 . 4a; -5 _ lOx-32
2a^^ ~ x+li' 2a;+10 ~ ^S^c'-^r'
57a; -43 30a;- 7 . 28a;+5| 36x-7
19a;+13 ~ 13a;+2o 115.x-- 29 180a;+23
210a;-73 _ 21a;+7-3 . mx-a-b _ mx-a-c
310a; -66 ~ ^Ia;'+8" ' 'rix^^^^c^d "^ nx-b-d
11 3a;+y(4a;-a;2) _ ^ y(12a; + l)+ s/(12a;)
' 3a;-y(4a;-a;3) ~ "' ■,/(l2a;+l)— /(12a;) ~ "^^
12.
x^-{-ax' —bx+c x'^-\-ax—h
x^ —ax'^ -rbx+c ~ x^—ax+b
13 l/(2a--a;-) + V(2^-a^) _ A + ?>
|/(2a2_a;3)-6-/(2a-a;) ^/a-^"
14 T/(x3+a^)+A/(a;^-^^^) _
' ^/(x^-+a^)-V{x^-a^)
8a;3 + 12.x3 - 8a; + 5 4a;3 4-6a;- 4
15.
8a;3-12a;3+8a;;+5 4a;2-6a; + 4
SIMPLE EQUATIONS. 151
16.
17.
18.
18.
20.
f{x + l)-^{x-l) _ 1
f{x + l)-{-^{x-l) - '1
28+ya; _ 9 +3 ./a;
28-v/a; ~ ^ + 2^x
083.3 +a2^^.2 _ acx-\-d a^x^+abx—e
a^z^—a^bx^ +acx+d ~ a"x^ -ahx+e
5^/(2x-l) + 2i/{Bx-d) _ ^^^^
4:i/{2x-l)-2-i/{3x-S) ~ ''^'^'
^/2x+y'(3-2x) _ 3
\/2x-\/(W^^,:) - T"
2^(3a;+3) + r (7^+ 8) _
• 2^/(3a;+3)->'(7a;+"8) -^•
22. 33{13-2i/(a;-5)} = 3{13+2;/(a;-o)}.
23. (i/»+l){i/(«a:+l) - ynx) = {i/n-l){i/{nx+l)^y'nx}
l/jz+c) + ^/b i/cc 4- 1/«
24.
26.
26.
V^+2.3 _ ]/«+38. 1^2^+17 _ 1^2^+27
l/x+ 4 ~ i/a;+ 6' f2x+ 9 ~ ■^2a;+15*
yx+2a _ i/x+Aa, 3a; - 1 1 + ]/3a;
iT^n* ~ v/^+36' v^Sx + l ~ 2
v/a--l/(ft-a;) _ _ ■/il+J'^^
l/a+^/{a—x)~ ' \/x — -Jb
28.
aa; + l + v/(a-a;3-l)
29 ii:y{ijiv^(i-^)}_.
30.
31.
'a+x fe+l . l+.6-+a;2
62 1 + a;
v/(2«x+x-3) ~ 6-1' l-ai+a;-^
5x* + 10a;2 + l rt^-flOa^+l
x* + lUa;»4-5.t; ~ 5a* + 10a- + l'
~ 63 ^ 1-a;
Art. XLIII. Various other artifices may be employed to
simplify the solution of equations.
152 SIMPLE EQTTATIONS.
ExA3IPI,ES.
1. Solve 24-7(4..^ -0^4-8) -2;r=0 : here there is bnt mie
surd, and it is convenient to make that svird one side of the equa-
tion and transpose all the rational terms to the other ; this gives
n/(4.t3 — 9a; -1-8) = 2a;— 2 ; squaring both sides,
4^2— 9.(;+8-4.7;2_8x+4, .-. a; = 4.
2. V{a + x)-\-\/{a—x) = 'l\/x. V/e might square this as it
stands, but the work will be simplified if we first transpose, thus
V{(i-\-x) = 2 Vx~ '/(a — x)', ^iow squaring,
a-\-x = 4:X + a — x — 4:V{ffx—x^), or
x = 2\/ (ax — x^). Again squaring,
x^ = 'iax — 'ix- , whence x = 0, or —
5
3. Clea^' of ra.'icals
f'^x -}- ^^y + j^z = 0. Transposing,
f/x+f/y = - i^/z : cube by formula [0] ,
•''^+?/+Bi^/.r//f]^«+]^/.y)= —z; and substituting /or
■^x+^y l.s val'i;> — -^/z, t^'is becomes
x-\-y — Q-^^xyz= —z, or
a;4-.y+z = 3^^a;//z; .*. caLlng again,
(aj+y + s)^ = 21xyz.
a+x+\/{'2rix + x^-)
a+x
Dividing and transposing, we have
division in leflhand member,
+ 1 = (^,_1) ... _^=^^i_(/,_i)n,or
(«+a-)2 ' a+x
a
— 4- 1 = f!-r.
SIMPT-T5 EQT'ATIONS. 153
5. Solve A/(4a-3 4-19)+v/(4a;2- 19)= a/47 + 3.
We have the ideutity
(4.c2 4-19) - (4.r2 - 19) = 38 = 47 - 9.
Now dividing the member? of this identity by those of the given
equation, we have
\/(4a:2 4- 19) - v/(4r2 _ 19) ^ ^47 _ 3. Adding this to the give«.
equation, then
2v/(4«2 + 19) = 2a/47, .-. 43:5+19 = 47, and a. = ± ^7.
6. f/{25+x) + xy{'lo-a') = 2.
Cubing by formula [6] , (See Ex. 8), we have
25+x+25-x+G]V{252-x^)^8, or
1^(625 -«3)=_7, or (62o-a;2)=-343;
.-. x^ - 525+453 = 9G8, and a; = ± 22|/2.
Exercise liii.
3 V{x + i) + x/(x-S) = 7.
2. y(3^ + l) + ,/(4.r + 4) = l.
S. ■,/(2a-+10) + i/(2a:-2)=6.
4. i/(i}ix) — -i/(nx) = ni — n.
5. V '{hx) +V{ab + Ox) = y/x.
0. ^,+ y(,+ 3) = ^_^-^.
7. V{ax-{-x'^) = i\-hx).
8. ^{Mx-2G)= A.
0. y^x-y^{a+x)= ^^
10. h + x-x/{h^+x"-) = c^.
11. ■^/(8 + .r)-|/x = 2y^(l+3;).
12. ■i/(2a;-27a) = 9|/a-|/(24.
154 SIMPLE EQUATIONS
13. ^/(l-,.)-f-^3/(i^^.) = ^/3.
14. ^{3 + x) + ^{3-x) = -^7.
15. f^{.c + l)-f(x-l) = f'n.
16. ^{a+x) + f(a-x) = ^b.
17. f/{l + ^/x) + ^{l-j/x) = 2.
18. ^/x-^/{a-^/{ax+x^-)} = }^^a.
19. Clear of radicals -^/a-{-^b--^c.
20. Solve x+i/(a^+x^) =
21 . Clear of radicals -\/x+ •\/y + y^a; — \/''n.
Solve the following equations :
22. y\i + x) + ^/{\+x^-^{l-x)]=i/il-s),
24. ^{l+x+x^) + v\l-x+x^) = 7vx.
25. i/{a^-x'-)+xi.^'{a^-l) = a^{l-x^).
Oft = -'^ — a.
VM+c n
27. 'v/(2x2+5)+\/(2x2-5)= \/15+v'5.
28. ^/(3.c3 + 10)+^/(3^•2-10)= /17+ a/3.
29. V(3x2+9)-\/(3.^•2-9)=^/34 + 4.
30. ^/(3a-3&+a;2)^ ^(2a-26+a;2)= A/a+ v'j;.
81. V(4a3 -352 - 2x2)4-/(3rt2 -362 -x^) = a+x.
82. Clear of radicals, ir(2x) - 1^(27/) -i^(2z).
33. ■i/(rt+-T) + \/{«- -x) = 2.r -: i/{ '/ + ^'(^'2 +^t') y.
84. y(^. + 2,«)+,/(..-^-2„x)= -^J^^^-^^■
86. y{(2a+x)2-l-i2|+^f(2a-a;)2+i2}=2«..
SIMPLS EQUATIONS. 155
Art XLIV. Sometimes a factor can be discovered, and the
principle of Art. XL. applied.
Examples.
X -a
Factoring we have
{x^+ax + a^)(x-^-ax+a'^) ^ (^_ j)(^.., _^^,^ + ,,,)^
x — a
or x^ — ax + a^ = (x — a)(x— b) ;
.*. (a-^h — a)x = ab — a-, and x = (i — —
6
2 Sabc b.i- a"h-^ b^x 2n^-|-^
^^ ~ H ^ {a-\-byi ^ ^^^ a ' {aVb)^'
Transpose — and factor, then
a
ab \ ^ ah
J "" [ a\ {a+bfl I
f Q I ^ "'1
= X ' oc + . r
= x I 3. +
(^bj2]
ab
a + b ~
x+a x — b X — c b-^c
'{a—b){G-a) {a—b){b-c) {b-c){c-a) {a- b)[b-'.:)[c -a)
Add term by term the identity (Th. iii., page 5-1).
x—a x—b x—c
{a-b}{c-a) ^ {a-b){b-c) ^ (b c){c-a)
2x _ b+G
{a—b){c — u) ~ (a—b){b — c){c—a)
_ 1 b+c
''• -^'^ T' b^c
156
SIMPLE EQUATIONb.
4. (x-j-a+b)^ +{a-\-h)^ — {x+b}^ - {x-]-a)S-{-x^ +a^+b^ = abc.
The left hand member vanishes for x = i), and .". by symmetry
for a = 0 and 6 — 0; /. it is of the form mabx in which m is
numerical.
Put x = a — b, and m is found to be 6,
.•. the equation reduces to
Qahx = abc, .". and z = ^c.
jx — a\ 3 x — 2a + b
5. r = -77-; — ; let X — b = 7n, x — a = n, and .".
\x — bj x — Ab + a' '
vi — n = a — b, then we have
m^ n — {m — n) 2n — m
n^ ?H + (wi — n) ~ 2m — n
:. 2m^—nm^ = 2n^ —7i^m, and
2{m'^ — 71^) — mn[m^ — n'-) = 0, which is divisible by m^ — «',
.•, m^ —7i~ — 0, or m-{-n = 0 ;
'Bntm+n = 2x — a—b — 0, :. x = ^{a-\-h).
1 ^3_4a; + 2 1 a;3-4.i-+3 _2_ x^ -4x-\-B 5
^- y x^-Ax-1 "^ (5 jj2~4a;_'3~"9"' ^2"T4^36 = 18'
Let 2/ = .«^ —4^, then this equation becomes
1 V/+2 1 v+3 2 ?/+3 5
•^- -f TT- ' 5-77" /^ = Tu' or by division,
1 . _1_ J^ _1_ 2 ^ _ 5
"3" "*" v/"^ "*■ T ''" ^^ ~ y ~ ^;rr6 = i8'°^
112
-^ — — - _ = 0 ; this may be written
y—1 y-'d y-Q
1111
^ + ^ =0' ••• 5^-15+8</-3 = 0, or
2/ = 2i .-. ^2_4;c^-2|, or «3_4^ + 4 = 4_|_2i,
• and a; — 2 = + §. We might assume (j3 — 2) ^ = ;y, when the ^iven
equation would take th^ form
SIMPLE EQUATIONS.
//-2 1 y-\ 2 y-
- 1
5
y—5 0 y-7 y ' !/ -
-10 "
lb
157
And reducing as before, we should find
</ = 6| = (x-2)2, .-. X- — 2= +^, as before.
Exercise liv,
i- —f^ = x'^ -{u-b)x'^-\-{a2 — ab)x-{-a^b.
2. fi±±!.^ = x-^ + 2a{a-b)x+{^a-b)x--'-2a^b .
x + b
a^ + uo+b--^ ~ a^-b-'i ^ ll^b ~ '^^■^'
i. ~r~ _ i- _ 1 _ i- = 2ab\:x^b)xK
a-\-b-{-x a b X
1 1
{x-b){x-c) ' [a+c){a+b)
1 1
(a4-c)(j;-c) ^ (a+//)(a;-6)
. 6x- 'dab a-//- 6^0? 2a — b
^- r + 7 TTT = oo; -
x^+2«-c ^ — rt
x* -llj;2a>+a4 ~ X''^3ax-~a^"
. 1 jjj — rt\ ^ 1/ . I ,i ■<^-»
o«. — — — - i( -ui + u)= .
2 \^ + rt/ ^+a
3. x^-{a + b+c)x+{a-2-{-b-^+c^)x-}{a3+b'^-\-c3]
= [x—a)(x — b)(x — c).
io. 1 + 1 , 1 = Ua+k+c)^ - i-fiL+A+ M.
ax Ox ex 2 \ofa; acx abx'
,, l-r/.c 1-^.^ \-cx 12 2 2i
11. — ,-- + + — ,— = — -{- — 4- — \x.
be ac ah \a b ^ ^
]P,. x^-\-{b-\-c)^ + ^b{b-^t)x = b^.
158 SIMPLE EQUATIONS.
14. x—a-S^(abx) = h.
15. 11x^ + 10x^-4:0x^116.
X ac c
16
+
a + 6)3 ' (a-b)^ {a^-b^){a + b) {a b)^
,„ „ a — h 2cx^ a — b l — cx
a+b l+6'.f; a-iro l-i-cx
2x-{-a
19. ' + ' '"^
a;2_lia;+28 ^ x2-17.6'+70 x^-Ux + AO
20- -.-4-^ +
x^-6x+5 ' x-2-14.i'+45 ft;2-i0x + 9
„, x+a x-b x-{-c
(a^b){c-a) ~ {a-o){b-'c) ~^ {h-c)(c-^)
a-\-c
{a-b){b-c){c-a)'
22. {x-aY-\-{a-b)^+{b-xy^=x^-ai.
,,C+2rt\3 /«+2.c, 3
23. x\ + a [- =%i.
\x — aj \a — xj
21. {x+a)^ -{a + h)^ + {}> - j^^ ={x + a){x + b)ia + b).
25. x^ -{x-b)^-{x-a-{-by-^--a^+{x-a)^ + {a-h)^+b^
■b)c^
20. (x+ 0^ - {x+by^ - (-c - 6)'^ - (2^()-! + (:e- o)3 + (a + Z>)3 4.
«-f-a x — a a^
^''^' x^+'tx+a'^ ~ x^-(ix + a^ ~ a(Ti^'-^x^~Va^)'
28. {x + a+b)'^ -{x + a)'- -{x + by+x* - (a + b)-^ +a^ +h^
= V2'ib{x^ + {u + by-}.
a — x b — x c — x Sx
29. -^—r. + i:¥—7.. +
a^—bc b^ — c<( c^ —ab ab-j-bc + ca
30. x^{b -a^)-{-((,^{x-b''i) ■\-b'^{a~x-) +ub.,{ab.c- 1)
= (a-x2)(63-a4).
SIMPLE EQUATIONS.
31. (l+x + X = j= =^^.{1+U^+X*).
^ ' ax— i.
32. J|^^ - ^~^ -1.
159
x-'b '2f+6 + c/
34 i/(.c2 4-27^-+ 180) --i/(a:2+26ic + 168)= ^f^±i^\
V\x+12 /
35. {(--a. r| (.<:2+2ax- + 6-^)}3 + {3;+a- ,/(a;2 + 2aic+i2)|3'
= 1'±(^ 4-a)3. (Sec page 17, Ex, 1).
36. {x+«+/(x2-2rtx-2Z»2)}- + {:t + a--/(a;^-2«ar-2i2)}2
= 2_ft3 ..2rt(a 6).
-b] " x--a-2//
38. (5x-7)*-(2,c-4)3 = 27(a;3_l).
39 J_ ^^-6-^-1 JL a:2 - 6a; - 4 ^ a;2-6a:-7
3 \,;2_tj^_4 + y a;a_6ar-9 ~ y ■a;--Cx-i6
14 4
45 ' x^ — 6u; — 9
^^ 1 x^'-'lx-?* 1 a;2--2x-15 2 a;2-2.r-35
5 ^•-•_2.r-8 ^ y :i;-'-2a;-24 13 a;^-2x-48
_ 2_
~ 585"
41. {■ + a-h ( 2 „2_i2);3_
a;+a-6- v(,«2 a^ ~ b^)^ = 8{.c-\-a-by.
1 1
42.
a)2 ^ -' • .-i)S
1 1
+
8x- 57 0,-68
39, ~j- -!- 5-
\ x-i-A x-to
l60
SIMPLE EQUATIONS.
44. 51 - ._[_ Tl^ +863 =
, x-1 a; — 4 /
^^"0^=^ - .-3-)-
45. ix-]-a){z + Sa){x+ia){x-^Ga) = x^ + Qa^{x^ +7aa;4-6«s).
46. _i_ . 2 _ 3 _ _6__
x+Ca x-oa~ x+'2a x+a
Exercise iv.
1. a{h — x)-tb{r — x) = h{a-x) + cx.
2. (a + hx){a—b)~{ax-b) = nl{x+i).
3. {a-b){x-c) + {a-{-b {x-\-c) = 2[nxi-ad).
4. (a - ^')ix-c) - (a + i)(a; + c) + 2a{b+c) = 0.
5. (a_6)(rt _c)(a4-.r) + {('+h){a + c){a-x) = 0.
6 ( ~b){a-c-^x)+{a + b){a-^c-x) = 2a^.
(solve in {a;— c}).
7 (»;+fl (a+6 — a;) + (a— «()(^ -a;) = a()H-f 6).
8. r;?(a + ' — .T) = n(a: — a — i).
9. {m+n)(m — n—x) + m[x — n) — n(x — m) = m^ -w'.
_^ m — x n — x p—x
10. [- — + ^^^ =3.
P -^-=0.
6 — a; c — x
12. -y— + + -^=0.
Oc ca ab
1 — ax 1 — bx 1— ex
13. --T— - + + — =0.
be e.ii. ab
(Deduce the solution from that ox I>;o. 12;,
a— hx h — ex c— ax
14. — ,— + + —— = 0.
be ca at)
^ a — 0 a-{-b a—o
SIMPLE EQUATIONS. IgJ
Sahc an^_ {2a+b)b^x _ (b + Sac)x
' a-\-h "*■ (a-^l)3 + a{u + by^' " "^
,, 10 4 9 2 / 1,
,8. 1 + 1 = ^4zf + ^ _ 1.
a; d ox 12, 4:x
19 I. _^. i! = 2(5.^-12) _ 17 10
3 iJx 8x 2U '
X
20.
21.
l^Tlf 2^+^ _ 7£-f-266 4.C+17
y "^ 7 " x-t-21 21
23.
24. r: ^ = 1.
6- ., —
e-
d-
X
25. (.^-lXr-2)-(.c-3)(^-4) = 3;
(x-3)(x-4) = (..-2)(x-G).
26. 2(x-4)(3a:+4) + i2.c-3):;::3x+2)-6(.c-2)(2u;-3)=0.
27. {a-x)[b-x)=x^ \ {a-x){x-b)=x- -c'^.
28. (a-a;)(/-'+-c) = 63-a;- ; (-c-a)(u;-i) =.c- -a^.
29. {a+x){b+x) = {a-x){b-x);
{ax + 6)(^x+a) = {b-ux){a - bx).
30. {a-x){^b-x)i-ia-c-x){x i/+c)=0.
Ifl2 SIMPLE EQUATIONS.
31. {a-x){b-x)-{c-x){d-x) = (c + d)x-cd.
82. {x-a)(x-b)-(x-c){x-d) = {d-a){d-b).
33. {{a^ -b-2)x-ab}{a--ia+b)x} +2ah''x =
{{a+b)^x+ab}{b-{a-b)x}.
34 {x+l){x-\-2)(x+S) = (x~ 3){a;+4)(a;+5).
35 {x+l)ix-\-2){x-\-d) = {x~l)ix-2){x-S) + 3(z+l){ix-}-l)
36. (a;+l)(ar+4)(a;+7) = (a;4-2)(.^+5)3.
37. (a;+2)(x+5)2 = (a;+3)2(x+6).
38 (a;-l)(a;-4)(a;-G)-a;(a;-2)(aj-9) = lB6
39. (a+a;)(6+a;)(c+a;) - (a-x){b-x){c -x) = 2{x^+abc).
a; — (i
41 x{x-a)^ -{x~a-^ b){x-a+c){x-b-c) = (a^ + hc){b + r').
42. (a;— a+&)(a;-&+';)(a; — c + rf)-a;-(ac-a4-^/) = 5f(c^-«).
43. (a; -a+&)(a;-& + (■)(«'- «+''0-^(^-" + '')(^-c+^)
= bc{d — a).
44. (ic-2a)(a;-2t)(a;-2c)-(x~a-6)(a;— &-c)(a;-c-«)
45. x^-{x-a+b){x-b + c){x-c+a)
= {a-i-b + c){a'^ + b^+c^)~2{a^b + b^c+c^a)-3r,bc.
\ « / \ a; / \ X I x^ X
47. {x+a){x+b)+(x+c){x+ct)r=(x+b)(x+d)+(:r+d){x+c).
48. (rta;+i)(aa; — c) — a(&-a;)(</a;4-i)=«^(a — t•)(^~^)-
a(«a;— c)(6-— x).
2a;- 3 3x-2 _ 5x^,-29x- 4
49- 2._4 + lc_8 ~ K3-12a:+32*
^"'ZL _ 3a; + 2 _ « 2 -3^+2
■ SfT^+Y) "" 2(a;-l) ~ 6a;3-6
SIMPLE EQUATIONS. 163
., ;b:-7 _ n(.v-+i) ii-^+3_
■' :Lx-d ~ 2[x-\-S) ~ 2a;2-oa;-27*
7x-r, 8«-7 lOx+7'
ro I 4- - — =5
• 3x-'2 ^ 3x-l ^ y*2-lb;4-:i
2.V+7 iix-{) h{x-Jj _nx-9. 5x- S '2.r 4-2
^^' 'dx-'7'^'2x - 5 ^ yx -25 " 2x-6'^' ii^c-25 '^. -dx-f
4.x^-3z 3.C 4x3+2.1- x — a x-b ^
54. —^. -:; =— ^^- . ; =0.
55.
66.
<2,{x-l) x+Q _ 3(5a;+16)
^'^' ~a:-^7 "^ a;-4 "" ~6x-2d'
68
z,^ ox + h bx ax (ax^ — 2b)b
61
cx — d {hii + fhn)x — {hq-{-dp) a c
vix — p nx — q {mx—p){nx — q) ~ m n
po tn n p in n p
02. _L. + - = + + — -.
x — a x-b x-c x — c x — a x—b
J 64
SIMPLE EQUATIONS.
go ax - 2a ax — 1h^ a ^ _ x
ax—2,b ~ ax + 2a '• "J"*""™"-^ ' ^^T
1 a + —
ax X
2x^ -3a; +5 _ 2
7a;2-4.j; + 2 ^ T'
??2uj2— ?2a;+^ ~ m ' mx^ ~nx- +mx — q ~ mx — n
65. iz^ + 1 =. _^ _ J_.
^+x ^ 4 ^-f-o; 4 '
f^-l 2 2 , |x+f
l-x b 3
G6. J^l Zl_ - 21 7^
a— 98 a; -94 .'^+44 a; -52
^,,7 8 9 1
67. -—5 + — TT = ^ +
a; — 6 "^ a;-ll a;-7 x-12'
9 9 2_ 2
a;_51 a;-15 ~ x-81 *" a;+81*
_5_ _4 8_ 1 .
^^' x-Q ■*" a;-9 ~ a;-7 "^ a;-10'
a;— 6 X— 3 a; — 2 a — 5
w. — n rf — h
69.
70.
w — n
a-h
~ X — h
x—n
h+fl
x — a X — m
a + b rt + c b+fJ c-i-d
x — b x—c x—{a-\-b+2c + d) x-[a+'ib + c-\x>
71. {x-a + b)^ -{x-a)^ +{x-b)^ -x^ +a^ -{a-b)^ -b^
= (a — i)c2.
72. {x+a+by -{a + by -{x+hY-{x-\-aY+x^ ^a'^^b'^
= lQabxl^lx^a + b){x-ira + b).
{m-n){x-a) {n-r){x-b) {p-m)(x-c) ^
^^- b+c + t:^ + a+b "=^'
SIMPLE EQUATIONS. 165
ax—1 l>x~l cx — 1 dx
74. z^t—tta + n
n^r + h) ^ h2{c+a) • c^{a+b) ab + be -j- ca
x~2a x-2b x — 2c
h + c — a "^ c+a-b a + b—c~'^'
x~2a x-2b x-2c Hx
b-^c-a c+a—b'a + b—c a-\-b+c
a-x b~T c — x rf
(7. .o — 7-_ +
a^ — be h" —< c ' c^ — ah a 4- b + c*
x+2ab 2at)-x _ x~2ah ji:^2ah
a-\-b-\-e b-\-c — a ~ a—h-\-c a + b^^'
a b a — e bA-e.
79 4- = • _ — — .
x-\-b—^ x-\-a — c x+b x-\-a
80. — + — ^ + — +
x~m x—n x—p
q [pd. + {n—p)c-{-{in — 'n)h— ma]
x — q ~
81.
(a;-2)(x-5)(a;-6)(a;-9) + (a+2)(a-4)(rt-5)(q-ll)
X
(6+l)(^> + 5)(i + 8)(^ + 12) . ... ^., ^^^
^.v_L_A '^ n / ^ (.T-4)(a:-7)(a:-ll) +
X
(a8-l)(a-8)(«--10)4-(& + 2)(6+3X^> + 10)(fe+ll)
X
Art XLV. Employing the language of algebra, the princi-
ple illustrated 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) = 0.
Thus, if a is a root of the equation /(a;) = 0, then /(a) = 0.
By Th. I., if a* -a is a factor of the polynome 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 which will make the corresponding polynome vanish. Sup-
po.se/(a;)" = (a;-a)<?)(x)"~^ = 0, we are required to find a value, or
166 SIMPLE EQUATIONS.
values, of x which will make (a; — «)?)(a;)"~^ vanish. The pciy-
nome 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)<^{xf'^ will vanish if a;-a = 0,
quite irrespective of the value of f (a;)"~^ Also, if ^(a:)"~^ = 0, the
I)olynome will vanish, irrespective of the value of x — a. It fol-
lows, therefore, that if /(ic)" can be resolved into two or more
factors, each of these factors equated to zero will give one or more
toots of the equation /(x)" = 0.
When there can he found two or more values of x which satisfy
the conditions of given equations, tliey are sometimes distin-
guished thus : a^i, JCg, iKg, &c., to be read " one value of aj," "a
second value of a;," "a third value of a;," &c. Thus, if
(x — a){x — h){x — c) = 0,
.'. XY=a, x^ = h, x^=c.
Examples.
1. Solve 2a;3-13a;2+27a;-18 = 0.
Factoring,
(x-2)(a:-3)(2a;-3)-0,
2. x^-{a-^h)x+{rt+c)h = (a-ic)c,
:. x^-{a + b)x-\-{a + c){b-c) = 0,
:. a;2-{(«-fc)+(i-c)}a;+(«+c)(6-c) = 0,
... |^_(a+c)}{a;-(&-c)}=0,
.". X'^=a-\-c, x^=b—c.
3. x-{a~h)-\-a^b-x) + h^{x-a) = 0.
.-. x^{a-b)-x{a^-h^)-^ah(a-b) = 0, \
.-. {x-a){x-b){a-b) = 0.
If a — b = 0, the given equation holds irrespective of the values
of a; — a and x — b, and therefore of the values o{ x; but ii a~b is
not zero, x^ =a, x^ =f>-
SIMPLE EQUATIONS. 167
4 _ ia^+b^)x-(a^-b^)_
.T+1 a-{x—l) /■'^+-'-\ " "^
0.
X =
■1 ~ b^x + l) •• \x-l} h' ~ ^'
' ' a;^ — i 6 ~ ' * ^ ~ a — b'
x^-\-l a ^ a — b
-^-^ + = 0 .'. a;o = —-r
Xg — 1 h ^ a + 6
(a-a;)g+2(a-a;)(6-a;) + (6-.r)2 _ 2(49) -34
\ri-x)'-i{ii-x){b-x)^{b-x)'^ ~ 3(34)-2(49)
((a-x)-{-{b~x)]^
= 16,
, — -''•2 = 0
[{a-x)-{b-x)]
{a-x^) + {h-x^)
.-. r — 4 = U,
a — b
{'l — X..)-{-{h — X.-,)
^ -^_-^ ^~ +4 = 0, .-. ajs = 4(5«-36).
g (g-cr)(3;-6) (x-5)(a;-fi) _
• {e-a){c-b) "*" (a-i)(«-c) ~
Subtract term by term from the identity (See page 53),
{x — a){x—b) {x — b){x — c) {x—c){x — a)
lc-a){c-b) "^ {a^b){^c) "^ {h-c){b-a)
.. {x — c)(x — a) = 0, :. x^—c, x.^=a.
7. Find the rational roots of x^ - 12..-3 + 51a;3 -90x+56 = 0.
Factoring the left-hand member by the method of Art. xxviii.,
(ic-2)(a;-4)(^--^-Gx + 7) = 0
a;i=2, x-3=4, ora;"— Ga;4-7 = 0.
Since x^ — 6^+7 cannot be resolved into rational factors we
know that it wiU not p'ive rational roots, .". Xj = 2, Xg = 4 are the
only vaiueti that meet tiie condition oi the problem.
168 simpJjE equations.
Any literal equation of the second, third, or fourth degree, ana
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 factor
should vanish, or else that the ppoduct of the remaining factors of
one of the expressions should be equal to the product of the
remaining factors pf the other expression, and it is sufficient if
one of these conditions be fulfilled. In symbols this is
li {x—a)f(x)=J^x—a)ip(x), :. x^ = a or f{x) = (p (x).
(B). If an equation reduces to the form [mx + n)'^ =c*
:. {mx+n)2-c2=0,
c—n
m
— c — n
lit,
(C). If an equation reduces to the form
(vix-^n] 2 ii''^
\px + q) ~ b'^'
qa — nb — qa — nb
thenic, = — i ' «o = — , , (See Exs. 4 and 5 above).
^ mb—pa ^ mo -{-pa ^ '
(I)). If an equation appears under the form
{a — x){x — b) = c, (1)
then Xy =\{(i-\-b^r), x.^ =\{ti-\-b — r),
in which r'^ = {ci — bY - 4^'-
From the identity (a — x)-\-{x — b)=:a — b
we get {a-x)^+2{a-x){x-b) + {x-by = {a-b)^ (2)
(2) -4(1) .-. {a-x)^-2{a-x)[x-b) + {x-by
= ia — by^ — 4:c~r'^ say
.-. {[a-x)-{x-b)]'^-r^ = 0,
:. {{a—x^)-{xi-b)]-\-r = 0, and .-. jj^ =4-(rt+ft + '-") ;
or {{a-x^) — {x^—b))-r = 0, Q.n(i .-. x^=^{a-\-b- r).
{_)nx^-\-n) — c = 0 and .". x^
or {mx.,-\-n)-\-c = 0 and .". x^
SIMPI.E EQUATIONS. IG'J
ft , 1 1 11
X a ax
... = Applying .(.4),
1 ax
.•. X — a = 0, or rtx = l>
1
a
0. {x-{-a + b){x + h + c) = {x-Za^h)y'lx-da^1b-c)',
x+a-{-b 2x-3a+2i-c
x — oa + 6 x-\-h + c
X — Aa-\-b — c
~ 'da-\-c
2(a; — a+6) x-a->rh
Page 122, (5).
*■ x—Qii-{-b 3a4-c
,-. (J) x^-a — h
\{x.^—3a-\-b) = 'd<i'{-c :. x^=9a-b-\-1c.
10 ^ ^ — — = — > .'. — = n
x^ - 'Ix b m{x + -l)'->rn[x^-1x) ma+nb ^^
But (C) cau be applied if //t aud ?i are so determined that
vi{x+2,)^ + n{x^ — '2x) is a square.
Tills requires that 4im[vi+n) = {2i)i, — n)^,
.*. S)n = n.
Assume m = l, then n—S, and (1) becomes, on substitution and
redaction,
(g + 2)2 a__ ^
(c3x— 2)3 - a + 86~' ' '^^y
_ 2(L-M _2(/-^l)
•■• ■^'^ ~ V-1 ' •^■'~"'l + 3/-'
(a; + l)^ _ _1 (•g^+2.c + l)3 _ _«
^^' (x3 + l)"(x-l)2 ~ b' •'• {x'' +l){x^ ~'lx+l) ~ b'
For .<;- -|-1 write xz
(.t-g4-2.f)3 _ a_ ^ (z-t-2)3 _ a_
xz{xz — '2xx) ~ b ' ' z[Z — '1} b
170 SIMPLE EQUATIONS.
This 'equation was solved in Ex. 10, hence z may be treated as
known.
^ , X2 + 1 .a;3_[_2a;+l 2 + 2
cut =2, .•. -s X — — r = Fj-
- — , = >c, a formed solved in (C).
12. (a-a;)4 4-(6-a04-c. ,
In the identity
LetM = ffl — X, v = x — b, .•. if+i' = rt- 6 and M^+'iJ* = c,
.;. {a-h)^^c + i{a-by{a-x){x-b}-2{a-x)^x-b)^
Write z for {a — x){x - b)
.-. z2-2(a-i)2z + (a-6)4 = i|c + (a_i)4}=:«3^ say,
,'. ■!z-(a-&)2}2 = j2
.-. by {B) Zi = {a — b)^-t;z2={a—by + t,.: z is known ;
But {a — x)(x- b)-z
.'. by(D) a:,=l(a + h+)^;x^ = U^+b-r) (1).
in which r^ = {a — b)^ - 4z,
or (a-b)^ - 4{{a -b)^ +t} = -it- 'B{a-b)2 1 ^^
and«2 = i{c + (a-Z^)4}. (3)
Hence x is expressed in terms of a, b, and r,
r is expressed in terms of a, b, and i,
t is expressed in terms of a, b, and c,
i,nd the expressions for r and t are cases of (B).
13. (a-a;)(6 + :c)4 + (a-a;)4(6+a:)=a6(rt3-f/;3)
Let a — a; = « -z and 6-|-a; = ?i + z .". n = ^(u + b) (!)•
The equation reduces to
(»2_z2){(w 4 z)3_|-(n-5)3}=ai{a34-6S)
/. (n2-22)(2w3 + G«z3) = «i((f3+i3)
SI?.tPLE EQUATIONS. 171
z~ may now be fonud by (/)), and from the result z may be
found by (/>'), and from (1) x--\{ti "h)+z ;
82^ = -I (a - b) 2 or' I ( 1 Oab -a^-b^)
.'. x = 0, 01- a -b, ov l{n-h) + ^x/(3()ab-'6a^-dl>^).
14. {y{„^x)+ y{a-x)\^V{o + x)+ y{a-x}=2cx.
Divide tlie terms of the identity
\/{a -f x) 4 - y (<f -x)* = 2x
by the corresponding terms of the equation,
•'* \ U- J ~ f-i' ■*' c-x ~ \c-i/ '
(r + l)4_(c_l)4
•• ^ - «-(c4-i)4 + (c_])4-
15. f/{'>-xy- + f/{{a-x){b-x)} + f/{b-xf = f/(a2+ab + b^)
Divide the terms of the identity
f/(a-x)^-f^{b-x)^=a-b
by the corresponding terms of the equation.
Cube, using the form (u—v)^ =u^ —v^—Buv(ii — v).
a-b
(a-x)-{b-x)- S^{{a-x){b-x)}
ir(«2 +«*+>)
(a~/>)3 _ _8ab{a-b)
~ a^-hab + b^ a^j^ab + b-
.-. ^Ua-x)ib-x)}=
:. (a -x){b — x)=
a form solved m (D).
^^- V{a-x)-^/{x-b) ~ ^^
Assume -i/{a -x)=z ^/ {x -- b)
.: {a-x) + (x-b} = iz^-{-l){x-b).
a-b
,*. X- b = .,-7^-
172 SniPLE EQUATIONS.
The proposed equation now becomes
{x-b){z-iy
(Z + 1)4 C
a form solved in Ex. 11.
•• (22 + l)(«-l)2 a-//
17. {x-2){x-5){x-Q){x-9)-L(;j + 2){y-i){y-5){y-ll) +
{z+l)[z + 5){z+S){z+l2}=x{x-i){x-7){x-ll) +
{y-{.l)(y-l){y-8){lf-10)-\-{z-j-2){z + B)iz + 10){z + ll).
Let a: ' = 3:3-11.7:, y' = y^-%y and «' = 22 _|_132,
.-. (a:'-|-18i(a;'+30)+(?/'-22)(7/'+20)H-(z'+12)(z'+40) =
^'(:«'+28) + (y'-10)(2/' + 8)+(z' + 22)(5' + 30)
... a;'2 4-48a;' + 540 + ?/'2-2?/'— 440+2'2 + 52s' + 480 =
a:'2 + 28a:' +)/'3-2//'- 80+z'2 + 52z' + 660,
.♦. 20x' = 0, .-. a;2-lla; = 0, .". iCj =0, jCg = H.
Exercise Ivi.
What can you deduce from the following statements ?
1. A'B = 0. 2. A-B-C=0. 3. (a-i)^ = 0. 4. 12a;!/ = 0.
5. What is the difference between the equation
(a;-5?/)(a;-4?/+3) = 0
and the simultaneous equations
a* — 5^ = 0 and a; — 4?/ + 3 = 0.
What values of x will satisfy the following equations ?
6. x{x~a) = 0. 7. «a:(a;+i) = 0. 8. {x-a){hx-c) = 0.
9. fla;2 = 3ra. 10. x^ = {a-\-h)x. 11. .tO-rS _a2) = 0.
12. rt2a;3 = ft2a;. 13. a;^ +(a-a;)2 =flr2.
14. a;2+(a-a;)2 = (a-2.r)2. 15. {a-xY + {x-bY =a^ ^h^ .
16. {a—x){x-h) + ah = Q.
17. (a-x-)2-(a-a;)(a;-?>) + (a;-ft)2=a2+flZ, + 63.
18. x^-{a-h)7-'-(0' = 0.
19. a-3-(rt+6+c)a;2 , a&-f 6c+m)a;— a^c = 0.
SniPLE KQUATIONS. 173
If ar must he positive, what value or values of z .vill satisfy the
following equations ?
20. (x-rj)(x-Jri) = 0. 21. a:2 + 29x-30 = 0.
22. a;2-17.c-84 = 0. 23. 3a;2 + 10x+3 = 0.
24. rJ -13x^+36 = 0. 25. a;3 -2a;2 - ox+6 = 0.
Solve the following equations :
26. (a-xy-+(x-by={a-h)^.
27. {a-x)^-{a-x){x-b)-h-{x-hy = (a-h)i
28. a2(a-x)^=b2{b-xy. 29. a2{b~xy^ =b2(a-xy.
80. {x~a)^+(a~hy + {b-x)^=0. 31. (x-1)- =a(rc3 -1).
oQ rt — a: x — a o^ «+&-« a — c-\-x
x — b c-{-x a — c — x ~ fl-f-c— a;
34. (x-a + b)(x-a+c) = {a-b)^-x^.
35. (x-fl)2-i24-(«-f6-a;)(6+c-a;) = 0.
36. (a+/>+c)j;3-(2a+6+c)a;+« = 0.
87 ^+^~^ _ fl + 6 — c
c a;
38. (^-x)2-f(a-6)3 = (rt + 6-2a-)2.
39. ar(a+i-a;)+(« + ft + Of = 0.
40. {n—p)z^ + (p—m)x+m — n = 0.
., rta;2— &a;+c _ c .„ ax^ — bx-^c __ a-h-^e
nix^ — nz+p p mx^ —vx-tp m — n-}-p
43. 4a;2 +0-2 _ ^2 _ 2(a + h)x = {a- x)(/; + x) - {a-\-x){b-x).
44. (2ffl-fc-a:)2+9(«-&)2 = (rt+?)-2a;)2.
45. {2a-\-2c-xY = {2b+x){Sa-b-{-Sc - 2x).
46. (3rt-5Z^+x)(5rt-3i-a;) = (7a-6-3a:)2.
47. (3a-5+.r)(33 + />-a:) = (5a+3i-3a;)2.
48. a{a-b)-b{a-c)x-{-c{b — c)x^=0.
174 SniPT.E EQUATIONS.
49. (ab-^hc + rn)(x^ + x-\-l)-\- [n—b]^ = {2ac -^b2){x^ + x + 1)
-T (a — r)^x.
no. (,r + l)(.'; + 3)(.r-4)(.r-7) + r.r-l)(.'?;-3)fx+4)(r + 7) = 96.
51. (x-l){.c + S){x-b){x-i-d) + {x-{-l)ix-'6){x+5)x^v}-t- IB
= 0.
52.
1
a;-f — = 3i.
X
53.
1 a+h a-h
X a—0 u+b
54.
1 a
X b
b
a
55.
a + x ^>+^_^-^
b + x a-\-x "2
66.
a—x x—b
X— b a — X ~
13
67.
a -X b-\-x
b-i- X a — x
m n
n m
58.
a X m
X ' (. ~ n
59.
x^ -\-ax+ a^
x^ —ax+(i^
60.
x-+n^
61.
X- +^'^
X- —(ix + a~
62.
ia — xMx — b)
5
63.
a—x x—h
x—b a—x ~
m.
n
64.
65.
66.
(x + a)^+{x-b)^ a^ + b'
{x+ay-{x-by 2ab
(a — x)^ —(x — b)^ iab
{a-x)ix-b} (a^-b-)
(a-x)^ + {a-x)(x-b) + (x-h)2 _ 49
{a-x)^ -{a-x){x-b)~+{x^b)^ " 19"
2a^ + n{a-x)-\-{a + x)^ c + 1
67. o 9 I — 7 — I — TTT ^ = ?• (Also for c = 5).
'za^-\-a{a+x) + {a—x)^ c — 1 ^ '
68. (5-a;)4+(2-a;)4 = 17.
69. x4 + (a-x)4=c; a;4 + (a;-4)4 = 82.
70. {n-xY^{x-bY = {a-bY. 71. {a -xY^-{x-lY ^c.
72. x-^^{a-xY^a^ ; ^- "• -^ , 6 - a;) -^ = 1056.
73. (a-a;)^(a;-fc)2 + (a-a;)2(a:-6)3 =an^(a-h).
SIMPLE EQUATIONS. 175
7-k. {a-x){h+x)^ -^(a-x)^{b + xy- + {a-xy^{b-{-xy^ +
^"- {a-xy^+{x-by' - 20 ^"-^''^•
(a-x)^+{x-hy _ 211
"^G. -^^_^)4 + (^_i)4 - 1,7- («-^)-
7/.
78.
70.
(a-.X-)4_|,(a;_Z,)4 ^ a4+fe4
(a — a;)5 + (a; — ft)^ a*— i
(a-x)^ (b-x)^ ff3 ^3
80. —, 4- ^ = -T + —
b -X a—x b a
a—x x—b a b
{n-xy + (x-b)i' _ a^h^
{a + b-'2xj^ ~ [a+b)^'
(a-x)-^-{-(x-b)^ a^-b'^
{a + iy^x)"~ ~ 0H^T)3'
82.
83.
(a-x) — (ic-i) ~ («— a;)(x — i)
^- A__Z; 1 = c(a—x)(x — b).
87 {«-^)^M^-W ^ _ J
(a-xj-^+liC-i)* (a-x-)(a;-i)'
88. ^l+x2)» = (x3-3)3.
85.
86.
176 SlilPi^E EQUATIONS.
x^ + 1 _ a_ QQ (a; + l)^(a;S4-l) ±
2^iy^4-l) - 6' ' {x-l)Hx^--x-\-l) " 6
91 (^-1)'-^ _ i!. g.. _(5c^±^+l)i_ ^ ^
• (a;2-a;+l)"2 ~ /> ' "' (a;+l)3(a:2 + 1) 6'
93 >!+llL _ iL 94 i^tm = ^
x-(a;+l)2 ^ ' ■ x(a;-+l) b
g- ic(x + l)3 (J a;2_^ 2.4.1 x^x—1 _ a
~{x-l)^ ^ T' ' (a;+l)- ■ I^-l)-~ ~ b
97 ^^~a;^ + l _ _«_ 93 x(x^ + l) a_
(x^-l)^ ~ b' ' ' (x'^-l)- ~ b'
99 (^ + 1X^^ + 1) ^ iQQ (^ + lH.g^-l) ^ _^
* (u;-l)(x3-l) * 6' * '{x-l){x'^ + l) b
101 (-^ + ^1' = -^ 102 ^^+^)' = ^.
x^ + l b' x^-hl h
103. 2(a-a-)4-9(a-a;)3(x-i) + 14(a-x)-'(.i;-6)3-
9(a - «)(.« - i)3 +2(x- i)* = 0.
104. 4(a-a;)*-17(a-u;) = (a;-6)2+4(a;-/>)4 = 0.
Find the rational roots in the following equations :
105. x-t - 1 2^3 -H40.c--78u;+ 40 = 0. [lucx. z^x-^ ~Zx],
106. x'^-Qx^+lx^+Qx-S.
107. x4- 10x3 + 35x3 -o0x+24 = 0.
,108. 32x^-48x3- 10x3 + 21x+ 5 = 0.
109. x3- 6x3 + 5x4-12 = 0.
4 9 4 5^
a ic — 2a X— oa x— 4a
5 4_ _ 14 5
x-rO X— 4 X— 55 x — 40 x-
x+8a X
X — a X— 2a ~
a;+5t/ 2x — oc*
110.
5 4
X X-
111.
14
x+20 "^
112.
2x4- 5a
X
X — (I
<c — '6a
177
113.
x-^ri x-f2 x+4: x4-S _ x-1 x-S
x-\-'2 ' X x — \ ~ X — 2 x — 'd x — o
,., 1 31 20 8 20 31 ,
114. _ 4- _L — _L _ -U
x x-l ^ x-2 ^ x-'S ^ x-4: x-5 '
x — b
116 i/(^i±^Hv>izM =
l/{a^-^2x)-'-i/{a-^-2x)
??>-x3 j/( OT2£-f2H-i/ ( m 2 .r - 2 ^
117. '/(a.'2-fl.2)4-^(x2_i2) + ^(a;3_^3) = ^.
118. {V(a-a-)+V(&-ia;)}{V(«-iK)-V(6-x)}»«.
119 r('^-x)-f^(x-b) _ a + b-2x
■^^{a-x) + f/{x-b) a-h
120. V(a+x) + V(a-a;)=y(2a).
[Write' u lor -^(a - x), and v lor f /(x — 6)] .
178
SIMULTANEOUS EQUATIONS.
CHAPTEE VI.
Simultaneous Equations.
Art. XLVI. There are three general methods 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 multi-
pliers.
In applying the elimination-method the work should be done
with detached coeiEcieuts, each equation should be numbered,
and a register of the operations performed should be kept.
Ex. Resolve
v-j-v+x + y+z
= 15
u+2v + 4:X+8i/-{-16z=^.
57.
u+3v+9x-^
27:
/ + 81z =
179.
«-Mi!-hl6x+64//-f-
256z
' = 45S
,
tt-f 5t'+25x-fl
25?/-f62
:52 = 9^
■5.
Eegister
71
V
X
.'/
2
1
1
1
1
1 =
= 15 (1)
i
t
4
8
16
57 (2)
1
3
9
27
81
179 (3)
i.
4
16
64
256
453 (4)
1
5
25
125
625
975 (5)
(2)-(l).
1
3
7
15
42 (6)
(3) -(2).
1
5
19
65
122 (7)
(4) -(3).
1
7
37
175
274 (8)
(-5) -(4).
1
9
61
369
522 (9)
!7)-(6).
2
12
50
80 (10)
(8) -(7).
2
18
110
152 (11)
(9) -(8).
2
24
194
248 (12)
(11) -(10).
6
60
72 {!?>)
(12). (11).
6
84
9G (14)
(14)- (13).
24
24 (15)
(15)-f-24.
I
1 (16)
^{(13)-60(lfi)}.
1
2 (17)
^[(10)-{12(17) +
50(10)}].
1
3 (18)
(6)-{3(18)-4-7(17
l + iT;! :(?)}.
1
4 (19)
(l)-{(19)-f(18) + (17)-i-(lG)|.
]
6 (20)
SIMULTANEOUS EQUATIONS. 179
An examination of the Register will show how easy it would
have heeu to shorten the process, thus (10) is (7) — (0) which is
(3) + (l)-2(2); similarly (11) is (4) + (2)-2^a;; .-.(1^)13(4) +
8(2) - 3(3) -(1), &c.
A general systematic arrangement of *^he elimination-method
will be given in Part II. For two or three simultaneous ec[ua-
tions it may be stated as follows.
a^x+b^ij+c'i =0
<i<^x-\-l).2!/-{-c.-^—0.
Arrange the coefficients thus —
«1 />! Cj «j
tto ^3 Cg rtg.
Form their products diagonally from left to right downwards,
thus "l^2 ^1^2 ''l^S-
Form their products diagonally from right to left downwards,
thus — ^l«2 "^l^S ^1^3.
Subtract the latter products in order from the former, thus —
a^b^—b^a^, b^c^—Cyb.-,, c^a^—a^c^.
Divide the 2° and 3^ remainders by the 1° remainder, the first
quotient will be the value of x, the second quotient will be the
value of y.
[Writing R-^, A'g, h^ for the three * remainders ' respectively,
the general result is {tnx-i-nij)R^ =mR^ +nR^ .
Ex. 1. Solve 11x4-5^-68 = 0
6x-ry+Bl=0
11 5 -68 11
0-7 31 0
-77
155
-408
30
476
341
- lijT)
-321
-749
3
7
1«0
SI
JIULTANEOUt
Ex. 2. 1?
X
25
y
_ 1
22
X
80
+ —
y
= 17.
12 -5
15
- 1 12
22 30
-17 22
360
425
— 22
— 550
-30
-204
910)
455
182
1
1
Y
T
li
11
1
1
X
y
.-. x = 2
and y -
-5.
2° Let the equations be
ArrflTifTfi the coefficients thus
"1
Ji Oj — t/j —a, — r»j
^3 '^■2 —^2 -^'a ""^^z
«3 />3 Cg -(/g -^3 — '>3
«! 6| Cj — cZ, — aj — /^,
«2 ^2 ^2 ""^3 "'^'2 ^2*
Selecting the first three cohimns form the ^agonal products
from lel't to right downward;^-, thus :
SIMULTANEOUS EQUATIONS. 181
a, b^ c^
\
\
giving
uJk.c.^
a., 60 C3
a.,0._^ci
\ '\
«3 '^3 ''3
a^O^c^
\ \
«1 ^1 '^l
\
«2 ''2 <^2
Form the diagonal products from right to left downwards, thus:
«i ^, (-1 giving c, 6. a 3
/
«2 ^3 <^a '^S^S'*!
/ /
«3 ''3 '^S C3^l"»
/ /
a^ b^ c,
/
flo ''2 '^2
From the sum of the former products talre the sum of the latter
products obtaining a remainder, which call it^.
Similarly form a 2° remainder, Pt,^ from the 2°, 3° and 4° columns
a 3° " A'g " 3°, 4° and 5°
a 4° " i24 " 4°, 5°andG°
Thena; = J?2-^2?i, 7j = B^-^E^, z = E^-^n^,
and generally I
Ex. 3. 3x4-2?/-4^+20 = 0
5.c-7y-G2- 1 = 0
7«+5?/ + 5z-24 = 0.
182 SIMULTANEOUS EQUATION:
3
2
-4
-20
-3
— 2
5
-7
-6
1
-5
7
7
5
5
24
-7
— 5
3
2
-4
-20
-3
- 2
6
-7
-6
1
-5
7
105 -288 28 -500 (3x -7x5= -105
100 700 432 14 5 < ox -4 =100
84-20 500-504 7x2x-G=- tt4, &c)
196
-90
-280 392
i 960
-990
196 600
-90 10
50 672
- 15
480
-840
240 (
980
15
-4:X —7x7 = :
-6x5x3 = -
5x2x5 = 50
156 1282
-375
1235
-445) -890
"Hsf-
^2225
2
-3
ii
5
z
«
Exercise Ivii.
Solve the following systems of equations :
1. 2a;4-S// = 41 2. 5x+77/ = 17
3a; + 2// = 39 lx-5i/= 9.
3. 11a; + 12// =100
9;« + 8// = 80.
4. 18a;
15;c
-35//-f-13 = 0
+ 28?/ -275 = 0.
5. 3a; +7// =-7
5x + Si/= -36.
6. 3a;+16//-5 = 0
28// = 5x+19.
7. 5a;+3//4-2 = 0
8.^+2/7 + 1 = 0
I
8. 21a;
23//
+ 8// + 66 = 0
-28a; + 13 = 0,
9. 10a;-l-7//+4 = 0
6a; + 5// + 2 = 0.
10. 23a; + 15//-4i = 0
82.f+21//-6 = 0.
11. ia;+l// = 6.
3a; -4// = 4.
12. 4a;-
SIMULTANEOUS EQUATIONS. 188
13. ^y = l7-l. 14. ^x+pj = n.
i>j=lx-l. fa:+|^ = 19.
15. l-Sx-2y = l. IG. 7a:=:107/-f 1.
2-5x-Sy = G. nx=lGy + -l.
17. Bx-4:y+l = 0. 18. ■lGx-04:y = l.
l-7x-2-2y-{-7-d = 0. •19j;--lli/ = l.
19. 3-5x + 2i7/=.13+4|rc-3-5?/.
2ja;+'8^ = 22^+-7a;- 3^.V.
21.
23.
20.
1 1
X y ~
5
1 1
1
X y
6
22.
1-6 2-7
X ~ y ~
1.
•8 3-6
X y ~
5.
24.
X 5
T + 7 =
4^
a; 10
T + 7 =
21-
26.
|a;-^(?/ + l)-
rl.
i(a:+l)H-i(2/-
-1) = 9.
25.
27.
28. = -^^ 29.
Ax-Z ly-Q
80. l^i^2 := 8 31.
45-?/
[S4: SIMUIiTANEOUS EQUATIONS.
32. ' -^ = A. 83.
2x-y + l
Sx-y + 1
x-jf+'B
x+1 y+2
3 4
^x-y)
6
x-2, y-n
4 3 =
2y-x.
2
34. _^_i^(±lL ^ 30. 85.
•4x+-5?/ — 2-5
•8x+.-l2/+-6 _ 1
5x+3^-23 ~ T*
86 ?:^LzI(±_^ _ ^--y+^ _ 4.
"^ ' 3 4 '
3a; -47/+ 3 4a;-2v-9
' 4- — ■ '- = 4.
4^3
37. 20(x+l) = I5{y + 1) = 12(x+y).
38. (x-2) : (7/ + I) : {.c+y-V>) :: 3 : 4 : 5,
39. (x-5) : (.v+9) : (:«+?/ + 4) :: 1 : 2 : 3.
40. "^ = ''l±^. 41. (r-4)0y+7) = (.r-3)(2/+4).
a + l 2/ + 5
^•^-^ ^^-^ (x + 5)0/-2)=:(a-+2)(2/-l).
2(^ + 1) 5i/+7
42. {x-l){5>/-3) = B{Sx + l). 43. (a; + l)(2// + l) = 5x+9?/ + l.
(x-l)(4y + 3) = 3(7a;-l). (a;+2)(8i/+l) = 9x+13i/ + 2
44. idx-2){5y-hl) = {5x-l){y + 2).
(3x-l)(,// + 5) = {.i-+5)(7i/-l).
45. «^7/ = 37. 4G. 2.r + 2?/ = 7.
y+z = 25. 7.r + ;9z = 29.
z + a; = 22. 7/+82;=17.
47. l-3a;-l-9y = l. 48- 5x+3?/+2z = 217.
17//-l-lz = 2. 5a;-3?/ = 39.
2-9z-2-la; = 3. 8y-2z- 20.
49. ;^a:--:i7/ = 0. CO. Ilx+lhj = l0.
lx-lz = l. 2lx + Q.}z = 20.
lz-iy = 2. 8i^+3.:2 = 30.
SIMULTANEOUS EQUATIONS.
195
51.
x-hy-z = n.
y-\-z-x= IB.
z-[-x-y= 7.
62.
x+y+z = 9.
a;+2?/ + 4z = lo.
x+3//+y2 = 23.
53.
x+y+z= 3.
54.
7x+Gy+7z = 100.
2x+4y+ 83:
= 13.
a;-2//+ 2 = 0.
'dx+dy+21z--
= 34.
Bx+ y-2z = 0.
55.
3x+2// + 3z =
110.
66.
x-\-y+z = 9.
5a; -f ?/— 42 =
0.
x+2y + 3z = U.
2x-By+ z =
0.
x+3//+62 = 20.
57.
a;+2»/ + 32 = 32.
58.
x+y+2z = B4:.
'2x4-By+z = '
12.
a;-f2?/+2=33.
Sx+ y-r2z---
:40.
'2z+y + z = B'2.
59.
Sx + Bij+ t =
= 17.
60.
x + 2y-z= 4-6.-
Bx+ 2/ + 3z =
= 15.
^ + 22-x = 10-l.
x + 3i/ + 3z =
= 13.
z+2x-?/= 5-7.
61.
a; 4- 2.'/ --72:
= 21.
62.
x+y = nz+8.
3x + '2y- z
= 24.
y-\-z^2ly-U.
•9x+7y-22
= 27.
z+x = B^x-B2.
63.
'^x+y + \z =
= 36^.
64.
2lx+By + ^z = U{i,
ix+\y+lz--
= 27.
dlx+^y + 5iz= 17^,
^x+y+\z--
= 18.
2|x+3|?/+4|2 = loV.
65.
y+l
z+1 ~ •
z+3
x+1 " ^•
66.
Bx+y _
■ z + 1 = ^
By+z _
x+1 - ^•
82 + . ^_ ^
y+l
67.
■ 2/-Z
■It! = 9.
a;-//
I/+2 _ J
68.
^+' _ 2.
y + z
y±^ = 1.
x + z
196 SIMULTANEOUS EQUATIONS.
69. ^ - ^ = 1. 70. 1 + Jl + ji = 4.
^ y X y z
2.3 385
— - ~7 = '^- — + — + — = 4.
2 X y z
1 „ 5 12 10
4
3
c
y
2
3
X
z
3
1
y
2
xy
x + y
1
5
yz
y+z
1
~ 6"
zx
1
z-\-x
7
71. - = — • 72.
6 4
— + —
X y
5
+ —
z
3 8
— + —
X y
5
+ —
3
.3_^lr
10
X y
z
xy
hj-'dx "
20.
xz
2x-€7. ^'
15.
yz
4y — 02
15*.
73. (x4-2)(%-i-l) = (2a:+7)?/.
Cr-2)(32«^l) = (a:+3)(82-l).
(7/ + l)l2 + l' =(7/ + 3)( 24-1).
74. {2x-l){y + \\ =2{x + l){y-l).
(.« + 4)(2+l) =(.r+2)(z + 2).
(^-2)(2+8) -(y-l)(2 + l).
75. (.r4-l)(52/-3)=:(7.r+l)(2y-3).
(.Ix- l)(2+l) = (:c+l)(2z-l).
(//+3)(z + 2) =(3//-6)(32-l).
76. 21«+31y+42z=115.
6(2a;+2/) = 3(3x+s) = 2(2/+2)
77. 15(a;-22/) = 5(2a;-3«) = 3(y/-f2).
21x-|-3l7/ + 4l2 = l&.n.
78. Qx{y+z) =. Mj {z-k-x) = Bz{x ^- \/j.
79.
1 1 1
— + — + — =
X y z
= 9.
3a; + ?/+z = 20.
60.
.r+2-f8.y-33
3'^4-a;^-4^/ = 30.,
5u + y-\-z = li.
%u^Qx-\-z^AO.
5?t+8(/+;:2=50
4»4-a;4-2 = lL
3?c ^i' + ?/ = li.
SIMULTANEOUS EQUATIONS. 187
81. u-{-x + rj + z = lU. 82. «4-a-4-i/+2 = 2-i.
tt+2.c+2y + 2z = 267. u+2x+Si/-dz = 0.
ii-\-2x + 'Bi/-\-3z = o5d. 'du-x- 5t/+z = 0.
z<+2x- + 3//+4z = 410. 2i(-^Sx-Ay-^z = 0.
38. u + x+i/+z=GO. 84. u+x + y-^z^l.
M+2x-|-3^4-4z = 100. 2u+ix + Si/ + 16z = 5.
u+Sx+Gu + 10z = 150. 3jH-9x-+27»/ + 81z = 15.
u + 4x+10?/ + 202 = 210. 4«+lG.c + 64y/ + 25Gz = 35.
B5. U+i!J-lz = l. 86. lu-lx-h-ly-]^^^l.
i-c-iy--^» = i. i«+i^4-iy-^i^ = 37.
lx+'iz-lu = l. ^u-:^x+\y-^r. = l7.
Art. XL, VI I. The principle of sj'mmetry is often of use in
the solution of sj-mmetrical equations. For from one relation
which may be found to exist between two or more of the letters
involved, other relations may be derived by symmetry ; also,
when the value of one of the unknown quantities has been deter-
mined, the values of the others can be at once written down, &c.
1. {x-{-y){x+z) = a.
{x+y)(y+z) = b.
{x+z){y+z)=c.
Multiply the equations together and extract the square root,
•■• {x-\-y){i/+z){z+x) = i/{abc).
Divide this equation by the third.
.•. x+y = V_l^_£i, and therefore, by symmetry,
c
l/(cibc)
a
b
Kence we get
_ nb — bc-\-ca
2\/{abc}
whence y and z may be derived by symmetry.
188 SIMULTANEOUS EQUATIONS.
2. x+y-\-z = 0 (1).
ax-\-bi/ + cz = 0 (2).
bcx+cai/-\-ahz-\-{a — b)(h — c){c — a) = 0 (3).
<;x(l)-(2) gives (c — a)a;+(c-% = 0.
.-. y = \ L, and similarly,
b—c
2 = (a-b)x
b-c
Substitute in (3) these. values of y and z, and reduce,
.-. x{a — l){c—a) = (a — b)(b — c){c — a),
.'. ovx = {b — c), .'. y = c — a, z = a — b.
8. a[yz— zx — xy) = b (zx — xy — yz) = c {xy — yz — zx) = xyz.
Divide the first and the last equations by axyz ;
. ■. — ^ s; __ _ — _ — , and hence, by symmetry,
axyz
Jl _ ^ i_ i-
b y z X
1 _ 2. i_ 1
c z X y
,'. -i- — = — — , and by symmetry,
be X
L L - A
c a y
1 i_ _ _ A.
a b z
i. ((x-tby + cz = l (1)
a^-x-{-b^-y+c^z = l (2).
a^x + bhj+c^z = l (3).
cx{l) — (2) gives a{c — a)x+b{c — l>)y = c-l (4).
cx(2)-(3) " a^{c-a)x-¥b^{c-b)y = c-l (5).
6x(4)-(5) " ah{c-a)x-a^{c-a)x = h{c-l)-{c-l),
or a[a — b)[a — c)x= {c — 1)(6— 1),
... ^ = {l-h){l-c),
a{a — b)ia—c)
Tvlien4je y and z may be derived by symmetry.
SIMULTANBOrS EQUATIONS.
189
6. Eliminate x, y, z, u (which are supposed all different) from
the following equations :
x = bij-\-rz-{-cIil.
y = cz-\-du + ax.
t = du-^ax+by.
u = ax-\-l)i/-\-cz.
Subtracting the second equation from the first,
.•. x — y = by — ax, or
(1 +ri)x={l -f b)!/ = (by symmetry) (1 +c)z = (1 +d)u.
These relations may be also obtained by adding ax to both
members of the first equation, by, to both members of the second
aquation, &c.
Now divide the first equation by these equals.
1 h c d
1 + ^' ''
1
1 + b "^ 1+c "^ l + u'
And since = 1 — , we have
l-\-a 1+a
1 = I _L ^^
1-fd
Exercise Iviii.
1. Given ax -\- by = c
a'x+b'y = c'
2. Given hx — mj
dx-\-md = cy-\-ni
3. Given axi-by + rz — d.
a^x-\-})^y + c~z = d'^
and that a; =
b'c-bc'
b'a-ba'
derive the value of y
a(dm— en)
and that x = —i — -, — »
be — ad
derive the value of y.
and that x =
a{d-b)(d-e) .^ ^
, write down
d{a — b){a — c)
ai^x+l'^y -hc^z~d^ the values of ?/ and z.
190
SIMULTANKOUS EQUATIONS.
4. Tliere is a set of equations in x, y, z, u, and ic, witt corres
pending coefiacients (a tea;, &c.), a, b, c, d, aude; one of tiif
eciuations is
ic = lJl/ + cz-i-d>( + (nv, write down the otliei*3.
Solve the following equations :
K ^ , ?/ '/ z , X z
'" n n p m. p
6. x+aij + hz = )n, y-j- az 4- hx = n, z f ax+hy -p.
7. x + ay = l, tj-Y(jz~iii, z-{-cit=it, ti. + dw=p, w + ex = r,
8. Eliminate x, y, z, (supposed to be all different) from tb{
following equations :
X — hy -f .■ z, ,j — cz + ax, Z — ax + hy.
0. Eliminate x, //, z, from
■^ = i ''■' - \, J_
y-\-z '' ' z-\-x ~ '* ^^+y ~ ^'
10. Having given
* = % -f ' 2 + '^» + fi'',
y = cz + <ln+eir-\-(ix,
Z =dii + en--\-ax + i>y.
u = CIV -\-ax-{- by + cz,
ic = a x-x-liy-\-iz-{- d tl ,
Shew that -— + — r + v-j- + ,-,-, + t-t- = 1.
. Art. XLVIII. Eesolution of Particular Systems of Liueai
Equations.
Ex. L x+y-\-z = a (1)
y+z + ti = b (2)
z-^u + x = c (3)
u+x+y = d (i)
(l) + (2) + (3) + (4) S{H + x-\-y-j-z) = a + h-^c-\.d (5')
3(1) Si^x + y-i-z) = 3a (6')
H(5')-(C')} w =H-2<,+/>+r+'/.)
SIMULTANEOUS EQUATIONS. 191
The values of x, y and z may now be written down by sym-
metry.
The following is a variation of the above method, ai)plicable to
a much more gfineval system.
Assume the auxihary equation
.-. (1) becomes s — u = a, (a\
(2) " s-x = h, (7)
(3) " !i~y = c, (8)
(4) " .s-z = f/, (9)
(5) + (G) + (7) + (8) + (0) is = s-\-n + h^c-^d.
s is now a known quantity, and may be treated as such,
in (G) giving ii = s — a
" (7) " x = s-b
«' (8) " y = s-c
" (9) " z = s-d.
Ex. 2. 7/2 = «((/ + z), (1)
zx = b{z+x), (2)
xi/ = c{x+ij), (3)
111
(D^aijz, 1 = _,
y z a
{2)^bzx^ 1 = -7-'
2 X h
111
{Z)---cxy, ^ _ = _.
x y c
This may now be solved like Ex. 1, using the reciprocals of a
6, c, X, y and z instead of these quantities themselves.
Ex. 3. a,M+i,(x+?/+z) = Cj (1)
a^x-\-h^{y+z + u) = c^ (2)
a^y + h^{z-\-u+x) = c^ (3)
<tiZ-^h^{n->rx-[-y) = c^' (4)
Assume the auxiliary equation
« + x+v/+z = s. (5)
192 SIMULTANEOUS EQUATIONS.
^1) becomes 5 ^s — (/!^j —rti)« = Ci
— s - a = (G)
Similarlyfrom(2) b-^.'-'^^hT^ C^)
(5) + (0) + (7) + (8) + (9) (^A_+^:^+ ^-^';s__^_*,_j .
C-i Co Co Cj
= s -f-1 — "^ — + -, -— +7 — ^— 4- -, — * — (10)
From (10) we can at once get the value of s, which may there-
fore be treated as a knowu quantity.
6 , s — c ^
m (G) giving u=, _
and tlie vakie of x, y, and 2 may be obtained from (7), (8j and
(9), or they nvay be written down by symmetry.
Ex. 4. . aa;+6(?/+2)=6- (1)
aii-\-l){z + iC) = d (2)
oz-\-h{u-\-x) = e (3)
au + h[x^,j)^f (4)
Assume u+x-\-y \ z = s (5)
(l) + (2) + (3)-f(4) (« + 26)s =c + J + .+/ (6)
Hence s is a known quantity and may be treated as such.
From (1) and (0) hs—hu-\-{a — h)x = c,
bu — (a — b)x=.bs—c, (7)
Similarly from (2) and (5) bx-{a — h)y = bs—d, (8)
«' (8) " " hy-{a-b)z=:hs-e, (9)
'■ (1) " " Lz~{a-b)u = bs-f, (10)
b{7) + (a - h){8) b~ H - {a -by-y== ahs -bz-{a- h)d,{ll)
fc(9) + (a-6)(10) b-^y-{a-bfu:=ahs-hi~{a-b)f,(\'l)
SIMVXTANEOUS EQtlATIOXS.
193
b-{U) + {a-b)^12) . {b^-{a-b)^}n = abs{h^ + {a-hy-}
-a{b2d+{a-b)y^}-b{b^{c-d) + {a-b)^e-f)} (13)
The values of x, y, and z may now be written down by sym-
metry.
Ex. 5. a^ + a^x-\-aij+z = ^.
The polynome t^ ^-xt^ -\-yt-\-z vanishes for t = a, t = b, t = G\
.*. by Th. II., p. 46, for all values of t.
t^+xt^-\-yt-\-z = {t-a){t-b){t-A
= t^ -{a + h-\-c)t- + (nb->rbc + ca)t-ahc.
^. Th. III., p. 53, x= -{'( + b + c),
y = ah-i-bc-{-ca,
2= —abc.
Ex. G. x+y+z + n=-[,
ax + liy-\-rz -\- da — 0,
a-x-\-b-2y + c^z + d2!i = 0,
a^x + b^y-\-c^z-\-d^ 21 = 0.
Employing the method of arbitrary multipliers,
(4)+Z(3) + w(2)+n(l) a^x+ b^'y+ c^ z 4- d^
+Z<|2j +/62| +ic2 +id-2
+ma + '.'ji + mc +md
-\-n \ ■{■ n\ + n -\- n
To determine x assume
c^+lc- -{-mc+n = 0,
d^+ld^-j-md + n = 0,
n
0)
(2)
(3)
(4)
(5)
(G)
(7)
(8)
(9)
a^ + la^ +ma-\-n
But the system (6), (7), (8) has been solved in Ex. 5, from
which it is seen that
1= —(^/)-^c + d), vi = bc + cd-\-dh, 71=. —bed,
and a^-\-a'l-^am-^n = [a —b){a~c){a — d\ :
194 SIMULTANEOUS EQUATIONS.
.'. using these values in (9)
— hcd
(^a — b)[a— c)[a — d)
The values of y, z and u may now be written down by sym-
metry.
Ex. 7. -±- + -^ + -^- = 1. (1)
+ _^ + _J_ = 1. (2)
n — b 71 — c
■+ ^ + ^^— = 1. (3)
p — 0 p — c
Assume 1 - -^ ^ _ ^_ ^ tS-^Bt^~ + Ct+D
t—a t — b t — c {t — a){t — b){t-cr
But in virtue of equations (1), (2) and (3), the first member of
(4) vanishes for t=rti, t = n, and t=p, and .'. t^+Bt^ + Ct+D
vanishes for the same values of t, and .•. Th. II. p. 46,
t^+£t" + Ct+D = {t—m){t—oi){t-p),
m
X
■a
n
X
a
P
—
a
X
.'. (4) becomes 1 —
y z
t — a t — b t — c
^^ (t-m){t-n){t-p)
~ (t-a){t-b)(t — c) '
To obtain the value of x multiply both sides of this equation
by («-«),
t-a-x- y^tZ^ _ <^~-^) ^ it-m)(t-n)(t-p)
t-b t-c {t-b){t-c)
Now t may have any value in this equation ; let i = a,
(a—m)(a — n)(a — p)
~ {a—b){a — c)
The substitution (x)jz\abc) will give the values of y and z,
Ex. 8. x+a ^ j/^ ^ z_+c_ ^^^
p q r
Ix + my + nz = s ^ (2)
SIMULXANEOUB EQUATIONS. 195
By Art. XXXYIL,
x+a ij + b z-\-c l.v-\-imj-\-?iz-\-Ja-}-tnb-\-nc
V ~ 1/ r lp-\-t:iq-\-nr
/ov s'^-\-la + nib + nc
i^J = — r— = R, say
Ip + mq + nr •'
,', x=pB, — a, ij = qR — b, z = rR—c,
Ex.9. yz+z-^-+x^={a+b + c)xyz (1)
yz + zx _ zx + xj/ _ xy+yz .^.
{l)^xg» A + -L _^ ^ ^ a^b^e (^)
(4)
Page 122 and (3)
a
b
c
1
X
+
1
+
1
z
=
a+6-h(?
+
1
y
=
1
+
1
z
1
_ 2
+
1
fl3
a
c
2
a;
+
2
2/
2
H
z
2
(5)
(4) and (5) .-. 1_ — = 2a, — + — = 2?>,
X y y z
- + - = 2c. (G)
z X
111
(3)_.(G) -~ = a-b + c, — =<t-\-h—c, y = -n-^rb + c.
E.. 10. £+f + »+^ = 2. (1)
a-\-b a\-c
x — by — c^ ,ns
+ -^—i = 2. (2)
a—c a—b
(1) . ^i _ 1 = 1 _ '^Jt^
a; — a — 6+c a + r — b — y
a-\-b ~ a+c
(3)
^^^ SIMULTANEOUS EQUATIONS.
Similarly from (2) x-a-b + c ^ g-h + c-y
a—c a—b
(3) and (4) ... x-a-h-^c = "L+^U-b+c-y)
a — c , , .
But unless _ J , this cannot be the case except for
a+c a-b k ^
a-b+c-y = 0,
in which ease x - a — b-^c = 0 also,
giving ■x = a-t-b — c Sind.y = a-b+c. (5)
Tj- (i + b a — c
^ ^7 = ^[ZTf ■'■ a^-b2=a^^-c^ (6)
^*2_c2=0, ov {b + c)(b~c)--.0,
b = c, or b= —c.
But if i = 4-c or — c, (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, 2ax={h + c-a){y-hz), (1)
2hy = {c-i-a-b)(z-[.x), (2)
(a; + i/ + z)-'+x2 + (/2-fz2=4(«2+5=i-f2) (3)
(1) and page 122^5) ?_ = JL±1_ _ «+i/±f (A^
b + c-a "la ~ b+c + a ^ '
(2) " «» ?/ _ a;+z _ z+y-\-z ,„.
c + rt-i ~ 2i ~ c-\-a-\-b ^°'
<,4), (5) and " ...^H:^ ^
y
a-f/y + f b-\-c — a c-^a-b a-j-b-c'
(x+y+z)^+x2+y2+z^
(6+c-a)3 {a + b + cy + {b + c -a)2 -^(c-i-u - by ^(a-\-b -c)2
^ ' 4(rt''-
Ked>,otionana(8)= (:E±SH^)l+i^,M:.: ^
^ ^ 4(rt2 4-i2^c2) -^*
SIMULTANEOUS EQUATIONS. 197
111
Ex. 12. flx =&7/ = c«= — + — + -— (1)
u^ y z
rt _ 6 _ c _ a+b + c
{l)^xyz :. — = ~ - ^ •• - ^yz+zx ^"^
a 1/1-1 1 \ xy+yz+zx ■' ^
Usofrom(l)-;i^^5 — = h— +— = ^^2T3- 3
^ -' "^ yz xyz \x y z j x-yz'
(2)x(3)
Ex. 10.
(1)
then
a-\-b-\-c
y^z^ x'^y'^^z'^
a^x'^ = a+bJrC.
y+z-x z+x-
a ~ b
-y x+y-z
c
(1)
xyz = vi^
(2)
z x
y "»
(3)
a+b ~ b-j-c ~
c + a' - r s^PPOS®
xyz
= 7i
(a-{-b){b + c){c + a)
.-. r^ = {a + b){b + c){c+a)
Hence the value of r is known and from (3)
rx—Vi{b+c).
Ex. 14. y-^z = 1axyz (1)
z-^x = 1bxyz (2)
x^y = 1cxyz _ (3)
• ' 1/+Z z-\-x x-\-y x+y + z
2a '2b 2c a + b+o
xyz
h-\-c — a c + a — b a-\-b — c
2 xyz
:. X y z = (i_,.c_a)(c + a-^)(a + i-c)
., ^ , 1
.. x-y «-= (^ij^(;-a){c+a-b){a + b-c)'
(1)
198 SBIULTANEO'JS EQUATIONS,
Hence tlie value of x~i/~z- is known, call it -^ and substitute
in (4)
1 X
^
r ~ b-\- c —a
:.
rx = b + c—a.
in wnich
r2 = {b-{-c-a)(c-\-a-b){a + b-c).
Ex. 15.
y^+z- -x(y + z) = a
(1)
z- -\-x^—y{z-\-x) = h
(2)
x'-+y^-z{x+y)=d
(3)
(l)+(2)+(3)
'2.{x-+y'^+z^—xy-yz — zx) = a-\-b-{-c
(4)
(1) may be written
a-3 -j-y'^ -\.z' —x[x + y-{-z) =a
(5)
(2)
x^+y^--i-z--y{x + y + z) = b
(6)
(3) -
x^+7/-+z"-z{x + y-i-z) = c
(7)
.-. x + y+i
^_a — & b — c c — a
y-x z-y x — z
.-. (x+y + z)'
^_ ^a-bY^^-[h-cY^{c-aY
[y-x)--\-(z-yY-^{x-zY
a;3_}-2/^+z2 -xy-yz — zx
(4)
1{a^+o-+c^-ab-'bc-ca)
a-\-b-\-c
(8)
2(a3 + 63+c3~3a6c)
(9)
Write r2 for 'l{a^ + h^+c' -dabc).
(9) ''■-+y^^ = -^j^ (10)
Eeturning to (8) {x+y+zY = '^^^'^^'t+V+t "'"'"^ ^'^
(4) 2{x"-+y^^z"^-xy-yz-zx)=i^^±t^ (11)
BIJIULTANEOUS EQUATIONS. 1^9
i{(8) + (ll)} ^2+,/=+3^ = ^+^I±''.^ (12)
(5) and (10) a;-+v-+s2- -— -- = n
a + b+G
(12) =a^+b'+c^-a{a+b-^c)
= b'^+c^-a{b+c).
(5), (6), (7) are symmetrical with respect to {xyz\abc); (10) sho-ws
this substitution does not affect r, and consequently the values
of y and z may be written down at once from that of x.
Exercise lix,
1. ax+hij = c, 2. ax-^hy-c,
vix + ny = d. mx — ny = d.
3. cix + by = c, 4 a; ?/ _
7?/a; 4" n ?/ = c. «. 6
x+y = c.
5. - + Jl = 1, 0. 4 + f =. 1.
a
6 ' ' a ' b
i + « ~ ■^' b ~ a'
7. rt« + 6c- = ft ;/ + ac. ** 1 '' _
8. — + — ^1
x + y =c. .- y
b a
— 4- — = n.
js y
9. {a + c)x—{a — c)y = 2ab, x—c _ a
(a'{-b)y-(a-b)x = 2ac. ' y-c ~ b
x — y = a — b.
X a 12 ^+?/ _ <^< + ft+g
~^ ~ 17' ' ?/+i ~ a — 6 + c
a; + »i c 2/-1 _ a-b-G
y^n ~ ~d' x-\-l ~ a-^rb — c
11.
200 SI.Ui;L.TAIvEOUS E:iUATIONS.
13. ^-=^' = ±. 14. "+' + y+t , 2.
y-a + h c a+b ^ a+c '
V + b^ _ c^+a x—b y-c
x + c ~ h + a a — c a — b ~ '
15.
__f_ , V _ 16. x+y+z = 0,
m-a "^ m-b ~ ' {b + c)x+{a-j-c)y+{ai-b)z
X y . =0.
n -a 11 — b ' bcx-{- acy ^abz = l.
17. x->ry+z = l, x-a y — b z-c
1 Q __ "
nx + by + cz:=m, P <1 ^
_j^__ , V__ , t_ _ l{x-a)-\-m{y-h)-\-n{z-c)
l-a ~^ l-b ^ I- c ~ =1.
--« __ y_^^ _ ^Zi ^^- (t{x-a) = b(y-h) = c{z-c),
P q, r ax + by-{-cz~m^»
lx-\-my-^;iz = l.
19.
21. x-\-y + z = a + b + c, 22. a;+2/+2 = 0,
6x+e?/ + r/2 = a2-j-i2^_(.2^ ax + by-irCz = ab-{-br + ca,
cx-{-ay+bz = a" -{-b" -\-c^ . (l,-. c)x + {c — a)y-{-{a -b)i
= 0.
23. x-{-y-{-z = m, 2-1. aa;-i-% + c2 = r,
X : y : z = 't : b : c. mx= my, qy=pz
25. a:y + ?/2 f ?.<; = 0, ays 4- hzx -f- ra;?/ = 0,
bcyz-\-acxz-\-abxy -j- (a — Z>) (6 — c) (c — rt)x//2 = 0.
26. {a + b)x+{b+c)y + {c+a)z = ab-\-bc+cay
(a+c)x + {a+b)y-\-(b+c)z = ab-{-ac+b<;^
(b-\-c)x + {a + c)y+{a + b)z = a-''+b^ + c^.
27. nix + 7iy-\-jz + qu = r,
x y z u
abed
28.
SIMULTANEOUS EQUATIONS.
p.r„,L~) 2/{x+z) z{x + >/)
201
1 .
a + b--c
111
— + h — = a + b + c.
X y z
29. (rt - h){x+c) -ay+hz ^ (c- a){y + h)-cz-hax = 0,
x+y+z^'^{cc-^h + c).
30.
^2.
84.
ar + by = l,
31.
Z?/ + 7?7.r = n,
by-\-cz= 1,
72a; + /z = w7.,
cz + ax=i.
mz+}iy = L
x+y = a.
33.
VI n
2/ + z-^ = -p'
y+z = b,
In
z-\-x~y = — ,
m
x-\-z~c.
Im
x + y-z = _,
71
I 1\
y *\
2a.
.V
1 1
2
a
1 1
z a;
26.
1
z
1 1
X y ~
2
T
1 1
aj y
2c.
1
a;
1 1
7/ z ~
2
c
86. (^H-''y^ + (a-5)^ = 2/;e,
(c -\-a'^z-\-{c — a)y = 2ab.
y +
= h.
38. 2- + -4- = ^-"^
X y
c+« a+b
202 SIlHiLTANEOUS EQUATION'S,
39. X-+.'/— 2 = ''', 40. u-i-v — x = a,
y+z — v—h, o+x—y = h,
z+v — x=c, x-t-y-z=c,
v-\-x — y=d y-\-x — u = (l,
z + u — v=:e.
Exercise Ix.
Resolve
1. {a + b)x + (a- b)y = 2{a- -\-b-) 2. x + y = a,
{a-h)x + {a+b)y = 2{a^-b'^) x- - y^ ^h.
3. 2x — dy = m, 4. {a-b)x-\-{a-\-b)y=^a-\-h.
2x^ —Sy" =7i^-\-xy. X y 1
a +b a — b ~ a-\-b
a + b + 1 • ^ {a-\-b — c)x — {a — b-^-c)y
= 4«(i — c),
.c a-\-b — c
y a — L-i-G
X — a a-~h
y — fi (1+6
X «^ —b^
\_u - u)x-r
■y- a+b '
a—b-i^l
» + (« + %= ^_5 •
•c+y
a
x-y
b-c
X+C
ai-b ~
y+b
x-y+1
x-y-l
= a.
x + y+l
x+y~l
= b.
x-y + 1
x+yl
- a.
x+y+1
x-y-l
= b.
{r) + r)x-{-
■ [a — c)y = 2ab,
(a+b)y-
(a— b)x = ^Aac.
8.
10.
y a^-\-b^
x + y + 1 «-t-l
x — y + 1 a — l'
x+y + 1 1 + b
x-y-l ~ 1-6
11. / . - a, 12. — _ + .^ = a+b.
x+v-1 , a-\-b a — b '
\- ^ — 2a.
a b
13. {o + n)x+{a-c)y = 2ab, 14. a^+ax-\-y = 0,
b^-{-hx + y = 0.
SIMULTANEOUS EQUATI0X3. 203
15. y + z-x = a, IG. ".r-{-ll>/ + z = a,
z + x-y = />, • 7^-(.iiz_(_^; = /,^
x+i/-z = v. 7z+llx+y = c.
X IJ z " ' •
L ^ L. _ 1. =<2,hc {c-a){y+h)-cz+ax = 0,
y ^ X " '
— + — — — = lea. ^ '
z X y
19. r^ + -^ =^a + h, 20. ^ + _^ ?_^0
b-\-c c—a h-irc c —a a- b *
.'/ , z X y z
+ 7 =b+c, — — :^ — 4- 0
c + a a—b b-c c~a ^ a-^b' •
z X X ri 4
a-\-b^b-c ^ b-\-c^ c-a^ a + b~^^
21. ^ + JL_ + _^L_ = 1 22. -3L
= a.
« a— 1 «— 2 . ' x+y
X y z ■ yz
c-1 c — 2 "'■• jj_f.a.
oo ^ V Z X 1/ Z
a i> c 0 c (c
X u z 1 1 1
c a 0 a b c
24. — = -^ = — - = — , 25. ax = by = rz = dii,
a l> c d
mx + ny-\-pz-{-qii = r . y2_23_3,_^^
26. y-\-z = au, 27. 3;+// = 771,
a; + -: = /)?«, ?/-|-z = j»,
x-{-y = cu, . z+t( = a,
1 — x a u — x==b.
204 SIMULTANEOUS EQUATIONS.
28. 11x+9i/'\-z-u = a, 2d. .c + mj + a"z+a^u+a* =0,
lly + Qz+u-z = b, • x+bi/-^b2z + b^u+b'^ = 0,
llz+9u+x — y = c, x+cy + c^z-\-c^u + C^ = 0,
llu-\-9x+y — z = d. x+dy + d-z + d^u+d^ = 0.
80. x + y = a, 31. x-{-ly=a,
y-bz = b, y + mz = b,
z+u = c, z + nu = c.
u -\-v = d, u -\-pv = d,
v-±x — e. v+qx — e.
82. x-\-y-\-z = a, 83. x — y+z = a,
ytz + u-b, y — z+ii = b,
z + w4-f = c, z—ii,+v = c,
u-\-v-{-x- d, u—v-\-x = d,
v-{-x-\-y = e. v—x + y = e.
84. x + y+z — tc = a, 35. x+y + z — u—v = a,
y-\-z + u — v = b, y+z + it^v — x = b,
z + u-i-v — x = c, z-\-u+v — x — y = c,
u + v+x~y = d, u+v + x — y — z = d
v+x+y — z = e. v+xi-y — z~u = e.
86. 2x-y-z+2io -v = 3a, 87. v-2x+Su-2y+z = a,
2y—z — u + 2v — x='ib, x — 2^ + 3^- 2z + u = 6,
2z-u — v + 2x — y = dc, y—2z + Sx—2a + v = c,
2u-v-x-^2y-z = dd. z-2u + 3y — 2i--i-x = d.
2u — x — y-{-2z — u -~ 3e. u — 2t' + 32 — 2x + >/ = e.
Exercise Ixi.
Kesolve the following systems of equations :
i+y+y- ' Z/+1 >y-i
1+y+x^ ^ xM-x+1 ^ ^^jx-l\*
i-^x+y^ ' y--hy+i \y-il
{l+x)(l+y) _ l±a x + y _ a^-oc^
{\-x){\-y) " 1-a 4. i+j;^ - «2_f_^8'
SIIIULTANEOUS EQUATIONS. 205
(l+x){l-y) _ 1+^i X-1J _ h^-B''
{l-X){l+l/) ~ l^b ■ l-XIJ ~ 6-+p2'
l+xy ~ 64-c' ' 1— x?/ l-a2
x — y b — c J^—y 26
7 ^+^ _ 2rta g 1-f-a^ >c+y _ 2a
a;-y _ 2hQ l-xy x-y _ 26
!+.'•// ~ 6^-/3-' a; — y 1-xy ~ n
g y(l+a:3) ^ ^ij y + 2 = 2ra'/2,
Z(l+^2j ' X+2 = 2/^.cz/^,
y(l —x^) x + y = 2CXI/Z.
n i'+~ — ^' 2+a; — y a; + 2/ — 2 12. ax = hy = cz,
a " h ^ c~~' 111
■ = ! + — .
xyz = m^. X u z
13. y'- +z- -z(y + z) = n,
x-+y^-z{x+y) = c.
14 '2ax={b+c — a){y-\-z),
2/,y = {c + a-b){x+z},
(u; + //-i-2)'-'+x2 4-//2+22=4(a3+63+c3).
,- a;— 1 _ (i — l ^,, x^ -\-xy-\-u- x^-^y^ xy
2/ — 1 6—1 x^—xy-i-y^" a ~ b
x^-1 _ a3 1
S/3-1 ~ 6^31'
17. x^+x-'^s^z/'^^rt, iQ
j;2+j;^4-?/2=6.
.1
19. =cy +j = a(x^ + ^=) 20- x3=a(x^+^'-')-6.^y,
20b SIMULTANEOUS EQUATIONS.
21. 4rU-2+l)=(rt + 6)(a;-7/)2,
^c{y-^-\)-={a-h){x-y)K
h-Vc
3 3 *-
2,3 _„3 ^ _
-c
'+-f^+i/
')(^+2/)-
23.
a-+.r2
24.
a;2 -f ?/2
- a,
xy
1-^xhf- ^
xy
25.
a;(?/ + 2) = a,
?/(z+a;) = 6,
2(a;+?/) = (,-.
26.
{.r+y){x-\-z) = a,
{l/+z){>J-{-x) = b,
(2-hx)(z + //)=C.
27.
a;(a;+?/+2) = a-
-2/2.
28.
x^~{y-z)^=a.
y{xAry-r'z) = h-
-sx,
y^-{z-xY-=b,
z{x-^y+z) = c-
• a?/.
Z^ — (x — ?/)2 =c.
29.
^.2_|_^3 — rt2^
30.
1 1 2a
x^ -^ y^ = z^'
a:+?/ = ^2,
1 1 26
a:^ ?/■■' r^
x — y = cz.
1 1 1
x y ~ c'
81.
.(•2 —7/ 2 =^,2^
32.
2-1
•^ 2+1
(x-2,-)(2+l) = 2^,
a;— j/ = 6'z.
{x^-y-)(z+lYz=^ht.
KXAMINATIOxN PAPISRS. 207
CHAPTEK VIL
ExAjnNATiON Papers : Education Department and UNrsraRsixr
OF Toronto,
I.
1. State the rules for the addition and subtraction of Algebraic
quantities. Express in the simplest form
(b+c — a)z+ {c-\-a — b)i/-\-(a-\-h - c)z
{c-\-a — h)x + {a-{-h — c)y ■\- {h-\-c — a)z
{a-\-h — c)x-^{b-\-c — a)ij-\-[c-\-a — b)z
2. State and prove the Index Laws. Assuming these to be
general, interpret x~"'.
Find the products in the following cases :
(1) (x3 + 6x3j, + l2a;?/2+8v3)(a;S-6a;2y+12x?/2 -8^/3).
(2) (^a + h-\-r){b^c-a){c+"~b){a^h-c).
3. Prove the rule of signs in Division.
Divide : [Apply Homer's Method to (1)]
(1) x-6-22.c4+60a;?- 55x2 + 12x4-4 by x2 + 6.r+l.
(2) a;* + 9-|-81x-4 by a;2 _ g + g^c-a.
(3) a;"' - 1 1:
4. Find the square roots of
(1) 4a;4'"- — ^"" + TT^^"*
(2) -nj + — 4- — - 2— -
b^ c a^ c
- 2f + 2I.
1} a
6. Distinguish between an algebraic equation and an identity.
Solve
(1) ^(l-2.i)-H|^(l4-2x-) = 3,
208 EXA5IINATI0N PAPER3.
(2) "-Z^ 4.-'^ - 2 '^.
^ ^ x-j-2 + x-2 - _-x--d
6. A person bought a certain number of oxen for $o20. If be
had been able to purchase 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 ^ - -1 shew that «_!±M±8L^ ^ ^i^^.
(2) Find the value of x'' -200a;5+198a;4 +200x3 - 197a;2
-397a; when a; = 199.
8. Three towns, A, B, C, are at the angles of a triangle. From
A to C, through B, the distance is 82 miles ; from B to A, through
C, is 97 miles ; and from C to B, through A, is 89 miles. Find
the direct distances through the towns.
II.
1. Prove x" 4- a" = a:"""".
Simplify {a+h-\-c)- ~d{a + b + c)^c + 5{a-{-I>-^c)c^ -e».
2. Prove the rule for finding the L. C. M. of two quantities.
Find the L. C. M. of
a^Jl.])^j^c^-Qabc, and (a + h)^ +2{a-}-b)c+c-.
a c ac
8. Prove -^ >< "J = fc^*
/1-.t3 1-x \ I' l^x l-x^\
Simplify [^^^ + xz-^q-^.l -^ Ji^^-s - i:^]-
4. Eeduce to their lowest terms — ^r^^. — ^ — ^, and
5. (1.) li a^ —pa" -'rqa — r^O, then x^ —px" +qx — r is exactly
divisible by a: — a.
(2.) Prove that {a + h + c){bc-{-ca + ab)-{b-irc){c+a){a + b)i&
divisible by abc. Is there any other divisor ?
EXAM IN' ATI ON PAPERS. 209
la + b\ ?^ , , a^-b^
6. If^= M""'"'^^^"i.^+6^(v^-^-i--^)
7. Solve the equations —
3-2^ 5--2x _ 4a;- -2
(3 ) !!l±? _ ^±^ _ i^±^ _ l^-g+17
a;+i ~ x+2 ~ 2x'+7 ~ G^ + IO'
8. A pei'son going at the rate of jy miles an hour, and desiring
to reach home by a certain time, finds, when he has still r miles
to go, that, it' he were continuing to travel at the same rate, he
would be q hours 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 m_ mouths hence, and also $b due n
months hence. Find the equation which determines the time at
which both sums could be paid at once, reckoning interest at 5
per cent, per annum.
III.
1. Ifa;=10, y = ll, 2=12, find the value of
{ z^ -(y-{-z)- r X ■^r — r ; and subtract
{y—z)a'^ + {z-x)ab-\-{x — y)ii'^ from
{y — x)a^ — {y—z)ab — {z — x)h^.
2. Divide a + (rt+&)j; + (rt-ii/; + c)a;3 ^{aJ^h+ifx^ + {h-\-c)x*
+ cx^hy \-'f-x+x^-\-x^ ; and find the square root of
9 - 2-lx+58a;3 - llGu;3 + 129u;* - 140.(;5+100.i;6.
4x + 5 x^5 2.C + 5 a;2_io
3. Solve (1) ---^ 4- ^^ = — --, - -^^:3- + x.
£10
STAMIXATION PAPERS.
(2'. ^:,-fT,,^.T,^o, ^x + iy-lz=-ll
4. 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
sold the whole quantity at the lower price he would 1 ave received
$78.75 ; but 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 o + y b.
Find the square root of 1-f ^/(l— a^)
^^+^ 4a;— 1 7a; +1
7. Solve ^ ^ -+- 5 = — ■- — - ; and find the value of i
^X — < X — £i X — o
when ax^ — ^Qx + S\ =0, has equal roots.
^^^^a-b ~ -i/(.7c)- V{bd)
9. SliQw thfit a^{b — c) + b^(c — a) -\-c^ [a — b) is exactly divisible
by a+b + c ; and resolve the expression into its factors.
IV.
1. MnltiTplj a^+h^-c^+2ab hy u^ -b^ +c^ + 2ar, and divide
the product by a^ —b- —c- +'2bc,
2. Simplify
x+y ' " t 7(c+4 " I 21^2~ -^ «(a;2 - y^)l f
EXAMINATIOX PAPSRS. 211
3. Find tbeL.C.M. of4^2-9^'^ Ax^ -lOxij + 6;/^. and 6x« —
lSxi/+6>j^, and the G.C.M. of l+x'^+x-fa;' and 2x + 2x' +
■i. Obtain the square root of i — |i/^-, and find the value of c
when Ax* — 12x^i/+cx'^i/^ — 12x!j^-j-4:!/^ is a perfect square.
5. Distinguish between an eqnntion and an identUy. Give an
example of each. Wbat value of m makes (x - 3)2 _ (x - l)(x — 6)
= III an identity ? Can any value of in make it an equation ?
6. Keduce to its simplest form
l/(2+a;)-T/(l+x) ^ l + i/{l -1 ^ (1 +a;)}
i/(i+x-)-|/a: "^ 1 + y {i-fi-^i+^n
7. Solve the equations
. 1 V 2j; -h 5 2x—5 ix — 5
(2) 7i',y~5x=[x-5i/){x + '6y),
'!_ _ _^ T_
x—5y x+'6i/ ~ 83*
8. A person performed a journey of 22| 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 1 miles an hour. He
did tiie whole iu 1 hour 50 minutes. Had he walked the first
portion, and performed the last by carriage, it would have 1*.iken
him 2 hours 30^ minutes. Find the respective distancci; by car-
riage, train and walking.
9. Solvu
x-t^ x + 1 _ 4a?+9 12a;-}-17
x-fl ~ x+2 ~ u+7 ~ 'ex+To'
10. What value of y will make 2x*-\-3xy+Qy^ ezacily divisible
bya;-3?
If a and h are the roots "of the equation x^ -\-x +-1 = 0, show
that a3_ 63=0.
212 EXAMINATION PAPERS.
V.
1. Multiply
Prove that
{lx — i/)^ — (x — hj)^ is exactly divisible by x+y.
2. Express in words the meanine: of the formula
{x + n)(x + h)=x^+{a + h)x + ah.
Retaining the order of the terms, how will the right-hand
member of this expression be nffected by changing, in the left-
hand member (1) the sign of b only, (2) the sign of a only, (8)
the signs of both a and 6 ?
a Simplify (flt + t)'^ + («-&)* -2(a2-&2)2 ; and show that
(^a+b-\-c){h + c-a){a-{-c-b){a + b-c) = 4.aH3
when a^+b^ = c^.
a c ad '
4 Prove that y - "^ = ^•
Simplify
[~2air "^ ■^/ U"3T^"3) '■ a^-ab+b*'
5. I went from Toronto to Niagara, 85 miles, in the steamer
" City of Toronto " and returned in the " Eothsay," making the
round 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 6
hours and 30 minutes ; find the usual rates per hour which these
steamers make.
6. Solve
3 2 1 2.1 2
(1) = — ' = — •
^ ' x y a X y a
(2) a;2+5a:=5^/(a:3 + 5.r+28)-4.
7. Find three consecutive numbeis whose product is 48 times
the middle number.
ErA^nXATION PAPERS. 218
8. If OT and n are the roots of ax^+bx-^c = Q, then
nx^ +hx+c = a(x — m){x — n).
Show that if ax- -{-hx+c = Q has equal roots, one of uitm is
pven by the equation
{'I'l- -lalAx + ah-b" =Q.
9. If — = — aud-r- + Vtt =1, prove mat
YI.
1. Simplify
2. Divide a' — ^3_c3_3a7,c by a — b — c, and show, without
expansion, that
(1 +x+jr- ;. 3 _ (1 „ a; + x3 )3 - C.r^^* +a;3 + 1) - Bx^=0.
3. Resolve into factors a;'^ — Ja;^./" + ?/*, and
Ix"^ - 6?/2 - a;,/ + 19x + 33;/ - 3G ; and prove that
b^{c-^a)+c^{a-\-h) — c(-(b-'rc)+abc is exactly dis'*sihle by
b-\-c — a.
4. Apply Horner's method of division to find the value of
5a;6 +497x4 + 200x3 + 190x2 -218a; -2000 when x= -99, <ind
the vaue of Cx'^ +5x4-17x3 -Gx^+lOx- 2 when 2x- = -3x+l.
6. Find what
V(n+x)+V(a-x) , , 2ab
— -^ '—1 1: -'.becomes when x = .
\/{a+x)— \/{(i — x) 1 + 6*
6. If a and b be any positive numbers, prove that
1 a , a b
T + r+-a > '• T + -„■ >
.+.'-•■ - ^-
214 EXAMINATION PAPKSS.
7. Solve the equations —
(1) cr- + / = 5,_
1 I
5 5
•» + y = |.
(2) a;+22/+82 = U,
2a; + 3i/ + z = ll,
3x+?/+22 = ll.
(3) (x+l)(x + 3)(x + 4)(x+0)=rI3.
8. There are three consecutive numbera such that the sum of
their cubes is equal to 1G|- times the product of the two higher
numbers : find the numbers.
9. (1) Form an equation three of whose roots are 0, y (— 3),
and 1-1/2.
(2) If one of the roots of the equation x'--\-px-\-q = Q, is a
mean proportional between p and q, prove that
10. Two trains start at the same instant, the one from B to 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.
VII.
1. Give some application of the ''rule of signs" in Algebraic
Multiplication and Division.
2. Find the numerical value of the quantity
bc(c — a)(a — b) — ca{a — h)(h — c)-{-ah(b -c){c — a),
when a =10, 6 = -01, t; = 0; and pr6ve that if
H, = , then will {a-\-b) . '
a-\-h a-\-b — c-\-x
EXAMINATION PAPERS. 216
8, Inveptigate a method of finding by inspection the remainder
after dividing any rational and intogral function of x by x-\-a.
Show that the quantity
ifl divisible by each of the quantities x-\-i, x-\-b, a—^x. h-x,
4. Investigate the rule for finding the H.C.F. of 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.r4 _ 7x3 _ i3a;2 + I9a;_ 6 and x^ + 2a-3 - 1.
(2) {x+y){ax^-h!i-^)—xy{a-h){x-iry), and
{x-y)(iix^-hj^) +xi/{a-b){x-y).
5. Prove, by general reasoning, that the value of a fraction is
not altered by multiplying or dividing both the numerator and
denominator by the same quantity.
13 7 X i
Simplify (1)
12(2a;-3) 12(2j,-+3) Ax^'+d'
(2) 1 1 + ^L__^ I .
1 1
+
[{x+a)(x+b) {x — a){x—b)
Solve, with respect to x, the equations
,,. z-lS 2.r-24 lla;-34 7
(2)
4 ' 11 ' ii2 44
5x^+x-B 7x3 -8a;- 9 a;- 3
5a;-4 7a;- 10 35a;3-78a:+40
(3) x^ = ax-\-hy, and ?/3 = hx-'r ay.
VIII.
1. Define the terms " power," "root," " index," and '* coeflfi-
cient ; explain also the reasoning by which it is shown that
a — {h — c) z= a —h -^- c.
219 EXAMINATION PAPEK3.
2. Multiply (x2 +.<•// + 2/2) 2 by ix—y)'^.
Find the values of a and h whicli will make
X' +ax+b divisible by x-^p, and also by a; + j.
y. Divide x^+7/^-{-2x^ij^ by (x^yy, and
4. Investigate a rule for the extraction of the square root of any
algebraic quantity, 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 haK the number immediately following it.
6. Find the square root of
(1) a^x'^ + '2abx'^ + (b^ -i-2ac)x^ -i-c^x-'^ +2bc.
(2) ix'-i:<:" + j/+i./-K' + A^'-
6. If x^-r-ax+h and x^+a'x-b have a common measure, it
will be x-{- — o — » a,ud the condition that they may have a com-
mon measure is ib^a- —a'^.
Find the H. C. F. of x'*' +2)^ x^ -}-p* and x^ +2px^ +p^x'' -p^.
Find the L. C. M. ot 2\{x^-^x-20), d\{z^-x-ZO), and
4i(x3-10x + 24).
7. Find values of a and h which will render the fraction
Zx^-{ia-{-h)x+a + 2b^
bx" - (Qa+b)x- a^Ab^
the same, for all values of x.
d. Solve the equation 2 + |/(d;-i-l)(x+6) - -j/(a;-l)(x-f 5) = 0,
and account for the circumstance, that the values of x, determined
from it, apparently do not satisfy the equation.
EXAMINATION PAPBR3. 217
IX.
1. Prove that a{2ni-l){a'' +7r7TTl) - "(2" -M)(«^ +«•« + !)
= (rt — n)'.
2. If a, b, and c are positive quantities, and if a>^ and c>a- 6,
prove that
e - {a — b) =c — rt-fy.
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. 11 x^ -^ax- -^h and x3+7"- + 5' have a common measure ot
the form of x^+mx+n, then a^hq = {h-qY
4. Find the H. C. F. of
a'^-h'^-abxij-^ahx-^y-^, and a-x^ -b^y-^ -\-a-bx'^y-b^-X]r''*
;>. A and B are two numbers, each of two digits. The left-
hand digit of A exceeds that of Bhj x; the excess of A above B
is y ; but the sum of tlie digits of B exceeds the sum of the digits
of -4byz. Pi-ove that y^z = 'dx; and give an example of two
such numbers as A and B.
^ -^l a. - — = — , prove that each of these ratios
■^a ^ , a+b-\-c
7. Solve the equations
x±a x—a _ b-^x _ b—x
^ ' x^a ~ x+n ~ h — x b+x
(2) a(x2+y3) -6(3-3-2/2) = 2a
(rt2_fe2)(a;2_j,y2) =inb.
8. A farmer buys a sheep for $P and sells 6 of them at a gaia
of 6 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 ; what are the numbers.
'^'•° BXAMfNATION PAPERS.
X.
bya;+?/+2. ''^ ' -^^
_ 2. Prove that if x^^px^+qx+a^ be divisible by a:^-! it
IS also divisible by x^ ~a^. '
3. Explain the reason for introducing or suppressiug factors in
tae process of finding the H.O.P. of two algebraical quantities.
Why is the name " Greatest Common Measure " objectionable ">
Find the H.C.F. of x*_^.3_^2 _^_2 and Sx^ -Ix'^+s^.^^,
4. A traveller leaves A for B at the same time that another
leaves 5 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
hour tiU he has got over one-third of the distance between B and
A; he then rests for 40 minutes; after which he resumes his
journey, walking now at the rate of 3 miles an hour The tra
Tellers reach A and B respectively at the same time. Find the
distance between A and B.
5. Show by examining the square oU+b how the squave root
Oi an algebraical quantity may be found.
Find the square roots of
(1) 25x^~B0ax^+A9a2x^ -2ia^x+Wa^, and
(2) $ + ^- l±^JL]y2 + ±.
y ^- \y xr 2
OT
6. Show that a" = Va^, when m and n are integers, and m is
divisible by n; 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
(1) ^^^+^-^ _ 7^2 -8a;- 9
5a; -4 ~ " 7»~-T0~"'
(2) {Bx-l)^ + {ix-2y^=(5x~8)^.
EXAMINATION PAPERS.
219
8. Two regular polygons are so related tliat the number of
their sides is as 2 to 3, and the magnitude of their angles as 3 to
4 ; find the figures.
XL
1. State in words the several operations to be performed m
order to obtain the result expressed by the following algebraical
expression :
* jma^ -\-nb^
m-r'H
Also find its value when a = 6 = 4.
2. Two men, A and B, dig a trench in 8| days. If A were to
do more work by one-thu'd than he does, and B more work by
one-half tliau he does, they would dig the trench in 2|§- days. lu
what time would each dig it alone, at his present rate of work ?
3. Perform the multiplications in
(1)
/ 2 J + 3/y^ ) ( 2a:^ - 2^/' \Ux'-h Gx^/+ ^y^ ) ( 4a;* - Qx^i/ + Oy^ \
(2) ax^+ixy-fi!/^)i^x^--ixy-\-%y^).
4. Divide
(1) x^ + d + Slx-* by x^-B + 9x-^.
(2) x'*^ — {n-\-b+p)x^-{-iap-{-hp — c+q)x^-{aq-{-bq-cp)x — qc by
X- —px+q.
5. Show that a;"'" +^ — aj^^-i is always divisible by x±il, m and n
being any positive integers.
6. Define a fraction ; and from your definition prove a rule for
adding together two fractions with different denominators.
Add together the fractions,
a^ — he 6- — ca- c^—ab
(a + b)(a+cy {b + c){b-^ay [c + a){c + b)'
220 KXAillNAl'lON i-APi-KS.
7. Solve the following equations :
(1) ^'^ + 2a;4-2 x" + Sx+20-_ x^+Ax + 6 x^ + 6x+12
'~x + l "^ x+i ~ ^+2 "' x^ *
(2) (x^+y^)-^ = ^^ (a;2-^2)J^ ^ ^.
XII.
1. When 7n and to are whole iiumbers, and m greater than n^
a'" 1
show that — = a"*"" and that —^ is correctly symbolized by a~" .
2. Multiply (a -i)(a+6)(a-+/>2)(a4_^^4) . . , to (n + 1) factors.
3. Divide 1 — tc by 1 — 2a;, to 5 terms, and write down the
(?-+l)th term, and the remainder after (r+1) terms.
4. If the number three be divided into any two parts, show
that the difference of the squares is three times the difi'ereucQ of
the numbers.
5. Find the L. C. M. of 1 -8x+nx^+2x^-2ixA, and
l-2x-rSx-^+S8x^-2ix^.
6. What relation must there be between the coefficients /«, h,
■p and q, in order that
{x^ -^mx+JiY -\-j)X-^ + qx
may be an exact square for all values of x ?
7. Solve the following equations :
l+a;3 i-x'
(1) (rq-jy2 + (i_;,)2 - ^'
ax — h^ ^{ax) — h
(2)
^/(«x)+&
(3) — r = 1, -jr = 2, and -^— = 3.
^ ' x-\-y x-^-z y-t-z
8. Given a; +.V+Z = «•« = %. find (x+y+z) -^z.
9. Find a number expressed in the decimal notation by two
dibits, 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 = 2i, fc = 3i, c = 4^ of
2. Show that the vahie of the expression, in the preceding
question, is not altered hy changing a into a+x, b into b-{-x, and
c inte c-\-x.
3. Multiply (1 4- «i a;) (1 + " 2^) (1+^3^) •*• (l+^na-') 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 8^, his profit at the end of the
year would have been only §65. Find the whole sum borrowed.
5. Given ax-{-l'i/ = c, a'x + h'i/ = c', determine the value of
mx-\-ny, and find tiie conditions under v^'hich the value becomes
indeterminate.
a„ a^ a-
thenwilla, -|-nf„+r?,+ . . . + a„ =
7. Eliminate x and y from the equations
a a a
a; -r y = «
a. = x+dx^y'
/5 = y-^-^x'y^.
8. li ax' -^-hx+c-^O a' d 7j,r2 -f-5^x-|-Cj =0, then will
9. Find that number of two figures to which if the number
formed by changing the placos of tiie digits be added, the sum is
121 ; and it Lai^ bame two numbers bo subtracted, Ui© remainder
is U,
222 EXAMINATION PAPERS.
XIV.
1. Simplify
a(b + cy + h{c + ny-i +c{a + b)^ - {{a-h){a-c)(b+c) +
{b~c){b-a){c + a) + {c-a){c-b){a-\-b)}.
2. State tiie law of Indices, and prove it for positive integral
indices ; and assuming it to be general, interpret the expressions
x~"\ X , where m and n are positive integers.
8. Having given the equations,
x-\-]j^z — Q, « -f/z'+z' = 0,
prove that a^{yz—y'z') + h'^ [zx — z'x')+c'^ [xij ~x'y') = 0.
4. A traveller P sets out to walk from A to B, proceeding at
the rate of 3 miles an hour ; and, 82 minutes afterwards, another
traveller Q sets out to walk fro^^i B to A, proceeding at a uniform
rate. They meet half v/ay betwixt A and B. P then quickens
his pace by 1 mile 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.
5. How are equations classified ?
Solve the equations —
(1) nmx+dmnrzn^x-j-mu'^ .
(2) x*-x^+y^-y^ = 84:,
x-+x"y^+^j'' = i9.
6. What two numbers are those whose difference, sum and
product are to each other as the three numbers 2, 3, 5 ?
XY.
1. What is the meaning of the symbols a, a~, a'^ . . ?
Show a prion that a° = 1 ; 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.
EXAJnNATION PAPERS. 223
2. FiuJ the valne of
(b-cy +2{c-ay + (a-L)^ -3{b-c){c-a){a~b)
Avlien a = 1, b= — J, c = |.
3. Simplify the following expression :
(ac-b^'){ce-d-2)-\-{ae-c^)(bd-c^)-{ad-bc){be-ra)
4. P aud 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, t\YO miles in advance of A. When are they
together ?
Has the answer a meaning, when to— n is negative ? Has it a
meaning when m = n ? If so, state what interpretation it must
receive in these cases.
6. Show how to find the Least Common Multiply of two or
more algebraic quantities.
(1) x^ --ax-2a^, x^+<ix^ aud ax^-x^.
(2) x^-x-i/-a^x + a'-^ir^^^x-^+('x^-xy--ay3.
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 e:icpressions :
(a^b-c)^ -d^ {b + c-a)^-d^ (c+a-b)^-d2
(1) (^6)2_(c+cZ)2 + (6 + c)2-(a+f/)3 "^ {J+af^-{b+df'
rj.ij^yi-z^ + 2.xij a^-[-a^h a{a-b) 2ab
(2) x--y^-z--^2ijz aH-b^ ~ (a+h)b ~ a^-0^'
7. U ax-~hij + o{x-y) = {a - b){a-{-h - c),
by - cz+aiy-z) = {b- c){b+c-a),
cz — ax+b(z - x) = {c - a){c + a — b)
then will a^{h-c)+b''{c-a) + c\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.
i?rove that 11 (x+y){P- Q) = 9 {x-y) {F+Q),
224 EXAMINATION PAPEKS.
XVI.
1. Show that
{{nx+hijy + {a7j -bx)' \ {(nx+hij)^ - (ay+hx)-} =
(a<i-64)(:c't-2/4); and that
2{a-b){a-c)-{-2{h-c){b-a)-i-2{c-b){c-a)
IS the sum of three squares.
2. If s = a + i-(-c-|-(i;c. to n terms, then
s — a s — b •"' — <^,p 1
-1_ J. + &C, = 77 — 1.
S .V S
3. Show that a — b, b — c, and c—a cannot be all three positivo
or all three negative.
4. Extract the square root of
4.c« + Ox^ - 12j:* + lGa;2 +9 - 2xiGx^ - 8a;4 + ^x^ - 12).
5. Gis-en ab - \{fi-{-b){p + q)-{-pg = 0,
find the val'.ie of p — q, and show that if eitlier a or b is equal to e
or d, then p is equal to q, unless a + b = c-\-d.
6. Find the value of — , having given
y
7. Prove that {a — b){b—c){c — a) is a common measure of the
quantities
(„2 _t2)5_^(/;2 _c2)5 +(^.2 _^3y5^
8. Find the conditions that a^x+b.^y = c.y, a^x+h„y = r^, and
a a;_|-/i T/^Cg may be satisfied by the same values of x and ij.
9. Two persons, A and B, start at the same instant from two
stations (c) miles apart, and proceed in the same direction along
the line joining t}\e stations with velocities (a) and (b) miles per
hour. Find ihe distance (x) from the stations where A over-
bakes B, and interpret the result when a z 6.
EXA^^^•ATION papees.
225
XYII.
1. Express? in symbols the result of snbtractinfr from unity the
quotient obtained by dividing the sum of a and h by their product.
2. Multiply to;^ether x + y/a + Z', x—\/a-^h, x-\-i^/a-h and
X— Va- b ; and divide 24^f3 ...'■2SLa-b + ^a'^c — 5ah~-\--llahc — o\ac^
+ G63-22/>2c4-lG6c3 + 8c3 by 'da-1h+4.c.
3. If x+a be the H. C. F. of x^-\-}:x-\-q and x- ■\- p'x+q',
their L. C. M. will be {x+a){x+p - a){^x -^ p' - a).
Show that the difference between
X x x ^ a be
-f 7 + and + ; + -^ — -
x — u x — b x — c x — a x—o x — c
IS the pame whatever values be given to x.
4. Prove, if the four fractions
bx+nj+(Jz cx+chj-\-az dx+<nj + hz ax -j-b]/ -\- rz
b+c-i^d—a c + d-ru — b' d+a + b~c a-}-b+G - d
are equal to one another, their common value will be ec^ual to
-—'- — as long as a + b + e+d does not vanish.
6. What do you mean by solciivj an equatioit. Show that 3 is
a root of the equation
8 + if(x-2)
6. Eliminate x between the equations
x3 +-73 + 3 x + — ) "= "h and
7 If 4. — _ — = — — , , a, 6, c are jaot all different,
a b c a-{- 0 — c
8. A cas'k, 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 b gallons of wine andc gallons of water ?
226 EXAMINATION PAPES3,
XYIII.
1. Multiply together tlie factors
1 — x, 1+x, l+x^, 1+x*, and 1+x^,
and sho-w that if 7i is any uneven number, the S'lm of the nth
powers of any two numbers is always divisible by the sum of the
numbers.
2. Find the numerical value of the expression
c */a+ */c
b */a— -v/c
where a, h, c are connected by the equation a''b — c)^ — c(b+c)'^=0.
3. A has a younger brother, B. The diffei'ence between their
ages is §• of the sum of their ases. By adding twice -B's age to
5 times ^'s, we obtain the age of the father ; and by subtracting
twice -B's age from 5 times .4's, we obtain the age of the mother.
Show that the age of the mother is y\ that of the father.
4. Find the H.C.F. of
z^-{2a+b)x^-ha{2a + l>)x-a^(a+b), and
xi-{2b+a)x^+b{'Ih+a)x-b^{b+a).
5 If J_ 4. _ = — , shew that
be a
^a^],_c)s^2{b + c-a)^ + {c + a-b)3 = 2(b+c)\
6. Show fully how the rule for finding the square root of a
o-iven number is obtained. If n-\-l 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
x^x^-+y^+z^)+y"z"-+1x{7j+z){yz-x^).
7. Find the value of the expression
x—v ^ a+b b
"t £- when x — , y = —
14-xy a-b a
ESAJIINATION PAPERS. 227
8. Solve the equations :
(1) ^{x-2a) -Ux + Sa) + l(x-ea) = 0.
(2) ^/(2a;3 4-l)+V(2.r3+3) = 2(l-«).
9. Divide 21 into two parts, so that ten times one of them may
exceed nine times the other by 1.
XIX.
1. Multiply together
Divide this product by
and extract the square root of the quotient.
1 1 1 , , .
2. If a;-fv+2= — + — 4- — = 0, shew that
^ X y z
(x^+y^+z^)-i-{x^+y^+z^) = xyz.
8. Find the H. C. D. of 20x4+a;3-l and 7oa;* + 15a;3 -3a;-3 ;
also of (x+yY —x^ -y'' and (x^ —y^Y'
4. Given that rti-(«4-fe)(a;-f2/)H-4a;2/ = 0,
cfZ-(c+J)(a;-2/) + 4a:£/ = 0,
find the value of (x- — ?/)3.
fi. Having given
x^ =7/2 4^2 —<2,ayz
y- =z^ +x'^ — 2lzx
z2=x^+y-—2cxy,
a;2 ^2 ^2
Show that j3^^ = 3;_r^ = j_^s-
l+x+/(2r4-x'^) _
7. Determine a; in terms of a and 6 in order that x^-^2ax^-\-
Sb'-x^ — 'La^x+Ah*^ 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 meu and women by 10. How
many men, women, and children are there in the company.
228 EXAMINATION PAPERS.
XX.
1. Divide (l-{-w)x^ — {m-\-n)x>/{x — y) — (n—l)y^ by
x^ —XjJ+ll^.
2. If x^+px^+qx+r is exactly divisible hy x^-^-mx+n^fherx
nq — n^ = rm.
3. Prove that if m be a common measure oip and q, it "will also
measure the difference of any multiples of p and q.
Find the G.C.M. oi x^ -2->x^ + {q-l)x- +i>x-q and
X"^ — qx^ -\-{p — l)x^ + qx — p.
4. Prove the rule for multiplication of fractions.
Simplify ^Jji-^^z:^ X y'-^::^^^^ x ""^^y)'^:
a a a3 2a3 _^,3._.ai3
a2_|.i2 a2_^2 -r (^a-h){a^+b^) a^-b^
5. What is the distinction between an identity and an eqmition ?
li x — a = i/-\-b, -prove x — b = y-\- a.
Solve the equation
16x-lS 40^-43 _ 32a;-30 20a; -24
4a;— 3 "*" 8a; -y ~ ~8x'^ "^ 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 the inference ?
Eliminate a; and 2/ from the equations ax + by=c, a'x-\-h'y = c'.
a"x + b"y = c".
7. Solve the equations —
(1) ■^/(n^x)-^-^{ii-x\=m..
(2) 3a;+?/+z=13, 3?/+2+a;=15, 32 + a; + y=17.
8. A person has two kinds of foreign money ; it takes a pieces
of the first kind to make one £, and h pieces of the second kind :
he is offered one £ for c pieces, how many pieces of each kind
must he take ?
EXAMTVATTOy pvrrRS. 220
9. A person starts to -walk to a railway station four and a-half
miles off, intending to arrive at a certain time ; but after walldng
a mile aud a balf be is detained twenty minutes, in consequence
of wbicb be is obliged to walk a mile and a balf an bour faster in
order to reach tbe 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;4+279a;2-2892aj2-58Gd;-312 when x= -289.
(o) Show without actual multiplication that
{rt + /)+c)3 - („ + Z>+c)(r/2 -ab-\-b2-bc-\-c''-ac)-3ahc =
d{a-{-b){b-\-c j[c + a).
Note. — In. Ex. 6, p. 87, after proving that a-^h-^,' is a factor,
we may proceed as follows to discover the remaining quadratic
factor :
Tbe quadratic factor must be of tlie form
m{a^+J>- +c^) + n{ob+hc+ca),
in which m and n are independent, being either zero, or a positive
or negative number. To determine them put c = 0, then the
given expression gives
{a^ + b^-\-Bab{a + b)}^{a-\-b) = n^ + b'^-^2(tb,
but also ='m{ri^-\-b^}+nab. :. ?« = landw=2.
,*. a»-¥b34.c^ + 3(a + h)(h-}-c)(c+a)}^{a + b-{-c)»
a^+b'+c^ + 2{ab + bc + ca) = {a-\-b-i-c)^.
230 EXAMINATIOJJ PAPERS.
XXI.
/ 1 . 1 \ /I
1. Find the value of ic' — — + ~r) -f^' + {'1 ]^"^~P
■wheu rt = i, h = ^, x = 2y Simplify
2. Find, by symmetry, the sum of {a-\-b+c)^ —(a-^h — c)^ -
(a-b + c)^-{b-a + c)^, and oi {a^-Aa^x + Sa'^x^ -2ax^ + dx'^)^
and {a* + 4:a3x+Su^x^ + 2ax^-{-3x^)-.
'6. Explain and illustrate the signs >, <
Prove: .-c^ +?/2 >2.r//, (.c+^Z+z)- >3(u;?/4-//z+z.v;), and
x^+l/^-\-z^ > Sxyz.
5 3
4 , ^ h h
4. Determine the value of . <;+?/ — s + 3a? >/ z , when a; +ij' -z
0, &c. : of a'' +lax'-^ + Sx' — Sa^ — {x* + lax^ — 8x^ — 3a-), when
x= -1.
p mp
5. Show that (a"*) «" = a7 .
Simplify j (-^y *l~' X ( yj ' X */ (256), and divide
a; — Qax -f-Scf ic + Sa^aj" — 2a by a; — 2rt x+a .
G. If u = }j ix+'~-\ and v = il >/+ ~] prove that
7. Gold is 19J times as heavy as water, and silver lOi times.
A mixed mass weighs 4,160 ounses, and displaces 250 ounces of
water. What proportion of gold and silver does the m^.ss con-
tain ?
8. Shew that l-{-px-\-qx"-\-rx^ is a perfect cube if ^^2-3^^
and q^ ='di)r.
9. Solve the equations :
IX— 2 ix + 2
(1) "^x^ + ^^^2 = 4-
(2) {x^ + i^'^y-+x^yHx-'-y'r^+x''-;/^ = 32S, x---y^ = 3.
x^ 2x+y y^+x
(3) — + -^ =20 - , x + 8 = 4:ij.
•BXAMTNVTrON PAPV,nS. 281
10. A person bnys two bale?! of clotli, each containing 80 yards,
for $240. By selling the first at a gain of as maeh per cent, as
the second cost him, and the second at a loss of as much per
cent., he makes a profit of $16 on the whole. Find the cost
price per yard of each bale.
SECONI) CLASS TEACHERS, 1880.
xxri.
1 Find the value of x'^+u;^~166x^ -lGGr^+81x + 8l when
J-- —-7 ; and the value of x^ — Zpx"^ + {'dp'-^ +q}x—pq wheii
jC = ii-\-p. (Arrange the latter result according to powers of a).
■2. What is the conditioa that x-i-b shall be a factor of
ax'^ +bx+c ?
Find the factors of
{a). (rH^-rt6) + 2(63-«/;) + 3^3-/;2)4.4(a-/;)2 ; and
(b). {ax+b){bx + c) {cx-\-a) — {'ix + c){bx-{-a)(cx + b).
3. What must be the relation aoiong a, b, c, that ax- -^hi--\-G
may be a perfect square ?
(a). Extract the square root of
(^a~b)^-4c{a''+b^){a-by+4:{a'^+b*) + 8a^l,'.
(h). If 5 be subtracted from the sum of the squares of any four
consecutive numbers, the remainder will be a perfect square.
(Prove this.)
a c <• li I n
■4. If T~ = "T = —F-^^^~l' ~ — = —
u <( J h in p
{a+r-{-e){h + l-\-n) ah + cl-J-en
prove that jir+dJ^k+^u:+p) = l^'dJn+f^-
■ ab{x^ -;,■■') +xi/{i>^-h-')
5. («). Reduce , , o , — jt~; T^TTirT to ifc.-i lowest terms.
(i). If xi/-{-ijzi-zx=l prove that
X >i z 4xiiz
+ v--:f^ +
\-x' ^ l-y' ^ 1-a- ~ {l-x''){l-y-){\-z^)
202 EXAMINATION PAPERS.
6. Prove that
2{x+2-f/(x2-4)}
J- 7' -^
(b) {b+c - a)a'' + {c + a — h]b^ + («+/> — cy =
i .1. 1 s a 2
(a + h-\-c)ia + //" +C'') — 2(t<" + i" +c') .
7. Solve the equations —
(a), (i- c)(^ - a) 3 + (c - a)r.y;- /;)» + («- i)(^-c) 3 =0.
(b). x+y = 4x// ; .y +~ - %2 i s +^ - ^2^-.
((■). x+>j+z = 0.
ax-i-bij + cz = 0.
bcx + ca>j-\-cibz-{-(a — b)(b — c)[c — a) —0.
x—1 x—B
FISST-CLASS TEACHERS, 1870.
XXIII.
1. Investigate Horner's method of division.
Divide x^ --^x*" -SW +'25x^ +Bx^ -8x^ + 19x'' +Sx + 10 by
3a;* — 21^;^ -i-Qx—Q, showing the " final remainder."
Find the value of 2x^-{-80'dx^ -3dSx-' + l60ox- -120ix+4:22,
when x= — -i02.
2. Iif{x), a rational and integral function of x is divided by
,, ., ■ {f{a)-fi0)}x+a.m-0Aa)
'£-+7>x + q, the remamder is ., >
^ -" a — i:>
where a, ^ are the roots oi x-+px+(j = 0.
Examine the case where p- =4:q.
a. Show without actual expansion that
'^^b-'cy+b-^ic - a) +c^a - b) '^
EX.UIINATION PAPERS. 233
4. Find the value of x aucl y that will render the fraction
.. „ , , 7- :t— ; 3-7 the same lor all values oi z.
5. Show how to find the sum of n terms of a series in Geo-
metric progression.
(1) Show that the sura of n terms of the series
l + /- + Vl + 20(l + r) + (l+3r)(i + ;-)^+ . . ., is n {l + r)\
.11 1
(2) Sum to infinity the series ^. p+A.p.o + g g.i a+ • • • •
6. Explain the notation of functions : prove that if
f (w) = l+mx+ "^^^^:^-^- +^^ ' *^eu/ {,n) xf (n) =f{iii-i-u).
Show that in the expansion of (1 -\-x)" the sum of the squares
1-2-3 . . . . 2/i
of the co-efficients = rrrrs w"
(1-2-3 • • • • n)-
7. Solve the equations —
/-.s x — a x—b x—c
^ ^ bT'c "*" a'+c "^ a + b ^ ^•
(2) x4-10x3 + 0ou;2-50.c+24-0.
_^_Jl 1
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 j)roduct of each term into the sum of all
that follow it.
Find the sum of the products of the jSirst n natural numbers
taken two and two together ?
X >/ z
9. If — -y + z, -r- — z -i^ X, -^ = X + y, prove
1.1 1 \+a \+h 1 + c
^•'^ V ' IT ' T ^ T^^) ■ l^^c. ' l — ra
c- y- z-
(2)
a(l — he) b{ 1 — ca) c(l — ab)
V'l-bc Vl-ca '^1—ah ^ 1— be Vl - ca Vl-ah
234
EXAMINATION PAPFRS.
10. AB is divided in C, so that AB, BC=AC^ ; from CA is
cut off a part CD equal to CB ; from DO is cut off a part DE
equal to DA ; from ED is cut off a part equal to EO, and so on
ad inf. Show that the points of section coritinually approach a
point C such that AC = BG.
14. Ehminate ;c, y, z and w from the equations
a^x + h^y-\-c.-,z+d.^u = 0.
a^x-{-b^y-\-CgZ + d^u = 0.
a^x-i-h^y-{-c^z-hd^^l-0.
12. A railway train travels from Toronto to CoUingwood. At
Newmarket it stops 7 minutes for water, and two minutes after
leaving the latter place it meets a special express that left CoUing-
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 CoUingwood 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 TEACHEES, 1877.
XXIV
'4-
x~{y—z)+y-{z — x)-{-z^(x — y)
- 1 X
x^y- +x^y^+x^z^ +x-z^+y^z'--lry^z^+2x-y^z-
ax+m -\-l ax-\-n ax + m ax-i-n+1
2. Solve (1.) —^^.^4- ^ o = ~ ~ — ^ + —5 i-
^ ' ax + m—1 ax-\-n—A ax 4- m — J, ax-f-n—l
(2-) fr^/x+fi - y.'c = 2.
3. A, B, and C start from the same place ; B, after a quarter
of an hour, doubles his rate, and G, 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
IXAMINATIGN PAPERS. 235
observed that tlie total distance walked by the three, had they
continued to walk uniformly from the first, is 6| miles. Find
the original rate of each.
■i. {l} Investigate the relations that must exist between the
con.stant3 in order that Ax'+By- -{-Cz^+a/jz + lixz+cxi/ shall be
a perfect square.
(2) Find the conditions that the values of x and y derived from
the equations ax-'rbi/= —-^ — = c'^ maybe rational.
5. li x'^+px+q -And x^-{-mx-{-n have a common factor, then
will {n — q)'^-\-u{)n — p)^ = iu{))i—p)[n -q).
6. Prove («'")" = ^t""*, whether m and n be positive or negative,
integral or fractional.
Show that (a^-^+a;-")"'" =x"^^ " x (a;'"-" + x"-'")m«
7. (1. ^^-r = -T tlien \---- — = M
^ ' b d c'^'^ + rf-" \c-d/
(fj'd" 5"c" «"c" — h"cl"
1
of these fractions = — (rt"4-/^"+c"+ti").
8. If X be very small, show that —
(l4-2a;)^+(l + 3x-)^
T — = 2 - 4a;, very nearly.
2 + ox-(l+'J:x-)
9. Prove that 1 - 71^ + ^ 32 ~ 1^ 2^ P"" +■ = ^
10. If a debt $a at compound interest be discharged in n years by
a
annual payment of S— , show that (l+r)"(l— mr) = 1, where r
is the interest on $1 for a year.
233 EXAMINATION FAPiSAS.
11. Solve— (1.) 8a;'^-2a:?/ = 55.
5 5
1 3 ? n-t-(7 i
(8) rt-5-a;" — irt'i'ajspi = (a — ('^)-a;''
FIEST CLASS TEACHESS, 1878.
XXV.
2 / ' /— — \ 2
1. Simplify fV"±i-_V-j;-) - (ViL_^(Jl
'. X a A- X \ a x
X'
a -^ X \ a X ■ a{a — x)
x^^j-{ll-zY y^-{z-x)^ z^-(x-y)^
2. Divide — - 1 — — --^4- — +-^ by .r-a ;
J
shew that ( - %a^) =^{ v (Or; 1 -f v ( - Or/)}.
???. n r x^ (/- 2-
8. If — = — = — ancl-:r+TT+— 7=1, prove that
X y z a^ b^ <:-
^ ~^b^ ^~^^ 'x^'+y'^z^
4. Fiiid the relations between the roots and eo. efficients of the
equation ax^+lx-\-c = 0.
If in and n are the roots of the equation ax^ + hx-{-'--^0, show
that the roots of the equation acx'--^-{2ac—b^)x+nc = 0 are
VI n
— and —
n m
6. Solve the equations :
(1) x^-\-2\/x^^^^=2x+8.
x^ w3 ; X y
(2) —-— = 104, — - — = ^.
^ y X ^ y X ''
(3) xz = y'\ x+y+z=12, x^+t/^+z^ = 91.
EXAMINATION PAPERS. 237
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 ;jth, 5th, rth terms of a G.P.,
shew that
12 3
Sum to infinity the series — 4--^+— :t4- &c.
8. Find the amount of ^f at compound interest for n years, r
being the interest on ijl for one year.
Supposing $p to be withdrawn at the end of each year, what
will be the amount at the end of n years ?
9. Detei mine the number of combinations of n things taken r
together.
The number of combinations of.?i things taken two together
exceeds by 6 the number of combinations of w — 1 things taken
two together : find n.
30. (1) Find the limit of (l-|-^)*when a; increases without
"imic.
(2) Find the (r-fl)th term in the expansion of (3 — 5a:)
a;2 _ 3^. _ 3
11. Determine the limits between which lies o ,0 , o ,1 for all
possible values of x.
FIRST CLASS TEACHERS, 1879.
XXVI.
7. Prove that 2{{a-hy+{h-cy-{-(c-ay}=7{a-h){b-c)
{c-a){(a-by-^{b-cy + {c-ay}.
2. Extract tlic square root oi ab — 2ai/{i-ib — a^), and find the
simplest real forms of the expression
l/(3+4v'-l) + y/(3-V !)•
238 EXAMINATION PAPEHs.
3. Solve the equations :
(1). 2z^ + x^ -llx^+x-^2 = 0.
(2). x^+'y^+z-=a2
yz-\-zx-\-xxj = b^
s+ tj- z=c.
(3). y(a;3+5a;+4) + i/(a:-+3a;-4)=-a;4-4.
4. Prove that the number of positive integral solutions of the
c
equation rta;+% = c cannot exceed -y + 1.
In how many ways may £11 15s. be paid in half-guineas and
half-crowns ?
5. If xy = ah[a-\-b), and x''^ — xy -\-y'^ = a^ -\-b^ , shew that
: X y\ I x y
a 0 j \ 0 a
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
0, 10, 14, &c., amount to 96 ?
7. Find the relation between p and q, when x^ +px + x = 0 Las
two equal roots, and determine the values of m which will maKe
a^ + m/tx-\-a^ a factor oix* —ax^+a^x^ — a^x+a^.
8. lu 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 r — 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-
pausion of (l+x)" is ]^h-(|_^)2, n being a positive integer.
10. Sum the following series : —
(1.) l-f3.c-f-5:c2+7a;3 + &c, to n terms.
1 1
(2.) Q^ o+o — To-f &c. to n terms, and to infinity.
11. Shew that
nhc(<i-\-b-{-c).
EXAMINATION PAPKS9. 239
be, — ac, — ah
/>2 - c3, «2 4- 2ar, -rt^ - 2nh is divisible by
c", C-, (a + i)
FIRST CLASS TEACHERS, 18S0-Geade C.
XXVII.
1. Ifiuax^-j-2bxi/ + c)/^, kii + lv be substituted for a: and nnt + uw
for ?/, the result takes the form Ait^ +2Bhv + Cv'^ . Find the value
nHB^ -AC)^{b^-ar) in terms of /,•, I, m, n.
2. Iiesolv(sa{b — c)^-\-b{c — a)^-{-c{a-b)^ into factors.
Prove that — =
uvw XIJZ
iiu = x{Bi/^-Cz3), v^^i/{Cz^—Ax^), w = z{Ax^-By^).
S. Extract the square root of
(a-/>)"-J(6-c)2+(/;-c)2(c-a)2-f-(c-a)3(a-&)3,
HU(.i thfl cube root of
^ia-b)^+{b-c)^+{c-a)^-S{a-b)^{b-c)^c-a)^\.
4. EHminate x, y, z from
a b c
ax-\-by-\-cz = \ — = — = —
' '^ X y z
k{x^- -\-y^ +z^-)+2{lx^-my + nz) -\r h = 0.
5. Simphfy "^J^^^^, {l/(4 + 3i) + v/(4-3j)}3,
/-l+;v^3\2 -l-i-yV3
and [—^ ] + 2 — + ^'
in which j= v/( — 1).
6. Given the first term, the common diflference and the number
of terms of an arithmetical iDrogressioft, find (i.) the sum of the
terms, (ii.) the sum of the squares of the terms.
240 3EXAMIXATI0N P.'.PESg.
7. Solve the equations
(i.) (a^.r)3 = (x-Z.)3.
a b
(11.) ax-^bu=~+~=-\.
—1 —1 —1
(iii.) x{y-\-z )—a, 7j{z+x ) = b, z{o'+ij ) = r..
8. What value (other than 1) must be given to q that one of
the roots of x^—2x-{-q = 0, may be the square of the other.
If a, b, c are the roots of x^ — yx'^+qx — r, express •
2a 2 /; 2 _|_ 2/; 2 1: 3 _}. 2c 2 « 3 _fl4_^Z, 4^_C4
'2iih + 2i/c + 2ca-«3 _ /,3 _'c 2
in terms of p, 5 and r.
9. A vessel makes two runs on a measured mile, one with the
tide in m minutes and one against the tide in n minutes. Find
the speed of the vessel through the vrater, 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 A, B, and C, measured from the point 0, are a, b,
and c ; their distances measured from the point P are x, y, z.
Prove that whatever be the positions of the points 0 and P,
x^-i^b - c) +y- (c - a) +« -'(a -b)-jr{b — c){c - a){a - 6) = 0.
APPENDIX.
Section I. — Elementary Theorems on Polyncmes.
(See page 39, et seq.)
Theorem I. If the polynome/ (x)" hs divided by x — a. the
remaiuder will be. /"(«)".
D'AIcmbert's Proof. f{xY is the div-idend, x — a is the divisor*
let/i(x)"~^ be the quotieut, which is necessarily a polynome o*
degree »i— 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,
/(.;)"= fa; -«)/•,(.^•)"-l+/^\
But R does not contaiQ x, hence it will remain the same, not
merely in form but in actual value, whatever value be piven to x..
Take the case x = o, then {x—a)f.y[xY~^ vanishes for its factor « — a
does so, heuco R=/(a)". 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 pi'oof " Division can be per-
formed only when there is an actual divisor, therefoi-e in assum-
ing R to be the remainder of/(.c)"-=-(x— a) it is assumed that ajis
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 ai-gument 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 quantit}'. 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, jf^ - a" is divisible by * », if n
be a positive integer.
By actual division — ; = ^""^ — a •
x — a
:. orJ" —a" is divisible by .c— a if a;"~^ — a"~^ is so divisible,
hence x""^ - a*^^ " " x — a " x''~'^ — ci"~'^ ....
Thus vre can reduce the exponent unit by unit until at last we
arrive at, x^—a^ is divisible by x—a iix—a is so divisible. But
x—a IS certainly divisible by aseK, .'. x^ —a^ is divisible by a; — a,
. x^ — «3 ig aigQ divisible hj x — a, :. so also is x'^—a^ and thus
we may go on to any positive integral exponent whatsoever.
Theorem. Writing f(xY m polynomial form arranged in
ascending powers of x,
/(./;)" = A^ +A,x + A^x^ +A^z^+ + A„x^,
:. f{a)-=.A^ + A,a-\-A2a^ ^A^a^-^ +4„«",
••• fix)'' -/{nr = A,{x-a)~\- A^(x^ -a^) + A.^{x^ ~a^)+ . . . .
+^„(-«"-«")-
But every term of this polynomial is divisible by x — a, and the
highest power of x in tlie quotient is a;"~^ got from the term
-^ni-*" — "")> so the quotient may be represented by.fj(x)"~S
.-. {f(x)''-f(a)-}~{x-a)=J\[x}''-^
f(xY f(a)
x — a -^ I ^ { 'x — a
Theorem II. If the polynome /(.«)" vanish on substituting
for X each of the n different values a^, a^, a^, . . . . «„,
thQnf{xY =A{x—a^){x—a^) {x-'-i-n),
in which A is independent of x and consequently is the coefficient
of.«" in./'(ic)".
Since /(ai)=0, :. f{xY ={x-a^)f [xy-^. In this substitute
a, for X, :. siucef(a^Y =0, it becomcd 0= (ctg — «i)/, (as)""^- Of
this product the factor a^ —a^ does not vanish since by hyjjothe-
sis a.2 is not equal to a^, therefore the other factor f^ia^Y'^ must
vanish that the product may vanish, and consequently /i(^-)"~^ is
APPKNDrX. 243
divisible by x—a^. Let the quotient be denoted by f^{xy~^,
fixY' ={x — ai){x—a„)f^{xy-~-. Substitute a 3 for .c and proceed
as befure, and it will be j)roved that x—a^ is a factor of /(a;)".
Continuing to n factors we get a quotient independent of x, since
each division reduces the exponent of x by unity, .v finally
f{xY=A{x-a^){x-a^) (.6--«").
Cor. lijXx)" and <p{xY' b(!th vanish for the same ?? different
values oi j-,j\xY is algebraically divisible by <p(j;)"'.
Let Mj, a2> ^3' ^m be the m different values of x for
which the polynomes vanish,
.-. f{xY ={x-ay){x-a^) (x— a,„) F(a;)"-"'
9.n^<p{x)"' = A{x-a^(x~a,^) {x — a„)
.'. f{x)" ~0(x)"' = F{xy-'"^A,
which is an integial function of x since A does not contain x.
Theorem III. If the polynome/(x)" vauisli for more than
n different values of x it will vanish identically, the coefficient of
every term being zero.
Let flj, a„. a^ «„, a„^i be n + 1 different values of x
for which /(a;)" vanishes,
Substitute rt„^.i force, and since /(a„+i)" =0,
But none of the factors «ri+i — «n ««+i — «2' ^^* vanishes,
.'. A must be zero, or
f(xr =Oix-n^){x-a„)(x-a^) (a:~«„)
and the factor, zero, will be a factor in the coefficients of every
term.
Theorem IV. If the polynomes /'(x)", <p{xY' {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{xr-<p{xr = A,-B, + iA,-£,)x+{A,-B,)x^-h
{A^-B.)x^+ .... +(^,,-7;>"'
+ J,„+ia;"^ + i+^„j+.a:"^+2 +A„x\
and tins is a polyuome of degree n at most. Biit/(.c)" =(p(.'c)'" for
more than ?i different values of x, that is /(x)" — (p(:i')"' vanishes!
for these values, .-. by Theorem III. f{x)"- - (pix)"" vanishes identi
cally, and the coefficients ^o — i?o, J. i — />\, ^2 "^'2'
A^-B,^, ^m+2, -4,»+2, A,, are all equal to zero,
.•.A^ = B,,A^=B^,A.^=^B.^,...A^ = B,,,A,n^^=Q, A,n+2 = 0 ...
Note to Art. XVII. To find, where such exist, the factors of
ax" +bxu + Cxz-{-ei/^ +gi/z + hz^.
Multiply by 4a
Aa-x^ -{-iahxif + iacxz-i-icei/^ -\-4:agijz-\-4.ahz^ .
Select the terms containing x and complete the square, thus
Aa^x" -\-4:(ibxij-\-4:acxz + b-!j~ +''2bcxz + C-z'
- (63 - 4«e) »/2 - 2{bc - 2afj)yz - (c^ - 4:ah)z" =
{2ax+by + C^) 3 - { (i3 - Aae)i/^ -\-%bc- 1ag)ijz + (c2 - 4ali)z^ }
If the part within the double bracket is a square say (?>'-?/ + ?^)3
the given expression can be written
(2rta; + % + Cz)2-(?H7/ + ?7z)3
which can be factored by [4] . 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 ft and c are hoik even,
multiply by a instead of by 4a and the square can be completed
withoit introducing fractions. If e is less thau a it will be easier
to multiply by A.e 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 a;3 ^-. ry +2-^3 — %"+ 7.'/2- 822 into factors.
Multiply by 4
4^2_i..i.c^4_8//2-8//3 4-28//3- 12z3
Complete the square selecting tjrms in x,
(2.i- + // + 2z)--(3.v-4z)2 =
{(2x+i/+2z) + (3^-4^)}{(2x+//+2z)-(3»/-42)} =
(2u: + 4^-22)(2x--2^V+ez) = 4(a; + 2y-;^-Xx-^+32)
.-. tlic factors are (.<; + 2y — 2)(.e— ^ + Ss).
2. Ga2 -7((^+2«c-20634.G'k-486--.
Multiplygby 4 x 6 = 24
144«3 _ lG8a6 + 48«c - 480^3 4;153G5c - 1152c2 =
(12a - 7^+2'-) 2 - 520^2 +1564ic - llGGc-s =
(12rt— 7i + 2c)3-(236-34c)2=
(12«+166-32c)(12a-306 + 3Gc) =
24(3a+46-8c)(2a.-5/> + Gc),
.-. the factors are 3a +46 -8c and 2a— 5/) + 6c.
3. a:2 4-12^z/+2^-2+2G^2_ 8,^3 -933 =
(x- + 12x^ + 2a-Z + 3G^2 + 12/72 +2-)- 10^2-20^2 -1022 =
(x+6z/ + 2)--{0/+2)v/10}'-' =
{^+(G+x/10)?/+(yiO+lM <
{^ + (G-yiO).y -(1/10 -1)2}
4. 8a2 +10'(.'> - 14^"?+12«f^ - 8/^2 _ 8W+8c2 - 8cfZ.
Multiply by 3, not 4x3, since the coclEcients of tho other terms
in a, are all even,
9« 2 + 30a6 - 42ac+3Ga(? - 2462 _ g^^ci + 24c2 - %icd.
246 APPENDIX.
Select the terras containing a and complete the square
706c-84//^-256'2+60cfZ-36c/-^ .-=
(3« + 5/) - 7c+Gf<) 2 - (7i - 5c + 6d)3 =
i^3« + 12Z/-12c + 12c/)(3a-2i-2c) =
3(/i + 46 -4c + 4(0(3^' -26 -2c),
,'- the factors are «+46— 4c+4(Z and 3a — 26 — 2<?.
Work Exercise XXIX by this method-
Section II. — Indices and Scp.ds.
The general Index-laws are
m p
an .act
= an ' q
(1)
m p
m _ p
= an q
(2)
m
(ab)"
• m m
= rt« . 6"n
(3)
{a -f 6) «
m m
= « » H" 6 »»
(4)
(a^)7
iiip
(5)
Tho law connecting the Index and the Surd symbols is
oJ^ = lYiiV) (6)
[The indices ,}, ^, J, &c., are generally used to denote * cither
square-root,' ' any of the cube-roots,' ' any one of the fourth-
roots,' &c.
The surd symbols ]/, ^, 4/, &c., are by some writers re-
stricted to indicate the arithmetical or absolute roots, sometimes
called the positive roots. Thus
V4 = 2, but 4* = ±2, .-. 4^ = ± y4
Also, v'{(-2)2}= \-4 = 2.
^'27 = 3, but 27^'= 3 or 8/ =f^ j •. 8' = (P).3/^'^
4/16- 2, but 16*= ±2 or ±2j, .-. 16* = (1*)V16.
APPKNDrx. 24V
Witli this restriction the general connecting formula -^onld be
In the following exercises this restriction need not be observer.]
Exercise.
] . Wliat is the arithmetical valne of each of the following :
"■^ Cir-,^ -i/>* on' .5 ni nrr^ n a^ OO" fll" d"^ /OSX^
S6\ 27^ 3 6*, 32', a\ 8^, 27^ 0^, 32% 64', 81% (Sff,
(5:VA (IfV)', (-^S)-, (-027)1 4!)'^ 32"', 8l'"'
-2 -2 1
2. Interpret a-"^, a^, a- , (a-) , a^ , a % (a ^) *, a", «-*.
3. "Wliat is the arithmetical value of
36""^ 27~*, (-16)"% (-0016)""^ {i)~', U%f^, (y%)~K (5,-^,)~*
4. Prove («"■)" = («")"';(«"')"=: (a" )'^'; «-">= (a-^)"* ;
avid express these theorems in words.
6. Simplify a\J, c^ .c\nf.nrK n.n~^'\{'7ifi2i)^-{di;)^
a e a e 6 i i li
"V 1' >' "T" ^=4' (2|) -(6*) ^(i)
a c a e a;
C. Eemove the brackets from
(a«)% (/;)^, (cf ) % (/) % (.^)^ (/ -')
1 a « in 1 2 1 a 4 ft
(a=62) , (a 6 ) , (rt2c i) , (« -c ) (x ?/ ) .
7. Eemove the brackets and sim])lify
(.?• ) (a; ) (x ) ; X x x ;
h h -hi
i; « ; a; a? :
— 2 — 2 2 ^
{a;2 H-a;-2 }{.c<-2) --a:-^ }.
248 A|>PENDIX.
1 a 2.
(-a;) {x ^-x }
9. Deteimiue the commensurable and the surd factors of
12^ 24*, 18~*, (-81)*, 12^ 64', {^^J, (6J)~^.
(The surd factor must be the incommensurable root of an
integer.)
10. Simplify 8*+18*-50^; 72*-}-(^y-5/'-(Tl5)~*;
{(6+2^^)(6-2^)}'; (2* + 8^)'+(2--3V;
(2 +3 )(4 +9 -6'); (7'--3') (7 +3") , ■
[{{a^x){x+h)]^ - {{a-x)ix-h)fy^ ;
Express as surds,
11 a , X , p , c . /i
7( + ^ — w + 5 -25 — 'i + OT
12. ic , y , a , b
a 7n-3 72 — a
13. (aa;-&)*, (x3-4;r+l) 4 , {p-gx)
Express mth indices,
14. ■,fa^ \/c^, \/x:\ ^Vy"-", V(r/a;), |/a-*.
15. ■,f(a3+i3), ^/(«3+/,3)3, /^(«3^_i3)p^ .>/|(a_j)a;|,
;/(a - /«)"-\ V («" - 6" )™-"'
16. (/t {b~^f, {o'~h~^, (xfK (ct-'x)'^, {a-^x~^r^,
7 —i 14
{x y ) •
Simplify the following, expressing the results by both nota-
tions.
APPENDIX. 249
17. rt.a , «".« , a .a , a.a , a •}/«,« ^a^,a \/a
l i 6 i i — ± s — i *
a 0 c . a 0 c
^1 i
a a !yc^ ^x^ F ?/~" ■|^(24rt~3^ c(ah) —ac
i/a ^a Wc as — i 2V« , / ,v
^ • * *^ X y bc—c{ab)
1 1 H 3 Sn Sn
a +rt a'— a "" a~ ^ — a" «-+l + a~"^
19. -J ZTf -^ =1' —3;^ 3^. a + l + ar^
a —a a —a a a + a ^
on -n- •;! v - - I , 9 I , I , a:S . i 4 4
20. Divide a;-i/ by x" —y; x +a x +a by -r" .<; +« ;
i \ ^ h i i
x+;j+z—3x y z hj x +y +2'
* i i
2ai+2k-+2ca-a3_ft2_c2 by a +& 4-c
Exercise.
1. Express tlie following quantities L as quadratic surds, ii
as cubic surds, iii. as quartic surds.
2. Eeduce to entire surds,
x^/x,afa, V-f/b\ 81^3, 4f2, V^, W^> W^y H'^
^V(!).^V(r'fr(.V).
{x-yi' '^ix^-+2xy^y-')~\ {x-x~^)f{x^--\-l)\
250 APPENDIX.
3. Reduce to their simplest form 1
,/12, yS, v^50, if 16, 4i3/-250, V^, f^h V^\, 5ir(-320), ^
V(l-A)> l/«^ l/(('^^0> if"-', 3iif(54a;9), 4/(a;'?/'^9),
,/{a3(l_.,-2)>, ^|a2(«2_i^4}^ ^(,,i)^ ^,,.+1^ ;ya'"+«,
y«^"^^ >,,'"'-S v'(«2a; + a3), if(a3+2r.4;« + asx^),
i/{(x-1)(.a;2-1)}, f/{(«3 + 2r/u,- + a,---')(rt3+a;3)},
/(8a;2- 16x4-8), ir{(a;2-2+;c-2)(a-*-2x-3 + l;;,
l/2x_-2+2^j l/3.r3-6a;2+3.^.^ I /(a^ -aZ»)3+4a3j \
^\ *+2+a;-i /N\27a:3+18x- + 3N\ ^1^6 j
4. Compare the following quantities by reducing them to the
same surd index :
2 : VB; 2 : ^f 9 ; t/2 : ,f 3 ; i/lO : ^f 30 ; 2 /2 : if 22 ;
fl3 : >/a3 . eyx:^^i/; ^x : ;/// ; ^x^ . "jx^; i/a : ^yb : l/c ;
ya:^^b: ;^c ; '>/" : ^^-'^ : ^o'".
5. Eeduce to simple surds with lowest integral surd mdex
v^(r«), r(Vi), r(i/t-), rd/^-i), v(r^-^). ^(r-.-^"),
r(^-«^^^)> if (l/27), ^/(if 81), 4/(if 81), i/(«^/a),
^(ai/a), -/(.rifx), -^(^s y,)^ -^{5^/5), |/(3if3),
V(3if3), t\x^^x), y{«y(/>yc-)}, a;/(..-iyx-i),
6. In the following quantities, combine the terms involving the
same radical ;
3/2 + 5i/2-7v'2; i/8-V2; ifl6+8if2;
^16+ /2; a^/x-^x; al/x-hl/x;
8i/fl +5 '/a;-7i/a+ v'(4«) - 3 v/(4a-) + 4 ^/(9x) ;
|/a;+3i/{2x)-2v (3a;) + i/(4a;)- v/(8a;)+ v'(12^-);
APPENDIX. 251
Ix - 3 i/.r + 5^x - 2 Vj;3 + ^.c3 ;
AV{a^-x) + 2y\h''-x)-S^ {{a + hy-x] ;
V{{a-hyx] + y''J,{a+byx\-V{a-x) + x/{{l-aYx)-Vx;
^/{a-h)-r-^{lQ>a-lQb)^-^{ax^-hx")- V {^{a-b)) \
■,/(a3-|-2r(3/j + a/v3)_ ^ [u^ --la^h^ab"-) - -^ (Ub^-).
7. In the following quantities, perform, as far as possible, the
indicated multiplications and divisions, expressing the results in
their simplest forms :
V2.T/6; a/3, a/12; ^/U. a/35.|/10; y'ayiSa);
v/c. a/(12c) ; V{ex).i/{Sx); v'l/^Vy';^!/'-^'/';
i/a"+V«"'''; iri"-^^iri-^""^ yi2--^/3; ^/(ex)-!/^^^);
(«+x-)H-V(a+a;); (a-' -x2)-f-V(a-a;}; (a;^ -l)^if (ic+l)2 ;
(3v/8-5\/2+v/18+v/82+v/72-2a/50).-/2;
(7i/2-5a/6-3v/8+4 a/20) a/18); (N/54-i/3)(i/5-y 3) •
(y/24-l)(s/6- v/3); (3-v/2)(2-f3v/2);
(5v/3+v/6)(5i/2-2) ; (v/a- ^/i)(^ ,, + ^/>) ;
(ay'6-j-i\/a)(6 \/a — ay/b) ;
{V{x+l)+V{x-l)}{V{x+l)-V{x-l)};
{ v/(3« - b) + -^{Bb-a)}{^/{Ba-b) - ViSb- a)} ;
V(a+Vb).V{a- Vb); a/(iA:+ v///). A/(v/a;- \/v/) ',
ir{x-v^(a;3-l)}.^{a;+,/{x3-l)};
^{V«-l/(«^-^')}-#'{v/(a3-a;3)+av/4;
252 APPENDIX.
y(B + 3A/7).y(8-3v/7); (Va+i/b)^; {^a+Vh)^;
f \l'2a\ |/S6\j2
[{V(«4-a')(^-^)}+V{(a-a:)(:.+/.)}]2;-;4T;;;;-4,^)|
[\/{(r? + .r)(a:+&)} + y{(a-rc)(.c-M}]2;
{V(V104-1)-V,V.10-1)}2;
[V{a4-V(«2-a:2)}4-V{rt-V(«2_a.2)|j2.
(Vj: + V//)* + (Vx-V»/)4; (a^+(,?V24-62)(a2 -^6 A/2+i2) ;
(V^H- V/' + Vr)(V/> + Vc - ^/a){^/c + \'a - ■^h){ya-\-y/h - Vc) :
(^,,+^i + ^,){^a2+if6-^+^.'2-2if(6c)-2i3/(c«)-2ir'y^)}-
8. Fiud rationalizing multipliers for the following expressions,
and also the products of multiplication bj' these :
a-\-\'h, Vct + Z^Vr, (iVh — hVa, « + V(a^ — a;^),
V(a-ic) - V{a+x), V(«2 + v<-) +V(rt3 -Vc),
V{8+V(24+V5)}-V{8 + V(24-V5)}, Va + Vi + Vc,
3 + V2+V7, V6 + V5-V3-V3, ^a + ^h + ^c + ^d,
f/«2— ^o2, V«+^/c, ^a~f/b, ^Ja^^a, y^ + Vy^
fa + fb-{-f/c, a + ^b + ^c.
APPENDIX. 253
9. Eationalize the divisors and the denominators in the follow-
ing, and reduce the results to their simplest form :
1^^2-V3), 3-i-{d+y'6), 5--(v/2 + V7),
(,/3 + ^/2)--(^/3-^/2), (7,/5+5t/7)-^(,/5+ ^/7),
a^x + bVi/ 2V& i+3,/2-2\/3
cVx-eVy^ ^2+i/3- V5* i72 + ^73+|76'
V^6— 1/5— 1/3+\/2 2
2c H-aH-v^+x2^)
^/{n-\-x)-\-Via — x) 1
y^{{l+a)(l + b,}-V{{l-a){l-b)}
y{il+a)(l+6)}+i/{(l-a)(l-^)}'
(a - X)l/{b^ + ,/2)_(6-y) i/K+^
(rt+x)i/(t2 + i/2) + (6 + Z/)V(«^+*^/
V(l + a)^l/'(l -«)+;v/(l+i)- V(l- 6)
y'(l + a)+V(l-«) + v/(l+/>j+V(l-/.)'
x/(x+fl)-v/(3;-a)- v/(a;+/^) -|- \/ (aj -- 6)
{/{x^a)+\/(x-a) + ]/{x+b)+ V{x-by
y/a i/b j |«+a;\ I /^-^i ' |t + y/x Wx—Vy
1 1 Va -Jx
lY-f-,/(rT2_l) s/x~V^j Vz~Va
•vl«r-i/t«^-i)' 2_ il' ^^ ^^*
Vx \/// Va; Va
254 APPENDIX.
10. Fiud the values of the following expressions for ?; = !, 2,
3, 4, 5, respectively.
1 |/i+i/5r /i-i/5n
V^Sli 2 i 12/1'
J._ J (2+t/6)"+^- (2+V6) (2- a/6)"+^-(2-V6)|
2v/6t l + i/G ~ l-v/6 I
11. Show that
^ [{^•+l/>'-l)}*"t^+{x-s/(a;^-- l)}"-' + 2]
2(a;-l)
is a square for n= 1, 2, or 8 respectively.
12. Extract the square roots of
x-^-ij — 'l v^(xi/), a-^c +e + 2^\/ {'(c-\-ce),
a + 2c + e + 2^/{{a+c){c+e)}, 2a + 2V (a^ -c^),
Vx + 2 + Vx-'^, x + dx^ +x'^-^2x^/x-]-2x^ V X,
x^ -X1J + \i/^ + \/{4:xhj - Qx'hj^ +x>j^), 2x+s/ {%x^ - y^),
6-2v/6, 10+2a/21, 9+4\/5, 4-V15, 7 + 4v'3,
12-5v/6, 70 + 3|/4ol, 4-|/15,
9 + 2|/6+l(i/2+ V3), 15.25 -o\/.6.
13. Find the value of
(a + h)xtj . a \fa h-^/h
y(a;-+.v3), given a;=vK(rt^c) y = f/{a^e);
V(l+a;)-V(l-a;) . 2ah
V(l+x)+V(l-«:)' S^^^'' •« = .T^TP '
2«V(l4-a;2) . f Y \'\
255
:.h+c = 0.
ir c- IT iO+V5)x-2 x(l-V5)a;-2
15. bimpMy ^, _,^+ ^g^i + ^rr. , ^i _ v5)..+i-
Complex Quantities.
Quantities of the form a-{-h\/ — l in which neither a nor h
involves \/ — 1, are called complex quantities. The letter^' (or i)
is frequently used as the symhol of the diteusive unit V — 1, so
that a + by/ — l would be written n + hj. So also V —x=j\/x,
\/ — x.V —y =p V {xy) = — Vxij, and J 3 = —j
Exercise.
Simplify the following, writing^' for -/ — 1 in any result in
which the latter occurs :
1. v/-4, 1/-36, |/-81, V-S, iZ-ia, 1/-72, if-S,
^_5.^_6, v'-6.|/-8, \/-8.v/12, -/_8.|r-8,
V/ -5.1/ -20.
2. ,/-», i/-a;3, v/-^<3^ i/-^'^", N/(-a)^ ■i/(-«)-\
n/5. \/ — rt.
3. i-, i^ i*, i', i^ j'\ j'\ j''. p\ ./n /"+S /"+^ /"+»,
4. "i-'y, jVx.j\/ij, 5j, j-x/o, Ji/- a, jY-a^, Ji/a.^ -a.
6- V-p, V-p, V-J^' v/-i^ -/-i^", i/-j4".
y' _ 6 V-C y/B J|A_ l/ rt 1
*^' "V3 ' y-3' ;/-3' v^-T 736' TZl'
a a^ i/( — «*) — V - 1 aS
^_e3' .j/(_«)2«-i'
256 APPENDIX
7.
111-11 1 -1 1
J ji jS J ji. jin-rl jin^l yn-l
a^j X — y cj^
l/-«^' JVx j\/-y'^' ~ V - c^
8. V{a-h).V{b-a), ViSx-4:y).V{iy-3x), {3 + 5j){l + Aj),
(8-9;-i(8-7;), (7-i^/5)(7+i /lO), (v'3~jva)(V2-jV6),
(a + /./j(c' + .;), {u + {a-l)j}{a + {a + l]j\,
(l/«+iT/6)(i/«-J/Vc) (a+/y-)(a-/y-), (a;' + &) («;'-&),
( \/« +i }//')( y«-i\/^), {(t V Ij +CJ V x)iaY/'b -cjW x),
^/{l+j).V{l-j), V(3+4j)V(3-4y-),
v/(12+5;-)VU2-5;), (l+Jj^ (ya-jVi)2, (5-2;V6)2,
(«+^yr+(«-^jP, («+^;J^-(«-/>y'j^ («+^!;')24-(ry-^/)^
n+jvd\\ /-i+iV3f /-i-iv3|3^ /1+jy^ (^-"-^y,
{a + bj}^-{a-bjy, (l+j)5-|.(i„^-).., (l+jV2)^ + (l-iV2)«
[A{ 7(30- G a/5)- l-/5} + i;{N/15+ ^/3+ v/(10-2v''5)}]»
for all positive integral values of n.
4 64 21 5 1-20/75
9.
14.JV3 l-iV7 4!+a;V6 V2+JV3 7-2yV5
1+JV3' 1-f 1+i' 1-i' (i+j)^' 1-i' ^--i//
a+iVx jVo + V— ^ <t — /^' a-\-jV{l-x^)
a^\/x V-«-iW/ o/'+i' a-yv^l-a;'-^/
APPEXDIX. 257
y/(x-y)-V(!/-x) 1 _^J_ l±i,l-i
1 1 - 1 1 X+ llj X — If}
■ ■+
x + yj x — yj Vx+jVy Vy+j^x
a 4- hj ~a - hj V.c - jVy ^y -jVx'
y(i +a) -iva - «) -v/ii - «) -7va +«)'
10. y'(S + iJ} + y{S-AJ), ^/(3 + 4;-)- V(3-4j),
v'(4 + 3j)+v/(4-3;-), i/(H-2Jv"G) + ;/(l-2;VG)
v/(5+2;VG)±v'(5-2yV6),
v/ (2^/15 + 3q;>±: v/ (2 v/ 15 - 30;),
v/(y3+iv/105)+ v^d/^-i/lOo),
V{a+j\/{z--a^)}±\/{a-jV{x^-a^)},
A/{«-+.MV'<-^-'+2(/2)}±-^/{«3-J>y(x3 4-2r(3)}.
11. Prove that both A(-l+.y|/'3) and i(-l -;V3) satisfy the
equation — 171" = 0,
that (x4-u7/ + (f2z)3 =a;3^^3_^2n_|_3(a.4.ii.y)(y.|.j<,2)(2_^„.a;)
and that {x+y+z){x-{-wy-\-iv^z){x+w^y + wz) =
2..3_|_y3 _|_23 — 3x^z, iu which w represeuts either of the pre-
ceding complex quantities.
Hence, prove that
(i) {2a-b-c+{b-c)jVB}^ = {2b-c-a+(c-a]j^S)}^ =
{2c — a-b + {a-b)jy/'S}^ ;
(ii) u^ -^v^ -\-w^ —Suva- = {a^ +b'^ -\-c^ - Sabc) X
(z^+y^ -[-z^ -dxijz),ii u = a;v + by + cz, v = ay-{-bz+cx,
w = az+bx + cy, or ii u = (ix-{-cy+hz,
v=zcx + by-}-az, w = hx + ay + rz.
258
APPENDIX.
12. Prove that i {\/5 + l+j v (^10 — 2 v'5)} satisfies the equation
x^ + 1
x+1 "^-
Writing w for the preceding complex quantity, prove that
and {x ■^y-i-z)(x+tc^y — tc^z)(x - w^i/ — icz){x - u-z-\- w^z)
{x+W^y + ic-z) =x^ +y^ +z^ - 5x''^yz+5xy"z^.
Prove that {4a + (/j-c)(i/5 -1) + (64-c)J|/(104-2a/,5)}» =
{l(a + b){-l-tjV{]/5 + -2)} + {a~b){V5+jy{s/o~2)}]
X y5-4c}\
Section III. — Pure Quadratics.
Examples.
^ x+B{a — b) a{3x+9a — 7b)
x^dja'^b) ^ b{8x-la+\9b)'
m p >n-\-ii. p-\-q ,
Apply, if — = — , •"• = »
i:i- J^ n q m — n p — q
X 3.-c(ffl + &) + 9a3-14«i> + 9i2
•*• 3(a_^)= Qx{a-b)-^9{a^--b^)
Dividing the denominators by 3(a — i)
.-. a;{+3(rt-l-i)}=3.c(a+^)4-9t(3_l4«i + 9i^
.-. a;2=:9«2_l4a6 + 963
'a;-2rt + 4//\ ^ 5x-9a-\-8h
a;+4a — 26/ ^ 5a; + 3a - 96*
m p n — m q—f
Apply, if — = — , .'. = — , and factor the numerator
rr .; ' ^ ^ ' ^^ p '
(a;+4a-26)2-(x-2a + 4i)3,
APPENDIX. 259
12(x-\-a+b)(a-b) 12{,7-b)
(a;+4a-26)3 "5x+3ft-96'
x+a+b x-h'4:a-2b 3{a-h) ^ ^,.
"• x+Aa - U = 5^+3^7396 = i^r^^^T^- ^7 taking differ-
ence of numerators and difference of denominators. To the first
and third of these fractions, apply if
rn ij m p
n " q ' " n—m~q—p
x + a + b ^a — b)
" 3(a - 6)~4x'-4a — 4Z>'
.-. 4{a;2-(a + ^)2}=9(a-6)2,
... a:-2 = j|4(a + 6)3}+9(a-/.)3}.
^- ,/(3a;-2_l)_y'(3-a;2)- ^ '
Sx- — 1 rt4-6
^
3_a;3 -a-6'
3-a;- -{a-by^
4. 7/7V(l+a;)-wV'(l-a:)=\/{??i2+n3) (1)
Square both members and reduce
.-. (7n2-w2).r-2/nr?v/(l— a;2) = 0. (2)
Transfer the radical term and square both members,
.•! (7ra2_n2)2a;2=4„i2,j3(i_a.i3) ^3)
.-. (TO2^„2j2a.2=4,„.2„3 (4)
+ 2?nn
x= ~ ■ o. (4)
The above follows the usual mode ol solvluo: equations involv-
ing radicals, viz., make a radical term the right-hand member
gathering aU the other terms into the left-hand member, square each
260
APPENDIX.
member, repeat, if necessary, until all radicals are rationauzed.
This method is couveuient but it does not explain the difficulty
-\-2mn
that only one of the values of x in (4) satisfies (1) viz. — ^ ^.j-
— 2mn
The other value, —5— — ^ satisfies the equation
m a/(1 -\-x) +n \/{l~x) = |/(m2 4- ?,3).
The explanation is simple. Squaring both members of (1) is
really equivalent to substituting for (1) the conjoint equation
{)nV{l+x) - nV(l - x) - V{m''^+n^)}
{mV{l-\-x) + nV{l-x)-V{nt-^+n'')\=^0 (5)
which reduces to (2) above.
Treating (5) or (2) by transferring and squaring is equivalent
to substituting for it, the- equation
{m\/{l + x)-n\/{l-x)- \/{m^+n^}'^ X
{ms/(l -ha:) -'«|/(1 -x)-^ V {m^ +n^)} X
{m-i/{l+x)-\-n V{l-x) - V{m^+n^)} x
{m\/{l + x) + ni/{l-x) + i/'{m^+n^)} =0 (6)
which reduces to
{(m2 -n^)x-2mnv{l -x^)} {m'^ - n^)x + 2m?z ^/(l -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 A = 0 and then multiplying the
member A by rationalizing factors. Thus instead of solving (1)
we Freally solve (G), i.e., a conjoint equation equivalent to four
disjunctive equations. (See page 140, Art xl ) Now the values
given in (4) will satisfy (G), 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 sucli a set of disjunctive equations is proposed
for solution, the conjoint equation must be solved, and if there be
a value 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 ai e) .
Furtlier, 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.
Reduced to an equation this is
y(.,+4)-i/(:c-4) = 4 (8)
Rationalizing this becomes
{4-v/(a;-t-4) + v/(x— 4)}{4-i/(a; + 4)--,/(x-4)}x
{4 + V(a;+4)-f;/(a;-4)}{4 + |/(a;+4)->/(a;-4)}=0 (9)
which reduces to
{2t-8|/(a;-F4)}{24+8T/(a;-f4)}=0
ie. 9— (a;-F4) = 0, or a; = 5.
Now a;= 5 satisfies (9; because it makes the factor
4-|/(x+4)- v/(x-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+ ^/(x+4)+ v/(a;-4),
or 4+ \/{x+4:)-V{x-4)
vanish. There is, therefore, no number that will satisfy the con-
ditions of the problem.
[It will be found that as x increases, 1/(0; 4- 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 ]/(a:+4)— \/(a; — 4) is 1/8 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))-x/{{a-^){h-x)\ =
^{{a-x){h^x)] - ^ {{a+x){h-x)} (1)
Collecting the terms involving -\/{a+x) and i/{a,-x) respec-
tively the equation becomes
{x/{a + x)-V{a-x)}{i/{b+x)+\/{b-x)}=Q (2)
This is satisfied if either
y'ia+x)-i/{a-x) = 0 (3)
or Vib + x)+\/{b-x) = 0 (4)
The rational form of (3) is {a + x) — {a - cc) = 0 which is satisfied
by a; = 0 and this also satisfies (3).
The rational form of (4) is (b + x)- {h—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^ +Suv{u+v)
:. {a+x) + {a-x) + nf{2a{a^-x-)}==2cL.
.-. 2a(a3-a;3) = 0,
.". X = ±«.
Both these values belong to the proposed equation.
APl^NDIi.
The rationalizing factors of
are -^{a + x)-{-u^{a-x)-u--^[2a),
and |/(a+a;)-f oj-|^(a — x) — w|^(2a). See page 257.
The remarks on Ex. 4, will apply mutatis mutandis to equations
of this type.
'■ f[a+x)^--^{a2-x^y,+ ^{a-x)^
Assume -^(a+x} = u and -^[a — x) = v
.'. u^-{-v^ =2a and. u^ —v^ = 2x,
u-^ — V^ X
and .". o , - —
Also (1) becomes
u^-\-uv+v'^
u^-uv-\-v^~'^
u — v
Multiply both members by
u + v
M3 |;S 11 — V ^
T tO\
X u — v
u^ ->rv^~ "^ u + v^ ' ' "•' ^"' a ~ u-\-v
(1)
(2)
(8)
(4)
Again adding and subtracting denominators and numerators
in (3)
m2^j^2 c + 1
My ~ c -\
Adding and subtracting 2 (denominators) and numerators in this
M-— 2MV+t;3 3 — c iu — v\^ 3 — <T
or
w3 +2 WW + r2 ~ 8c- V " \« +«/ 8c - 1
.c^ 8-c
.-. substituting by (4), "^=''■^3^^113'
|3-r
264
APPENDIX.
8. \\/{x+a) + Vix-a)}^r{x-ha) - V(x~a)} ^1c {1)
Assume ?6=V(x+«) aud v=\/{x — a), and (1) becomes
(ii+v)^{i(,-v) = 2c or (u+v)^{u^-T^) = 2c (2)
Also u^-v^ = 2a or (u^+v^){u^ -v^) = 2a (3)
and u^+v^ = 2,x. . (4)
From (2) and (3), (u-v)2{i(2 -v2^) = u.-2c (5)
.'. (2)X(5), («2_t.2)2(„3_.j;2)2 or (,t2_^.2)4 =4^(2(1-0) (6)
Also (3)2 + (6),
or (u4+i;4)(if2 _r3)2 ^2(«2+2ac-c2)
Substituting by (4) and (6)
2xi/(2flc-c2) = rt2+2ac-c2.
Exercise.
1. {x-\-a-^b){x-a + b)-\-{x+a-b){x~a-h) = 0.
2. (a + 6a;)(/; — rtx) + {6+oa;)(c— ?jx) + (c4-aip)(a— ca;) = 0.
3. (a+bx){ax—b) + (6-|-ca:)(6a; — c) + (c + aa;)(c» — a)
= ^-(a2 + i2+c2).
4. (a + a;)(6-a:) + (l+aa;)(l-6a;) = (a + 6)(l+x-2).
5. (a4-.c)(i + a;)(c-,v) + (a+a;)(/>-x)(c+x) + (a— a;)(6+a;)(c + ^)
+ (a-a;)(6-a;)(c+a;)+(a-a;)(6+.t,-)(c-a;) +
(a + a;) (6 — x){c — x) = 5abc.
6. (rT, + a;)(?> + .T)(c + a;) +-(«+x)(6+.c)(c-a;) + (a+a;)(/;-Jc)(c-f-a;)
+ {a—x){b + x){c+x)-\-{a+x) (b—x) {c—x) + (a—x) {b + x) {c—x)
+ {a-x){b — x){c+x) + {a-x){b-x){e-x) = 8x^
7. {a+5b+x){5a-^b-{-x) = 3{a + b+x)^.
8. {a + nb+x){na+h+x) = 9{a + h-\-x)^.
9. (9a~7fc + 3a-)(96-7a+3a;) = (3a4-36 + a-)=^.
APPKNDIX.
265
ub cd ^ x — a x-\-a
iO. ,-T36^.-H,T3rfF->=0. 11. ,+!+,— , = 2c.
a+x x+b ax+b ex+d
' a — x~x — b' ' ■ a-\-bx~c+dx
a-x 1 — bx ^ a—x -k-x
14. :, = -, . 15. 1 = '. /—
1-ax b-x 1 — iix 1-bx
x+a + 2h b-2a + 2x a+U+x Sh-a-{-x
^^' x + a-'lb^b + 'Ia—'Ix a-4:b+x~ 3b + a-x
x-Loa + b x-n + b a—lb-\-x a+5b+x
■JO . JQ = . - _ _
x — 'da+b a — x+'db' ' la-b — x 5a-\-b-j-x
Sa-b—x 5b-da+x ^, 3a-2i + 3a; x-a+2b
a — 36 + a; ~ 5a — 36 4-;»'
3rt-264-3a;_ x~7a + 8b
a — 2b-\-x ~ 3a;— 5a +46
^^' ^-^6 + a;'^5a-36+^-' a-26 + a; ~3a;-3-i + 26'
22.
5a_G6 + u; 3a — 55+3^ ff + 6-a: 3(r?.-6-!-.'c)^
a+x ~ rt + 6+a; * ' 3a — 6 — 3a; ~ a— 56+a;
la + b — x 3(fl^ — 6+a;)
6a-{- 36 - 3a; ~ a - 176 + a;"
5a-b-\-x 2(2g— 6+a;)
2(rt + 26-sc)^ a+116-a; *
'ja-b-\-x a{a + 5h-\-x) x-hn-h a(x+a + ljb)^
27. fbZi;[:^= bij^i + b+x}' x-a+b~b{x+5a+b)'
25.
26.
29.
5fl-36+x\ 2 7a -96 + 3a;
/5a-
\56-:
3a+a;/ 76-9a+3a;
/a + 56+3;\ ^ a + 176+x
/a + 56+3;\ ^ _ a + 176+x
30- [^a+b+xj ~17a+6+a;
7a- 6+a;, 3_17a + 6-a;
31.
32.
\76 -a+a;/ 176 + a— a;
17a +6-X _ a 2 (a + 176-M) ^
0+176^^ 62 (17a +6 + x)'
2G6 APPENDi::.
33.
{5x + 3a-llb){x-a + nb) 5x+7a — 59b
g^ ( 1 +3x+\5x^){x^ + '3x+5) _9^
{l+2x+Sx^){x^-{-2x+'3)~ 4
35.
36.
37.
38.
39.
40.
N/(l+a;2) + y/(l-.e2) g
V{l+X^)-y/[l-X^)- b
^(l+a;2) + ,f(l^a;2) g
-^(lJrx')-f{l-x^)~ b
t/(l+^-") + t/(l-a;^ ) _ a^
V(l+a;2) + y(l-;e2) _ _«_
e/'(l + .«^)-y(l-u;3)- /;
V(l+^2 )+ Via;3-1) _ ji
y(>f3+l)+y(x3-l) a
41 . ■,/(4rt + A - 4x) - 2 v/ (a + Z. - 2x) = i/&.
42. ^/(3rt-2/; + 2a;) - %/ (3a -26- 2..:;) = 2 v a.
43. >v/(2«-/>+2.6-)- -/(10a-96-6a;) = 4N/(a-&).
44. -i/(3rt - 4/v + o;c) + ]/(«-«) = 2 \/(.t;+a).
45. \/(3a.-4//+5x) + i/(a;-«) = 2v'{2.c-26).
46. i/(5.c-3r,4-4i)+^(5.e-3«-46) = 2v/0c+aV
47. -i/(2a+5 + 2.f) + i/(10^<H-96-6.c)=2v/(2a + ?>-2a;).
48. 2'/(2rt+6+2vC) + i/(10a+6 - 6a;) = v/(10a+96-6.-,;).
49. |/(2rt-13^ + 14a;)+v/{3(/>-2a+2a;)}=2v/(2«-6 + 2x-)
50. |/{3(7a4-6+a;)-N/(«+76-a;) = 2,/(7a + 6-x-).
61. >/{{a^x){x-^b)) + V{{a-x){x-b))=^^{ax).
62. ^/{(a + x)(.c-4-6)[-i/{(a-a:)(x-6)}=2|/(M.
63. a/(«:c+x-)- \/{cix-x^)- v/(2ax-a2).
a/. ^
58
APPENDIX. 267
1 1 _ ^
^^- l4-^(l-«)+l-i/(l-x)- 9*'
x+ \/{ax) a-\-i/{ax) x — a
« — n/ {ox) X — ■\/{ax) ~ a
_ T/{(«+a:)(.-c + fc)}+A/{(«-a:)(.r-&)} |^
{{a-{-x){x + b)} -V{{a-x){x-lj)} <b'
i3rt-2/. + 2.r {x/g-h \/{2a-2b)}'
■^3(1 — 2b — 2x ~ '2b— a
59. ^{a-\-x) + f{a-x) = 2^a.
60. ^(a+x)2-if(a2_^.3j_,_^(„_^)2^-^3/,j3.
62. f(l+x)'-' +1^(1 -^•)' = 2i 1^(1 -^•-).
63. 1^(3+^) + i^^3-x) = 1^6.
64. |^(l+x)3+|^(l-x)3 = 5{^(l + a;) + i;^/(l-x)}2.
65. ^{U+xy- -^{VJQ-x"- j + f'iU-xy^ =1.
66. {^(9+a.0+p^/(9-x)}ir(81-a;3) = 12.
67. {^(14+.i-)2-if(14-x)2}{i^(14+a;)-i^(14-a,-)}=16.
68. {f^{57+xy- +^/{51 -xy-} {^{57 -x)+ ^{57 -hx)} =100.
69. 5{4/(41+a;)4-V('il--'^-)}^=8{V(41+.r)+]/(41-x)}.
70. {t/(x+5)+^{x-5)}m/(x+o)-t/{x-5)}=2.
71. {V(^+l)4-Ma'-l)}{v/(x+l)+v''(x-l)} =
26{t/(a;+l)-t/(^-l)}.
72. ^\±^^ + f^]^^ = a. [y + r^=«].
73. 2{ir(l+a-)2+^V(l_x-2)} = (c2 + l){^(lf.f)-v^(l-a:)}2.
268 APPENDIX.
74. f^{a+x) + f'{a-x) = ^c.
75. {f{a + x)+f{a - x)}^{a^ -x^-) =c.
76. ^{a+x)"- - ir(rt- -a;3)4-ir («-«)- = fc\
77. {#/(«+■«)'— if («-^)'}i]r(''4-^0-ir(«-^)}=c.
78. {^(a+:c)3 + ^/(«-x)3}{^(a+.^)-f.^(a-x)} =c.
79. (a+a;)|f(a-.T;)-(a-:«)irOf + .c) = ^{l^(a + a;)— |r(a-a;)}.
80. (a + ;^)|r('f+-«)-(«-^)lf {«-*•) -''■{#^(« + ^)-ir(«-u^)}.
81. {if(« + .t-)3-y?/(a3_.,3) + ^(a_x)2}2 =
c{if(a+x-)4-ir('^-.^')}-
82. {:t/(a+.'«) + M«-^-)}' = (<' + l){ /('^+-«)+ \/(«-:c)}.
83. {t/(-^-+«) -t'(.«-a)}{^/ (.*; + «) + i/(.c-a)} 2::=
c{:/(.f+a) + V(aJ-«)}.
Section IV. — Quadratic Equations and Equations that
CAN BE resolved AS QUADixATICS.
Examples.
1. x^ + {ab + iy- ={a^ ^h'^){x^- +1) +2{a^~b^')xi-l,
^. x^+n^b"- = {a^ + b^)x'' + 2{a''i - b2)x + {a- by
:. x^ + 2abx'^ +an^ = {a-\-byx'' +2{a^ -b^)x + {a -b)'-
.-. x''+ab=±{{a-\-h)x^-{a-b)],
or x^T{(i'-\-b)x + ab= +{a — b),
:. a;3qi(a4-^)cc+i(rt + 6)3-i(a-&)3±(a-6),
APPENDIX. 269
2 («-^)V(«-a;)+(a;-fc)W(a;-^)_^^_;.
{a — x)i/{a~x)-'[-)x — b)\/(x — b)
Writer — 6 in the form {a—x)-x-{x~b) and multiply by the
denominator of the left-band member,
.-. {a-x)-i,''(a-x} + {x — b)^V{x—h =
{a-x)'-\/{u-x)-\- {a — x){x — b){\/(u-x)+ y(x-b)\-L
{x-b)W{x-b),
:. {a—x){x-h){s^{a-x)+ /(x-i)}=0,
.". (« — x) = 0, or x — h = Oy
or s/ {a — x)+V{x — b) = Q.
x'l =«, o-'g = b.
The equation v'(a— x) -I- V(a;-6) =0 has no solution for the
sum of two positive square-roots, cannot vanish.
The solution x= •K''' + ^)) belongs to the equation
<^ya-x)-\/{jc-b) = Q.
ax-\-b mx—n
3. ,— J- =
bx-^ a nx — m
Add and subtract Numerators and Denominators
{n+b){x + l) {in+n)(x-l)
{a - b}{x — l) ^ {m-n)[x-{-iy
lx-{-l\ 2 _(a — 6)(m-fn)_
. /x-hl>'^_(a-6)(m-fn)__,
•• [x-lj {a+b}{m-n) ^'
S-fl _ Sj-1
••• a^i = ^^, x^= ^_|_^-
6-1- u; a —a;
Square both members, subtract 4 and extract the square-root.
270 APPENDIX.
•■• v''^^ = ^{'-±V(f2-4)}-^say,
a — X 2x — {a-b) 1 — e^'
.-if
■ — n
(a
Or thus, cube both members,
a — x h-\-x o
0 -\-x a — x
{a-xY + {h^-xY ^
2{a — x){b -^x)
f (/^+x-)-(a-a;) \ 3 _ 0^-30-2 _ (c^l) " fr - 2)
•"■ uT+^i^(^^-)j " c 3 - 8c + 2 " ^c^l) 2 (^t + 2)
1x-{a-b) c + l |c-2
* ■ a + h ^ c^\ 'nIc + 2*
1-^3 c + 1 Ic-2
(Prove «iat^:p-^3 =-]3 ^^T^g' ^^ 2. = .±v/(c3 -4).
5. -r, r-^vn: ( = — ^^— — —- Eationahze Denom.
{V{a-x) — V(b-z)]^ _ V{{a-x){b-x)}
[a — b) ~ c
{V{a—x)-Vib-x)}^ a-b
o^ ~'V{{a-x){b-x)}~ " c '
(V(a-a;)i- V(6-g)p_ a-b .
'■ \v{a-x) + V{b-x)) a-b + 4c ' -
V{{a-x){b-x)} I g-fe . .
c ~N«-i + 4c' ^ ''
Also from (A),
a-^h-2x a-b + 2c
V{{a-x){b — x)}^ 'e '
(^)
APPENDIX. 271
Multiply (B) and (C) member by member
a:2 4-20
6. x4 - 4 = -o- ,-, ; x« - 2u-A ~ 5^2 -12 = 0.
Find the rational linear factors of the left-hand member by the
method of Ai-t. XXVII., page 90.
.-. {x~2){x+2){x* + 2u>^--r3) = 0,
:. a;-2 = 0, orx4-2 = 0, or a;-i+2.c3 + 3 = 0.
The last of these equations may be solved as a quadratic giving
x- = -l±2V—2, :. x=±l±V-2,
.-. x,=2, x^_ = -2, X3 = l + V-2, x^ = l-V-2, '
a;5=-l + V-2, a;g = -l-V-2.
N.B. — In solving numerical equations of the higher orders, the
rational linear factors should always be found and separated as dis-
junctiie equations, before other methods of reduction are apj)lied.
Such separation may always be e£fected by the methods of Arts.
XXVII. to XXX., and unless it is done the appHcation of the
higher methods may actually fail. Thus, if it be attempted to
solve as a cubic the equation,
x^-dx-10 = 0
the result is x= {5 + ^-2} +{5-V — 2} , which can be reduced
only by trial. The left-hand member can however be easily
factored by the method of Ait. XXVII.. and the equation reduces
to
(a;+2)(.f2_2a:-5j = 0,
which gives x = 2 or \±yQ,
272 APPENDIX.
i'actor, (See No. 20, p. 89), rejecting constant factors,
.-. a;(a;-2)(.i,-2-2a;4-4)2=0
.-. x = 0, or x-2 = 0, ov x^--2x + 4: = 0.
The last equation gives x=l±:\/ - 3.
Exercise.
Solve the following equations :
1. {x+a + b)^=x^+a3+b^. 2. (x + a + b)^ =x'^ i-a^ + b'^.
3. (a—b)x^ + {b-x)a^ + {x-a)b3=0.
4. {a-b)x^ +{x-b)a- + {x+a)b^ =2abx.
5. (x-a)^ + {a-b)^ + {b-x)^ =0.
6. (x-ay ■h{a-by+(b-x)'' =0.
7. {a^-b)x'^+{x^-a)b't-{-{b^-x)a*=abx{a2b2x^ -1).
8. {x—a){x-b){a—b) + {x-b)(^x-c){b-c) +
(x — f) {;x — a)[c — a) = 0.
x^-1 x^^-1 ^
X—1 X^ — i.
11. ^,--p^o. 12. -^i3r-«-
13. x"" + 5a;3 - lOx-^ 4-20.r- 16 = 0. (See Art. XXII.)
14. a;4_3^.3_|_5^2 4.6.,.4.jt = 0.
15. (a:-«)4+a;4+a4 = 0. 16. 2x^ = {x-6y.
17. 4.c-2)2(a;+2) = 2. 18. (4x3 -17)u; + 12 = 0.
19. x^ + (ab^iy = {a'' + b^){x^ + l) + 2(a»-b^)x + l.
20. a;=(a;-169)2-{-17a; = a;2-3540.
APi-ENDIX. 873
21. 6x(x2 + l)3 + (2x-5 + 5)3 = 150x+l.
22. 2x{u;-l)2 + 2 = (x+l)3. 23. x^ = 12x + 5.
24. 5.«;-t = 12.<;3 + l. 25. (a;+-l)3 = 3(2x--l)2.
2G. V(.f2 4-»t3)-f V{(n-ic) 3 +;/t2}=v](;<; --^91)3 + (A/iV3 -«()}•
27 (a;+l)-* w 28 (-^+1)^ ^^
'* (j;2 + l)(.«-- l;3- -rt * * ;c(.<;3 + 1) " >i *
oj (x-+l)(.r^ + l) w-^ g2. (a;3-l)3 m
33.
84.
8G.
x3(x-+l) n x(a;- + !)(.<;— l)-* «
x{x-¥l)^ n{n — ni)
(^H- lT(I--l)-2 = 2>ii{2in-ny
(x'^ + ps 4m2
a;(x3 - 1)- ~?n3 — ?f3
(a;-l)(x3+l)^ _2{m -v)^^
{x^ - IXx-i-iy- ~ rim
x^ —1 2m
{x + l){x'^- 1) 2m -n
^ ^x^-l){x-{-l)^ m + n (x-\-l)(x^ + l) m + n
ax—b ax — b
39. a;3=^ . 40. x^ = .
ox —a ox —a
ax-h ax^+bx+e
41. x'^=j . 42. a;4=— — -; — ; ^^
ox— a a-\-bx + cx^
43. x^ = {x-l)'{x^-{-l). 44. a~x^ = {a-xy{a^-x^).
45. X3^(j._„)2(a;2_l).
46. aVix'^ + l)-xi/{x^+a2) = cx.
274 * APPENDIX.
47. ■^(a3 4-.i'3)-|-^(a3_a:3) = -^/(ae_3.6)2,
48. m{x+m ~-n)(x — m +7n)^ = n{x — m+n)(x-\-lyn — 7i)^ .
49. ni-{x + m + 17n){x-m-5n)^ = n°{x+llm + n){x-57n + 7i)^,
50. m^{.r-\-m + lln){x—m + ln)^=n^{x + nin+n){x + "m-n)^.
y/\x — n) + -^/{x — h) \x — a
''^^' V{x - a) -y{x-b)^ ^J^)
g9 V{x-a)-^V(x-b) la
\'{x — a)-\-V{x — b) la -a;
V{x-a)-V{x-b) "" -Nx - h
\a — x \h-\-x \a—x,\h-
53. ri— -^ =c. 54. , — +V—
^A—x ^J>+x sl/a — xX^ , (b —
- =c.
.a— .r ^ J) + X a/''—x a /^ — X
' r-r- + V -^ = c. 58. V I V :
0 -f-x a — X 0 — X a — X
59. Vr~+V- = c. 60. V; -e rrC.
b +x a — x b — x a — x
t,,"—x f. ,h-{-x „^ ^,a — x r,b — x
61. Vr-r-+V-^— =(?. 62. V, -V r = c.
63.
64.
h-\-x'^ ^ a-x ' "'' b — x a — x
V(a~x)^+ V{b-x)^
■l/{a-x) + y{b-x}
^/{a-x)^+i/{b-x)s
{V{a-x)+y'{b-x)}^
\/{a-x)^+ l/{b-x)^ _
^/{a—x)-\/{b — x) ~
gg {V{a-x)+^/(b-x)}^_
^ ' ■y/{a — x) - V{b — x)
^{a-xr+V{x-hy
^'' V{a-x)+]/{x-b} -"•
APPENDIX. 276v
^/(a-x)''-Vix-b)^ _
^^ ^(a-xy+K^{x + hy
^/{a—x)^ + V{x + h)^ _ {a + b]2
" V{a-x) + V{x + b) ~-iV{{a-x){.c-^b)y
70.
x^+{a-x^)V{a-x^)
x + V{a — x")
7ti.
x+V{a^-x^) ^ '
73. ^/{a-x:)-^~^{{a-x){x-h)\^^\x\hY=f{a^-ah^h^)
^ b-^/[a — x)-\-a\f{x — }))
^_ a\/{a-x) + bi/{x-h) _
l/{a — x) + y{x-h) ~
r-Q v/(a;-a)+ \/ (x+a) - -/ (2a) _ yX+c
^{x — a) — ^{x-\-a)+^{'la) ~ x — c'
78. ir(a-x)2-if{(a-x)(.c + ^)}+ir(^+M2 = -,^(rt2-«5 + i3).
79. {^{a-xY- -^[[a-x){x-h)-\ +^{x-b"~Yy^ = .
(r,.-i){Tf(a-a;) + i3/(a:-ft)}.
80. {f{a-xY+f'{b+xYy-={a + h){^^{a-x) + f/{bJrx)].
81. if(a-x) + ^(x-6) = if6-.
82. iK(a+x)3--|f(a-a:)3 = #^(2r.r).
83. f{a-xY-^^{{a-x){b-x)]+f/{b-xY = f/cK
84. i^/(a-x)---^{(a-a;)(a; + /.)}-^i3/(a; + i)2 =
276 APPENDIX.
85. {^(a-x) + f/{x + b)}f/{{a-x){x + h)}=c.
86. ^/(a-xy+^{x-br-=c{f/{a~x) + f/{x-l)}K
87. x + f/{a^-x^) = ^/^
X'^{a^ —x^)
gQ a^ x+f/{2h^-x^)
89. {a+x)^{a+x) + {a-x)^{a-x) = a{ir(a + x) + ^:a-x)].
90. (n + x)V{(t-x) + {a-x)::/{a + x) = a{^{a+x) + t/{a-x)},
91. 4/('-^6-.x-) + t/(.c-10)-2.
92. {^{a-x)+t'{x-b)}^"-=c{V{a-x) + ^/{x-b)}.
93. (a-a;i:^ a-;c)H-(a;-/>):y(.t;-&) =
94. {t/{a-x)4-4y{x-h)}^ { ^/(a-x) + i/(x~-h)} = r{a + b - 2x).
95. {^{a-x)+^ib-x)}{V(<.i-x)+Vib-x)}'' =
c{y{a-x)-if{b-x)}.
96. fl\/(l + .r-)-a;i/(;c2+a-') = e.
97. {a-x)iV{x-b) + {x-b)f/{a-x) = c{f/(a-x) + ^{x-b)}''.
98. {ir(a-x)+ir(6 + ^)}"=f{ir(a-a:)3 + |r(6 + ^)2}.
99. {^{a-x) + ^{b+x)}'=c^/{{a.-x){b^x)}.
100. -,3/(a-cc)^-ir(6-a;)3=ri3/(a+i-2a;).
101. V(^-^')+i/(^-^) = :!/'^-
102. ^(a-x) + ':/(x-b)=^c.
(a - x) i/(n-x) + (x - h)l/(x-b)
^^^- {^t^t/{x-b)+{x-b)t'[a-x) = '''
104.
105.
(a - x)^'h - x) + (h — x)^(a — x)
i/[a-x)-1/{h—x)
i/{a-x) + ^{x-b) c
i/{a-x)~t/{x-h) a+h-<2.x
APPENDIX. 27T
^^- i/{a-x)-t/[h-x) -'■
107. [a-xyif{a-x)-{x-hy:/{x-h) = c{':/{a-x)- ^{x-h)].
108. {a-xY;/{x + b) -{x-^hY:/{x-a) = c{i/{a~x)-^j^x+h)].
109. {'^{a-xy^';/[X-bY]-:y{{a-x)ix-b)\=c.
110. {!y{a-x)-l/{x-h)y{-:y{a.-xy-%f{x-bY\=c.
111. {■y(a-.'c)3-^(a;-Z,)2}3{y(a-a;) + ^(a;-i)}=c.
112. {^(«-u;)3 + y(a; + t)3}2=c{v(«-aj) + ^(.c+/0}-
Section V. — Quadratic Equations involving two or more
VARIABLES,
1. {x^-y){x^+y^) = a, I.
x^!/+xy^ = e. 11.
I+2IL .-. {x+y)^=a + 2c
.-, x-\-y = f^{a+2e). (Any one of the three cube-roots). III.
I — II x-+y^ ^ — ; .-. i^~^\ ^ ^ fi-^c
xy c ' \x+yi «-f-2c'
n TTT l/(^ ~ 2c)
Also ;.+j, = Y-^l
v/(a+2c) + v/(a~2c)
y =
2V(« + 2c)
\/(«+2c)-i/(a-2c)
2V(«+2c)
(Not any one of the siy. sixth-roots of a+2c may be used indiffer-
ently in the denominator, but only any cube-root of whichever
equare-root of a4-2c is used in the numerator. Thus if the radi-
278 APPENDIX,
cal sign be restricted to denote merely the arithmetical root, if k
be defined by the equation Ic^ — k-\-l=0, and if m and n indicate
any integers whatever, equal or unequal, the value of x may be
written
{^'2™ \/ia-\-2c)+k^^-^ ^/{a- 2c)} H-2:y ^a -f 2c).
2. 8x2-5a;?/+3?/2 = 9(x + ?/) 1.
lla;2-8x(/+5?/2 = 13(a:+?/) II.
1st Method. Eliminate {x+])).
.-. 104a;- - 65a;//+3%2 = OQa;^ -72a;^ + 45^»»
.-. 5.i;2-|-7a;//-6^2^0,
.-. x = %y or —2?/.
Substitute these values for x in I.
.-. 72.?/2 = 360y or 45^3 = -%
.-. y = 0, or 5, or -^,
and z = 0, or 3, or f.
2nd Method. Take the sum of the products of I. and H. by
arbitrary multipliers Jc and I,
hiQz-' -5a;?/+37/2)+Z(lla;2 -8X// + 5//2) = (9A- + 130(a:+?/). HL
Determine k and I so that the left-hand member of III. may,
like the right-hand member, be a multiple oi x+y. This may
be done by i^utting x- -y in III. from which
16A" + 24Z^0, .-. 2/.-=-3Z
;.. if yl' = 3, /=-2.
Substituting these values in III., it becomes
1x^-\-xy — y^=-x-^y
■'• (x + y)[2x-y) = x + y, or (x + y)(2x-y -1)=:0,
:. either x + y = 0, or 2x — y-l = 0,
;, y=^ -X, or 2a; -1,
279
Substituting these values for y in I., it becomes
lG.r2=0, or 10x--7x + 3 = '27a;-9,
.*. x = 0, or 3, or | ;
and i/ = 0, or 5, or — ^.
.rS-f-y'' flS + Z^S
a:^ + ?/^ ~«^+ f>^
I.-IL,
(a;- +2/-)- +x^y^
" (x^+y^)^ -x-y"^' {a^+b-^y -a^b^
xy ah
Write z for -;r~; — 7, and Z; for
a;2+2/3 """ ^"" a2 + ^2
z A: 1
"I- ••• l3^ = rrp' ••• ^ = ^or-^
xy ab a^ +/;2
or
x'^-\-y^ a^-\-b^ —ab
xy ctb a^ -\-h'^
o, or
I.
X* —x'-^y + x-y^ —xy^+y^
III.
'■ x^+xy + y^ a^ + ab + b-' a^—ab + b^
a2+ab-4-b^
II., .-. ^// = «^,or(«24.,,.)-^_^-^ IV.
v/ill.+IV.), .-. x+y=±{a-\-b}
V^(II.-3IV.) and x-y=±{a-b),
.a^+ab + b^
.*. r= ±a, ife or
280
y=+b; ±ci or
M--\-ah + is
lAV{%r'~ab + b^-)-j^{^a^-+ab^2h^^))V-^.f~^^^,-
4 (^.=^+^2)(x3-+y3) = a, I.
Pat 2- ^'J ' ^-^
X^ + l/^'
1-
2z' ~ b
%IZ^ — bz~
-{a-b):
= 0
4:az'^=b±:
7(8^3-
-8ai + &2-),
-b+r
■ say.
xy
b+r
' 4a
x + y
2rt4-?^ + r
a;-y/ 'N2rt-i — r
. x_ _ y^(2a4-6+r)+v/(2«-fc-r)
" Y ~ i/(2a + i + r) - V (2« - Z* - r j
_ {|/(2.< + /^+r) + N/(2a-^>-r)}''
2(6 + y)
in.
IV.
• ^io/?/\M32«^(2a+6+r)(4a-6-r)2j _ ^
~ 1?// (32(2a+?'+»-)(4«-^ -»■)'' ^
j^ _ {x/(2a + 6+r) + -i/(2a-6-r)}">
1024(2a+&+?-)(4a-Z>-r)3
X
APPENDIX. 281
2y{(2« + 6 + /-)(4a-6-r)»}
in which r= ±:v/(8a- —'6ah-\-h-).
The value of y may be derived from that of x by the first form
iii IV.
6. x"* =ax — bij, 1.
y* = ai/ — bx. n.
iC.I. — 7/.II. x'^ —y^ =o{x^ — y-)
y.l.-xlL xy{x^-y^) = h{x^-y^-),
:. either x — i/ = 0 from which a; = ?/ = 0, or ^/(« — 6) IH.
or x'^+x^y + x^y^-\-xy^+y'' = a(x+y) IV.
and xy{x- +xy + y-) = b{x+y) V.
(IV.+V.) (^+2/)3(x2+y2) = a-|-6 VI.
V. (a:-|-2/)4-(a;2+i/3)3=4i(;r+y) yil.
1/(VII2 + 4.VI). (a;+^)4 + (:c2+y3)2=2i(.« + y) VUI.
in which i= y{(«+6)3+462|^ jX.
^(Vn.+Vin.), .-. {x+y)^ = (26+0(x+</)
:.{x-iyy = 2b-{-t
:.(x+y) =^'i2b+t) X.
VI. .A ..X -^y -^^^2b-tt) ^^■
2.XI. -X2 .-. (x—yV' = ^("±3^ -^/m^t\?' - -
2a— t
f\2b+t) ^^^"-^'^ -f{2b+ty
V{2a-t)
X. and..+ </ =v:(2Ml)
V(2i+0
282 APPENDIX.
and ^=^^^^+±1:^2^^;
in which i= \/{a2 +2ab + 5b^).
*+'/ 2* 2?/
Let 2 = ', .-.24-1 = andz— 1 = — '— III.
x-y' x-y x-y
I. + 11. x'^+y'^ = >n[x-\-y)'^ + n{x-ij)^
... (- + l)4 + ,2 - 1)4 = 16(,H24.^„)
.-. (8«i-l)24 --6^2 + (8/1-1) = 0,
= 4
3 + y-|9-(8/M-l)(8?^-l)}
8?rt — 1
n &III. {2-\y(^x-y)^^-lQ>c* = lQn(x-y)*
IV.
2c
•"•'''~^";t/{16n-(2-l)*} ^^
and a; + V =4 tttt. ~! rr^T"
-' ^{lb»-(2 — 1)*}
<z + l) c(;s + l)
•'• '^-y{16/i-(^-l)4} - v{2+l)*-16m"ij*
C(2-1)
and y- ^.s i6,i _ /gUiW ' ^^'^ ^^® ^'^^'^^ *^^ * ^^ given by IV.
7. :c2+y3 = i(2»,,+«2^,
x^ -\-y^ = m.n.
.'. (a;+?/)--2x.v = i(27/?,+w»)
and (x- + y) 3 - 3x//(j; + ?/) = nm.
APPRNMX.
283
Let u = x-i-y and v = xij, and the equations become
tt^— 2t;=i(2m+n2);
«' —Zuv=^mn.
Eliminate r, .•. ?/3 — (2w+n2)»+2wn = 0,
.-. M* — (2/» + 7^3)^t2_[-2w/i t< = 0,
.*. jt^ — ?/? = ±:(»u — 7??.),
.•. « = «, (the value u = Owas introduced by the multiplica-
tion by n),
or u" + n}i -2??4=0,
.-. u=h{-n±V{n- -^^m))
.'. r = A(«2_„,) ox \{n^+Qm + Zni/{n^+Qm)}
.-. u and V are completely determined.
Also x-{-y = u, x — y=-J{u'—-iv)
If m = 7 and n=5, the above equations become
x^ +?/2 = 13, and »■' +y3 _ 35.
Solving, as above, gives
u = o, or 2, or —7,
2i' = 12, or -9, or 36,
.•. x-\-y = 5, or 2, or —7,
x-7/=±l, or drv/22, or +/|/23.
.-.. 2;= 3, 2, ^(2±s/22) or ^(-7+7V28j;
?/ = 2, 3, i(2+v/22) or i(-7=P;V23).
284 APPENDIX.
••• il-^=(f-?y')^
Testing this for rational linear factors it is easily reduced to
.: y = l or^(-2±-/2);
x=^ or i(-l + 4A/2).
H. (2x-y + z){x + y + z) = d; I.
{x+2y—z}{x + y+z) = l; 11,
{x+y-2z){x+y^z)^4:. in.
Let s = x+]f+z and the equations may be written
{s + x~2y)s^9 rV.
(^s + y-2z]s^l V.
(s-8zV = 4. VI.
1V.+3.V. {4:S+x+y-Qz)s = 12,OY (5s-7z)s = 12 VIL
8 VII-7.VL {(15s-21z)-(7s-21z)}s = 8,
.-. 8s2 = 8, .. s=+l.
Substituting in I, XL and III. they become
2x-y+z=±9, x+2y-z=±l, x-^y-2z=±i,
.'. x^ +4, y^+2, z=+l.
10 x^+y^ = a;
xy-\-uv = c;
xu + yv = e.
Let t=xy—t('V.-
.-. (^x + y)^=a+c+t, .•.x = l{./{a+c-{-t)-i-V{a+c-t)}
{x-y)^=a-c-t, y = lU/{a+c-\-t)- \/{a-c-t)}
APPENDIX. 285
(u^vy = b + c-t, ii = l{s^(b-}-c-t) + r/{h-c-{.t)\
(u-v)^=b-c + t, vr=^{^/(b + c-t)-V{b-c-\-f)}
Also 2{xu+i/v) = {x+]/)(i(+v) + {x — y){u—v) = 2e,
.: ^/{{a + c + t){b+c-t)}+V{a-c-t){b-c-\-t)}=2e,
.-. {4.e^-{.{a-c-t){b-c + t)-{a + c-{-i}{b+c-t)}^ =
■[Ge-{a-c-t){b—c + t).
.-. {{a-by-\-4:e2}t2-2{a^-b^)ct-\-
(^a-^h)2c^—4:e-{ab+c^)-{-ie^ = 0,
_ (fl^-&-)c±2cv/ [{ab~ g^){(ff-6)2-4(c2-g2)}]
•'• ' ~ {a-b)-+ie'^
11. xy = nv I,
a;3 + j/3 + u3 + r3=63 jjj
x''+i/^+u^+i-^=c^ IV.
liet x+y = h{f+z)- :■ ^i+v=l{a-z). V.
Also let r = xy = uv yj,
(a; + 7/) ^ = a;3 + 2/3 + 3x?/(a;+ »/)
(w+v)3 = u^^v^+Buv{u + v)
.-. a(3z3+a2) = 4(fc3+3«r) VII.
Also {x + yf = x' +y-' +5xy{x^ -{-y^) + 10x'' y^x + y)
.-. a(5z* + 10a2z2irt4^ = 10|,.5^5^3,._|.i0f„.2| yjji
Eliminating r between VII. and VIII,
45a224 - 30a(a3 +263)52 ^^e - 20a363 _8066 4-l44,,c« = 0
.•.15az2-5(a3+26^)=+2s/{5(a3_|.5i3)2_i80«c-} IX.
a3 4 263-|-2y[^{(a3 + 563)2-36ac=}]
.-. r= i/ 3^ X.
286 APPENDIX.
VII. & IX. 12ar = a^-4:b^ + 3az-2
= 2fl.3 - 2b^ ± 2V [i{{a-' +5h^)- - BGac^}]
5(a^ - b^)±:V{5{a^ + 5b^)^ -ISOac^}
•■•'■- ¥Ort ■^^•
X. and XI. give the values of z and r which may now be treated
as knovv^u in V and V.
^+y~¥'' -^^)> anda;^ = r
.-. x-ij = ,^v{{a+z)-''-lGr\
a;=i(fl+z±V{(a + 2)2-16r});
v/ = i(a+zqFV{(a+z)^-16r}).
The vah;es of u and v may be obtained from those of x and y
respectively by changing z into - z.
Exercise.
1. 6{(7-x)2+,v2} = 13(7-x)2/, x2 + 477 = 7/2 +4.
2. 10a;3 - 9i/2 = 2a;^ Sx'^ -6//^ = Vdx.
3. a;y = (3-a-)2=(2-r/)2. 4. :c3+7/-2 = 8a; + % = 144.
5. a:2+2/2=cc+7/+12, X!/ + 8 = 2{x-\-y}.
6. a;+a;?/+y = 5, x2+a,-?/+//2=7.
35 2S
7. x-'»+.V^ = 7a:?/ = 28(a: + .y). 8. a^s+xv+z/^ = ^^::^ =— •
9. s:4 4-a:3?/3+?/4 = 133, a:^y+a:2?/2 +a;;!/3 := 114,
10. (x4-?/)(a;-+2/-) = 17a;(/, (cc-?/)(.v2 - (/2) =9ar(/.
11. 25(x^+y^) = l{x-\-i/)^ = 175x1/.
12. 2a;2-?/3 = 14(a;3-2?/^) = 14(a;-2/).
13. 2x^-3x7j = 9{x-3ij), S{x^-Sy^) = 2{2x^-dx!/).
14. 2u;3-a;?/ + 5?/2 = 10(.T+?/), a:2+4xv + 3?/2 = 14(a: + .v).
15. (2a,— 32/)(3a;+4?/) = 39(»-2y), (3a;+2y)(4x-37/ = (99(a;-2y)
APPENDIX. 287
16. {x-{-2!/){x+3y) = 3{x + y), ^'2x-\-y){3z+y) = 28(:x+y).
17. x+y = 8, x* + y* = 70Q. 18. x+y=^o, u;^+^^ = 275.
19. x-\-y = 2, r6[x^'+y^) = V21{x-'+y^).
•20. .*; + // = 4, ■il{x^+y^) = 12-2{x'^+yi).
21. x^ — 5xy-{-y^+5 = 0, xy = x-{-y-l.
22. x^+y = o{x-y), x+y'^ = 2[x-y).
23. 8(x-2 + //) = 3(a;+.?/3) = 13xy.
24. 10(.^3 4.^) = 10(a:+2/^) = 13(x2 +2/^).
25. .c2+.v=V, ^+y- = ¥- 26. 9(a;-^+?/) = 3(a;+^3)^7.
27. x+-c//+^ = 5, x^+xy+y3 = 17.
28. ^+// = 2, (x- + l)^+(y-2)^ = 211.
a;2 4-.c+l 31/.c2-x4-l\
29. 3(x-l)to + l) = 4(.+l)fa-l), j^5Tp^i = 3sj(^._y|-i)
1
30. x+y= — , x-y = xy.
31. x+Tz-fl^O, xc+i/s 4-2 = 0.
32. :c+j/ = l, 3(.c8 +?/«) = 7.
33. •4x^/2 = 5(5 -:«), 2(.^2 _|_,/2) = 5.
34. 21xy = 17, 9{x-+y^)=-8.
35. (.^2+^2)2-1-4x3^/2 = 5-127/, y(x2+2/2)+3 = 0.
36. z+y=xy, x^+y'^ =x^+y^.
37. a;4 - Qx^ V{y--x^) - 16?/2 = 9a;2^
(x2 +5i)2 =4{2+:cV(2/'+i--) -2/'}.
88. x(y3+3//-l) = 2i/2+27/+3, i/(a;2 + 3.c-l) = 2.c3+2a;+3.
288 APPENDIX.
^^ , V^ o,.3 ^ ,11 . / a; V
39. :l- 4. ^ = 2c-3, _ , iL ^ e
rtS ^ ^3 a ^ b \a b
40. rc^ +xi/^ =a, y^ + x^y = i.
41. .«+(/ = a, -I- ^ = c.
42. x--^ay'^= "^ ax^+y^ = {a--l)y.
a—1
43. x + y^=ax, x^+y = by, 44. x-\-y^=ay'^, x^+y = bx^.
45. x^ -y^=a-{x — y)', x^ -x^y+xy" —y^ = b^{x+y). .
46. (a;+^)(.«3 + 3^2) = ,,j, (aJ- 2/)(.c3+3//2) = «.
47. x-y- -y{a-x)^ =x[b-yY.
48. a;3(6-(/) = (/3((i-x) = (a-a:)^(i-?/)3.
50. x^-y^=a{x-^-y^), x^- +y'^ = h{x+y).
51. a; + ?/ = (i, x^+y^^bxy.
52. I^ _ !^ = — ^', x{c"--\xy) ^ ^^^
Ay Ax
a y{o^-xy)
53. x + y = xy = X'-\-y-- 54. x-y= — = x^-y^,
y
55. .x3(l + //2)(l+i/4) = a, a;3(l-2/2)(l-^4)=A.
x^-xy+y^ ~ a b
57. x^y-hxy'^= ^^ ,, a;'*(/+a;i/*= &.
59. (^ + ^)(x+i/) = «, f! + ^ = 6.
\y xj y s,:
APPENDIX. 289
CO. x^ +y2 = rt.r2//3 =xii{x-\-ij).
Gl. abxy = a{x'^-'ty^) = h{x+!j)^.
62. xy{x+y) = a, x^y^[x^ +y^) = b.
63.
(1 -f l.](x^-y^) = a, (1 _ J_)(a-3+^3^ = ft.
\ X y ' \ X y I
64. a;* + ?/4 = m(.r3 4-?/2), x^+xy + T/^ =:».
65. rt6(.c + ?/) = av/(a4-i), x^ -fj/S =a2 + i'.
66. x^ ty^=a{x + y). .r-i +//4 = 6f3r4-«.A"
67. a;2_|_,/2^f^^ a;5 4-w° =i^fx3 +v*».
68. a,-y = (j, a;-' + ?/-5=i(a;3-f ?/3).
69. {x-y){x-^ + u^) = {a-h){n^+h^), x- - y^ = a"^ - b'-! .
70. a;2— ?/2=af^ a-3 _f_y3 = i(^^_jy^_
71. a;+?/ = «, a;4 4-//4 = 6. 72. x + y = a, x'''+y^=b.
73. a:+?/ = a, x~ +y- =b^x-y'^.
74. x + ?/ = rt+^;, (a-612(a;4 4-_y4) = (^_y)3(rt4_i.^4j,
75. a + y = a, o(.'"^ + //4) =a://(.t;3+^3)^
76. {x+y)^=a{x''-}-y-), xy = c{x + y).
77. a;2)/^a;^3=rtt^ c^(x^-{.y^)=x^y^.
78. x^ =:(i(x--\-y-) — cxy, y^ = c{x~-^y^) — axy.
79. a;3-?/2=</2 j.3_^3^c4 I — -— |.
80. x*-y''=a2xy, {x- +y^)^ = h^{x'^ -y^).
81. {x->ry)x-y'^=a, x^+y^:=b.
82. (x + yjxy = a, x^ +y'^ — hxy.
290 APPENDIX.
83. x^+y^ = a{x+7jy, x^+y^ = b{x-\-9j)3.
84. x^ + x-i/'^+y'^ = a, x^ —xy-\-y- = 1.
xHl+x^y^) a l + xy
-\/{x'^y^)—x
86. x-\-y={x-y)V{xy), ^^^i^^,s^^y = a.
87. ^ + y = ^, i/{l-x)-V{l-y) = b.
88. x^+y^=a{x + y), x'^ + y^ = b{x^ + y^).
89. x^+y^ = a, {x + 7j){x^ + y^) = b{x'^+y^).
90. (x^+tj^){x'-^+y^) = axy, {x+y){x^+y'^) = hxy,
91. {x+yy-{x'^+y^) = a, (x"- -{-y^)^x^+y^) = h.
92. (a;-?/)(a;2-?/2)(;^4_2/4)^4rt^^^^
[x-}-i/)(x^-+y^){x^+y^) = b{x-y).
93. a;4?/ +a;;y4 = ^(a;^?/ -^xy^) = h{x'^ +//4).
94. a{x^-{-y'^) = ab{x-\-y) = bxy{x^-\-y^).
95.
a;^— 2/^ ffl^ — b^
x^ —y^ a^ — 6*
X'-y^~ a^-b^'
x* — y^~ a'^ — b'
x^ + y^ a^-\-b^
:r^-y^~a--b^'
x^-\-y'^ a^^b*
x^ — y^~ a^ — b^
96.
97 . .r^ = ^ax -by, y^ = %iy — bx.
98. (x+?/)(x3+i/3) = a, {x-y){x^-y^)=h,
99. i^?^)^^^'i!±^^^-l _ 8^2,
(x'^-\-ij^)\x^ —xy + y^)
{x-y)^{x^-xy + y^) ^ _,^^^
{x^+y'>){x^-^xy-\-y^)
APPENDIX. 291
100. {x + y){x^ -^y^)=axi/, {x — y){x^-y^)^bxi/.
101. {x + y){x^+y^)-^a{x-^+y-^), ^x- y){x- -y') = b{x'+y-).
iu2. (■^+y)Hx'+y^) ^ ^^2
{x'^+xy + y'){x-+y-)
{x-y)Hx^-yS) ^ ^^
{x'-xy+y-){x--\-y-)
103. i^'^ + 'J^){x+y)^ ^ o,^3 (x^-y^){x-y)^ ^ ^^^^
x--^xy+y^ "■' * x-—xy+y'-^
(j;-'-t-j;// + ^-;- {x- — xy+y-)-
105. xy(./; + ?/)(x3 + 7/3)=a, xy{x - y){x^ -ij^) = b.
106. 4x+(/)(a;+2^)(x4-8//) = a3, (x + ?/)2 + (a; + 2i/)a =6.
107. N/(:c-.cj;)-i-v/(y-a;^) = ^, >/(x — a;^) + v'(i/-2/-; = 6.
108. (x4-l)(y-l) = «(x-l)(y-rl),
(a-5 + l)(t/-l)^=i3(.^_l).(^..+l).
109. x+t/ = u, V{x-lc)-\-V{y-^^) = c-
no. a:+(/ = a(l + x7/), (x- + ^)4 = 64(l + .c4_y4).
111. x+y = a{l + xy), x^+2/= ='->^(l+^-'^=).
112. (x+l)(y-l) = «(a;-l)(//+l),
(x«-l)(y-l) = 6^(7/5-l)(.«-l).
(l+a;)a+?/) (14-a;)3(i4.,/)3_^
i^^- ^i_^^(i_^;-«' ^i_^3)(i_^3)
{c+x){c-\-y) {c^+x^)(c'^ + yi)_^
{x + m){y + iu _ (x^ + W^)(j/44-/< 4j ^ ^
292 APPENDIX.
^^Q {x+l){y + l) a {x^ + l){y^ + l) h
11Q ^+^ P ^(l+a;+a;^)
122
"n^/
y(l + a:^) y'il + x^)_
(.a;-i/)(x-^ — i)"~ 2(f6 ' (/(a;--lj~a-4-6
TOO (^+y)(l+a;.y) x{l-y')_
(a;-yj(l-sc^) "*' {x- -y-)[l - x^y'^)
,o(. (a;4-y)(l+a;//) (a;3+Z/^)(l + .^^y'^)
'^^- {x-y){l-xy)-'' {x^-yZ)il-x^y^)-'>'
(a;2 _ xy + y^{l-xy-\-uy^ii^)
{x-y)'{l-xy)^
129. a;4 -3^3^ + 5«^x+^2=o, 2/^ -^^i/" -2a"x = 0.
APPENDIX. 293
130. 2x(!/^-2x)^=a, 2/(y2_2a;)V(.v'-4cf) = 6.
(Hence deduce the solution of a;^—6a;2 -|-2 = 0).
131. 2xy{x^+y^y=a, {x^ -ij^)ix'' -hy^)^ =b.
132. V{x--^i/^') + V{{a-x)^+y^}=V{{UVB-y)" + {la-x)^},
6(a;2 -2/a) = a(6a;-2</v3+a).
Exercise.
1. {2x-}-y-4z)(x+y+z) = 24:, 2. x^-yf:=l,
{x + 2y-2z){x+y+z) = 6, y^-xz = 2,
{-2x+3y-{-5z){x + y+z) = nO, z^-xy^3.
3. {x+2y-'dz){x + y + z) — 2(xy+yz+zx)= -12,
(2x-3y+z){x+y+z) + {xy + yz + zx) = 61,
{Bx-y-i-2z){x-i-y+z)-5{xy-{-yz+zx) = 5.
4. x--yz = 0, 5. {x^+y^-hz')^ + (x+y)^=Sl.
x + y+z = 7, {x^+y^+z^)^ + [x+y + z)^=129.
x3+7/3+22 = 21. i^x+y)2-\-{x + y-^z)^^81.
6. «-— 1/2=0, 7. x+yz = li,
x-\-y-^z=21, y + zx=ll,
(a;-2/)2+(j/-s)3-f(z-x)2 = 126. 2+a;y = 10,
8. x+2/ = 8z, 9, «;+?/ = 52,
x-^-i-yS = 13iz^, x^+y^ = Sdz,
a;2+2/^+z2 = 184. a;3 4-^3 = 105^2.
10. x+y = 7z, 11. a:4-j, = 72,
a:2+i/2 = 25z2, ic2+y2^25z2,
x4 4-7/4 = 67423. a;« + ?/« = 20272z.
12. x+?/:y+2:2+x::a:i:c, 13. x+y:ij-{-z:z+x::a:b:c,
{'i + b-\-c)xyz = '^ {a + b + c)xyz=2{x+y-iz.)
^94 APPENDIX.
14. ax = by^cz= 1 4. J_ _|. 1. 16. s /^ 4. ^) = a,
X y z \y ' x]
15. {x-\-y — z)x = a, jx z\ ,
yl i =b,
{x — y+zhj = b, \ ^ -^ I
(-x+y-i-z). = c. xlJL + l.\==c.
U i/l
17. (y + z)(2x+y+z)=a, 18. x(7/ + z):y(z + x):z{x+y) =
{z+x) {x + 2y-\-z) = b, 6 -f c : c+a : a + b,
{x+y){x+yY^z)=c, xy+yz+zx = {a-{-b + c){x-\-y + z).
19. {a + b)x+{b+c)y + {c + a)z = {a+b+c){x-}-y+z),
a[x+y) = c{i/+z},
{x + y)^ + iy + zy + [z+x)2 =4:{a^-^b-' +c-^).
20. c(x+y)-i-b{x-z)-a{y-^z) = 0,
b{x-z) = {a-c)y,
X- +y^ +z^ = a^ +b^ + c^ .
21. x-\-y — az = x—by-\-z= —cx + y-{-z = xyz.
22. {a + h + 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^
23. xy -\- — = a; yz -t ~ == b, zx + — =* e.
z X y
24. y-\-z:z-\-x:x+y::h-\-c:c->ra.:a-{-b,
{^x+y + z){xyz) = {a + b+c){xy+yz-{-zx).
25. n- —■yz = a, y^—xz = b, z^-xy = c.
26. x-+{y-z)"=a-, y'^ + {z-x)- =b- , z" -^-^x-ryf =c^.
APPENDIX. 295
27. T-+:n/-{-y'^ = a', y^+yz-\-z^ = h^, z- +zx-\-X' =c*.
28. x^-\-y^ —z^-\-dxyz = a{x+y — z)
a;3_2/3_|_23 .{.Qxyz = b[x — y -\-z) ,
— x^ +y^ +z' + dxyz=: c{- X + y -\-z),
29. a-+?/ + 2flz=--0, 30. x-{-y-az = 0,
x"+y^-2bH^=0, y^x^^y'i)^i%^
a:"+y" + 2" = (-■". x^+y^ = c^.
81. x(y-l)(2-l) = 2a, 32. xfy- 1) = «(2- 1),
a;3(y3 _ 1)(23 _ 1) = 6f2= «3(,/S _1) ^ c3(23_ 1).
83. a;(j/-l) = a(z-l), 34. x{y-l) = a{z-l),
z^{y---l) = k^z'^-\), zHy-^-l) = h^z^^-l),
x*{y^-l) = cHz^-l). x^{y^ + l) = c'^{z^ + l).
35. x{y-l) = a{z — l), 36. (ir-?/^ =^^2(.^•+?/),
a;2(y3 _1) = 7,2(^2 _ 1)^ (a;3 _,^3 ^i2(a;+?/)3,
87. x—y = a, 88. a--}-?/ = rt,
u — v = b, u + v = b,
xy = uv, X'-\-H^=c^f
x^ — 2/'4-«' — ■"' =(•(« + ?>). t/-+r2 =e'^ .
39. xy = itv — a^, 40. xy = uv = a',
x+y+^i+x = b, x-^y+u+v = b,
41. xy = au = a't 42. xy = uv = a^^
296
APPENDIX.
x^+?/^+u^+v^=c\ (a;+M.)3+(y+i;)3=c».
43. xy = uv = a^, x+i/-\-ii + v = b, {j^ + u)^ ^(^y ^v^ =c*,
44. x!/ = uv, 45. xy = uv,
x + y-\-u-\-v = a, x-^y+u+v = a,
46. xy = 2(v, 47. xy = uv,
x+y + u-\-v = a, x + y->ru-^v = a,
X^ +2/- +2t-+t'2 = 62^ 3.3 _^^3 4.^3 -1-^3 = 6',
x^+y^-\-u^+v^=c^. a;4 + 2/4+?t4 + i;4 = c4,
48. xy-uv = 0, 49. a;3+?/2=a2^
x:(,+yv = a'^, u^+v2=m^,
x+>/ + ii + v = h, ux+vy = c",
x^-\-y^+u^+v^ =c^. vx + uy = 71^.
50. x+y+ii+v = a, 61. 2Kl+^^) = 2iC,
xy + 2iv = b-, u{l+y^) = 2y,
x^-{-y^=m-, r(l+M2) = 2M,
u2+y2=,j2, iC(l+r2)=2r.
52. a;+^+?< + ?' = a, (a;+?/)2+(M+r)2=^,3^
X la — u y 2b — u z 2e— M
63
y-{-z a-2u z-\-x b — 2ii x+y c—2u
ANSWERS.
Exercise i.
1. 9. -69, 1, 0, 1206, -29, 1^. 2. -160, 106, 41, 108
8. -T^, H. -25, 125, ^V -31, -4^V0, -1- 4. 9,8,
7, -^V 5. 176, 82, 254|6, -37H-7if3. 6. 18 each.
7. 146,14,-72,-270,396. 8. Each = 0.
Exercise ii.
1. -1. 2. -166542. 3. 100. 4. -2967511.
5. 968. 6. -162. 7. 10. 8. -8. 9. 0.
10. -20. 11. 706440254900. 12. 0 each. 13. Each 0.
Exercise iii.
1. 0, 16a4. 2. a, a^/S. 3. 2a., 0. 4. 26«e, -26n^.
5. 0. 6. 4rt4. 7. 6a4. 8. f. 9. c. 10. 0.
11. aH-(a+6). 12. a3c(6 + 2c) ^62. 13. a24.i2_1.c2.
14. 0. 15. (12a26-24a63 4.2863)^(36-a)3. 16. 0.
17. 0. 18. -b^c. 19, 20, 21 and 22, each 0.
25. 2{b + l)h, 4x^. 32. ^2 = 3^2. 33. i= y^(irf2),
85. 'jjr', xir + r'){r-r').
Exercise iv.
1. 2{bz+ei/). 2. 3{ax-b7j).
8. a^{z — z) — ab{x — i/) — b-{y — z).
4. {x-^y+z)(a + b + c). 5. (« + 6+c)(x3 + 2/2+z2).
6. 2{x+i/-\-z}x{a'-+b2 ^c^—ab-bc-ca). 7. 0.
8. 2(ox+hy + cz). 9. aS -f-62 .|.c2.
10. 2a"(a-26). 11. a-{-b-c.
Exercise v.
1. 2(V4-9;r4), 4a^h^. 4. -iia^-b^)^.
5. x^+Ax, -3ix4-4at2?/2+3iy4. 6. a^.
8. x^-6.v^+9x'>' + 2xi/ - 6xy^ - dx^y + lSx^i/^ +y^ - 6y^ +9y*.
9. 4x//fx2-?/3), 2(1 + 12;c3 +16.1-4).
10. -r'^c^. 11. a3_262, 8rt&(« + 5)2. 12. 2{a-c){b-d),
13. ix= + if/2 + iz3_|.i(^j/ + y-.+;.;g). ig_ {l+x-y-.
16. 4(.-r.y+?/z + 0.c)-2(«3+?/3+z2), 17. 3.2.
18. (a2+2i3_2c3)3. 19. I6x^y2^ 20. -4ab.
21. 4(a + 64-<*)^- 23. 4(l+a;3+a;'t +a;S).
24. {a^x"+b2y"')2.
Exercise vi.
1. l-4r + 10.r3-20.r3 4-25s:*-24a;^ + 16.c%
1 - 2x+Sx^—4:X^ +3.c4 - 2x'' +a;«.
2. 1 -4x + 8x-3 - 14.^3 + Ux^-Sx^+ox^ +6x'' +x^,
l+6x + 15x2+20a;3 + 15a;4 + G.eS-fa;«.
3. 4r/,2+//-' + c4 + l-4a&-4ac3-4a + 26c2 + 26-i-2c2,
1 +a;3 +ij2 -f 23 _ 2.C + 2)/ + 2z - 2xy- 2xz+2yz,
ix^+^y"" + 36z2 - ^xy + 6a;z - 4//z.
4. a;«-2./;-^'^ + 3a;-i!/3 — 4.i-3i/34.3x3;/4_2.r»/5+y6^
a^a;2 + 2a6a;3 + (2ac + 62)a;4 4. 2(rtrf + 6c)x5 4.(26(i+ c2)a;6 4.
2c(^x^+ci3a;8. 8. 3{a3 -^b^ +c^)-2{ab+bc+ca).
11. 4rt'^ +^63^2 ^^^c^x- +4:d-x- - 2a6a; — rtcx + 8ao?a;+ ^kx* —
2bdx'^ — cdx^.
ExEKCisE vii.
1. (a2_52)2. 2. iu:4+y/4. 3. a^^Sa"b2+4:b^.
4. a;* -2/*. 5. ^.-3, 6. 1G,.;^ 7. 0.
8. 4a4-9&*-16c*4-24i3c2. 9. b'^ -9c^ -4ta^ + 12ac,
9cS_4a2_62+4a6. 10. x^'-y^. 11. a;8+j;4^4+^8^
ANSWERS. lU
12. a^-a^i^+b^-l. U. «*+?/* + tV;=2/'.
15. a;8+O^.G_j.3a.4^2x- + l.
16. 'ia*x--Aa*x]j + aM/'^-a-x^-2a^:)^i/+2ax'^ +2az*ij-x^.
20. (.,:3+^--=_2xy-z2)3. 21. x^ - y\
22. ^-6u2+ 27^-1. 23. (m+p)3-(«+?)2.
24. 2.t-+x* + 2x<^-x«-l. 25. a8-Z>i6.
Exercise viii.
1. i-4+4a;3 4-3.^2- 2a; -12, a;3+*/3 -2.r?/ + 8ar2-8?/2+15z2.
2. x-i + 12x-3-f49.c-+78x-+10, x*^ +bx^ -a^+Sab-2i2.
3. a»+8«e — lUa4-104«2 + 10o, a;«+2x« -x^— 2.
4. x^ + ux^i/'-12x-iy^ + 5xy^ + ij'^.
x^ V2 2a; 2;/
5. x-2"-2a;"-a2-16a-63, — +^-j-— + — -1.
■(/- a;-^Y/ x
6. 7i2a;2+2na:y + ?/2 + 10Ha; + 10?/ + 21.
7. (u; + a)2+2^(x4-a)-3i/2. 8. a;*"+2x3» + a;2»(l-fr-6)-
af(a+i)+«6. 9. ia;8-u;'i?/3 + j/4_;c4^22/»-8.
5
11. a;4-8a;3+19j;2— 12a;+2.
12. (a; + 6)*-(a2-f-c2)(a;+i)2H-a2o3. 13. ai + ci.
Exercise ix.
1. 2(l + 3a;4), 2x>/^{3x^+x-y^). 2. 9i(a2 + ft2^r762)^
6(27a2-27a6+7/>2). 3. (x + y)^. 4. ba^. 5. 8a;3.
6. 8x3. 7. ,/3. 8. 27x3. 9. (2+a^3. 12. S{x-+y-')^.
14. (a3+63)(a;3 + 2/3). 15. 0. 16. 0.
Exercise x.
1. l-3.r + 6a;2-7a;3+6x*-3x*+a;6, a^ -b^-r'!f -3a^{b+c)
+ Sb^{a-c)+dc2{a-b) + 6abc, 1 - 6x+21a;2 - 56a;3 -j-
lllx* - 174.t-^ + 219x'* - 204a;^ + 144x« - 64x^
/I 1 \ ^ . / 1 1 1
10. — +— =^ — +—
\x y I \x y I
IV ANSWERS.
5. 0. t5. 45x^+1682a;*?/2_432a;3//223. 7. (rta; + i?/+cz)3.
Exercise xi.
a;^+7a;e?/ + 21a;57/2 + 35a;4y/ 3 + 85^,3^4 ^-2Lr3?/5+7a;.ye +t/7,
a;8_^8a;7^^28j;6i/3+5G;c5vy3 4_70a;4y4 4.56a;3?y5_|_28a;3.^6_^
Sxij-' +y», x^ 2 4.12^1 1/7 + 66^1 0//3 + 220a;9^-3 ^ 495a;8y4_j.
792^^2/= +924^6(/6 +792^;^^^ +&c.
2. The signs will be alternately positive and negative.
3. a^ - 5a^b + lOa^b^ - lOa^b^ + 5ab^ -b^,
a* — 8rt2i+24a-62 — 32a63-(_i66*, same as last, terms in
inverse order. 4. l+(!))ii-\-15m- + ^Q>n.^ + 15m^^Qm^ +
m^, >n^ +5/?i4 JL. I0»i3 + 10m^ + 5m + 1, Qhii^ + 192hi-'-j-
240^4 + 160m3 + 60»i2-j-i2Ht + l. 5. 120.
6. x^ -'ix^ii + Qx^ij^ -"^x^y^+y^, a^ -lQa^b"+^Oa^b*'~
ma^h^+^Oab^ - 3261", «i » _ 12rti ^^3 +60ai s^g _
160a9/^9 + 240a66i2— 192a36^5_j_64ii8.
7. 495a8i4_792a7i5,
Exercise xii.
1. l^-a;3+a;4 4-a;6+a;l^ 2. 1 +a; + x' 4-2^3 ^x^+^e+x^^
a.8^a.9_j.a;is. 3, a;4 + 2a;3 -85a;2-86x+1680,
2a;9-3a;6+4a;5+a;4+.x3_2x2-a:+2. 4. ic«-57a;* +
266x3-1. 6. 18^8 +21a;^ + 8^6 +a;5+63x='+96a;2 +
43x+6. 6. l-ia;2-ix4. 7. 6x^2 _ 4^9 _ 5^8 _2a;7 +
9x«-10x^+a;*-5a;3+5x2+.c + 4. 8. aj^^+g^^ +10x + ll.
9 a;4 + 3x3. 10. a;* -3^3. 11. a;4+8x3-8a;.
12. (1), -1. (2), -1, (3) -4. 13. -1.
ANSWERS. V
Exercise xiii.
1. 8x5-2^«-4.r + 2. 2. 5z^ - ix^ + Bx^ -2x+l.
8. a'^ + 2a^ + Sa^+ia + 5. 4. x^+2xy>/+3.rij^ + 4y3.
5. a^+Sa^x + Sax'^+x^. 6. 4x''-\-8x-^7, -13x-20.
7. 10x3 + 5x2 + 1, lOx+lO. 8. x^-xy + y^.
9. x3_„3, 10. x^ + {l-a)x^ + {l-a + b)x^ +(l-a)x+l.
11. 8a;3+2.r2+a'+li, 3Kx + l). 12. 5x- +lBdy + 12y^.
13. 6x*-a;4-a;3+x-2-a;+6, -1.
14. 2x4-3x3+4x2 — 5a; + 6. 15. a + b. 16. x+(/+2,
17. lOx-3, 10(x4-20). 18. wx3 + n.r2+a.
19. l+x-5fx3- 3x3+9x4. 20. 88. 21. -4.
22. -20. 23. 15?/4. 24. 85x+8. 25. 755.
Exercise xiv.
1. y3_2,/2-4?/-9, if// = x-l.
2. y^ + Sy + S, if?/ = x + l. 3. ?/4 + 81, ify = x-2.
4. 7/4+4?/3-43(/3+92(/-67, iiy^x+2.
5. 3j/5+302/4+1192/3+238?/3+249/y + 106, if y = a;-2.
8. (x-2//-'')-8.v(x-2i/)2_18(/3(x- 27/) -247/3.
9. (:r-?/)^-10?/2(x-2/)3-207/3(x-^)2_lO?/4(^_y).
10. (2x+i/)3+2i/2(2x+2/) + 5?/3.
11. 512y^-3y-y\\, ify = ix-^\.
12. (/4-24(/2+49i/-28, if2/ = x+2.
Exercise xv.
1. a^b, +au'^-j-a^n + b^c-\-bc2+ac^,
(a-i)2 + (i-c)2+(c-a)2, «(/.-c) + t(c-a) + c(a-i),
afc(x — c) +ic(x — a) +ac(x — i),
a6c(a26 + a36 + i2c + a62-t-af2+6c2),
Fl ANSWERS.
(a + h){c-a){c-b) + {b+c){a-b){a-c) + {c + a){b-c){b-a).
(a+c)^^b^ + {b+a)^-c^+{c + b)--a'',
a{b+cy-\-b{c + a)^+c{a + by.
2. abc-\-bcd-\-cda-\-dab,
a^{b+c-{-d) + b^{c + d+a)^c^{d+a + h) + d2{a + b + c),
(a^ b) +(a-c) + {a — d) + {b -c)-\-ib - d) -r (c-d),
a-(a-b) + b-{b-c)+c"{c-d) + d^{d-a).
13. xandy. 14. ax and by, x, y, z. 15. /audfe. .
16. X and y, also a; and —2, and y and —2.
'/7. a, b RJid —c. 18. it;2, — ^/-ands^. 19. b And c.
20. rt and c. 21. a and 6. 22. a^ and 2aZ/.
23. a-6 and abc. 24. a^Z^, a^c. 25. a;^, a^y, and x^j/'J
same ; x'^y, x^y~. 26. Not symmetrical.
'28. a^, a36, a-bc, abed; a^, a^b^. 29. a^, uH.
Exercise xvi.
1. 4(a3_|_7/i_j.c2), 2. 3(a3+i3+c-^) + 2(a6 + &c+ca).
3. 4(a2+Z^2^.c2^f/2). 4. 2(a2+i2_|.c3).
6. 4(a;2+?/2+22+?i2). 6. 2{a3-{-b^ -{-c^) + 6:Sa^b -12abc.
7. 14(x2_{_^24.22j_j_2(a;!/+?/z+2a;). 8. 24:abcmnr.
9. 2aZ;c(a+i-t-c). 10. aS^s + Z/^cS+c^aa.
Exercise xvii.
1. 115. 2. 2;aS-33a5 + 3ra-s. 3. 2. 4. -17-3533.
5. 1, 2{Qa-+l). 6. 0or2?/", 2?/", 0. 7. 36.
8. -(Z^2^a2)3_ (3^2)3. 9. _i5a4. 10. 3888a*6*.
11. a262(a + i), 12. 0. 13. 2a^ -3ai(a-6),
2A34-6a6(a+Z/), 2(^3 + ^3).
Exercise xx.
1. 3. 2. 1. 3. -l±2x/-2. 4. 2. 5. 36. 6. 11.
7. _l^^ 13. 2^=-q, q = (i. 14. jL>=-46, (^ = 14,
ANSWERS. Vll
Exercise xxi.
1. h=-S,r = S"2,d=-24:. 2. c= -20i (f=-13^, ^ = 00i.
8. h=-d, c=-10. 4. a = S, 6 = 0, c=-57. 5. a=-2,
c = 2-H, ^ = 0. 6. c=-10G^, d = 202^. 7. rt = 200,
b=-810, c = 639. 8. « = 4, c = — 27, d = 7, e = SO.
9. 399. 10. x^-{p + 3)x^+{2p + q+d)x-{p + q+r+l).
11. :c3-(2?-3)x2-(2/)-y-3)a;-(^-2+r-l).
12. rx^-{dr-q)x^+{dr-2q+p)x-{r-q-{-p-l).
13. a;3-5x3+7>;-x-r2. 14. a;3-(;)3_2^)x2 + (9^ -2pr)a;- /-.
15. x^ —2qx^ +{pr-\-q^)x + r^ —pqr. 16. ra^ — (j9^ + 3r)a:;2 +
(;>3_2^g- + 3r)a;-(pj-r), 83. -1. 34. 1. 35.-1.
SO. 1. 37. -1. 38and 39. a + 6 + t+c^. 40.-1.
Exercise xxiii.
1. 5?>4 + 15c4. 2. 6. 3. 3. 4.-{ili^6+c + rf) + ... + ...4---.}.
5. 0. 6. 56* — 30ai3+30a363_5rt36. 8.0. 9.0. 10.0. 11.1.
12. {a + l>-\-c + d). 13. -1. 14. a + b-\-c + d.
15. (rt + 6 4-c) (« 2 4- i 3 _i_ 034. ((^4. 6c +ea)+rt6c.
16. (rt.+6+c)2(a2+i3_|.c3) + 2rt6c(a+64-c). 17. a4-h + c-Jrd.
18. (a + 6 + c4-<^)2. 19. (a + 6 + c+rf){(fl+/>4-c-f<0--
(a64-<JfZ4-ac + 6c + 6f/ + ctZ)}+«6aZ. 20. a + 6-fc. 21. 3.
22. -1. 23. 0. 24. 0. 25. 0. 20. l+^x--kx^ + ^'sx\
27. 1-ix-ix^ -^\x^. 28. 1+x+x-^+x^.
20. 1— 2a; + 3a;2-4x3. 30. l-i-k-c-^x-^+^\x^.
Exercise xxiii. /'a)
1. (p-p'+q)^ = (p + l)(p^—pp' — q).
8. 9ip^-q){r'-qt)-{pr-t)^=9{3{p^-q)(qr-pt)-
{pq-r)ipr-t)} X {S{pq-r){r^ -qt)-(pr-t){qr-pf)}.
9. x"(4x3 + 3/>.c2 +3<^a; + r) -i- (a;* + 4/;x3 + 67a:-' +4rx + iJ.
Vln ANSWERS.
10. -4p. (4/9)2-2(6?), -(4;9)3+3(4j9)(6<z)-3(4r),
(4^)4 _4(4p)-2(6(^)+4(4j9)(4r)+2(6^)2 -At,
-(4/j)5 + 5(4/))3(6^)-5(4/>)2(4r)-5(4^j)(65)2 + 5(4j3)« +
5(6r7)(4r), (4jt>)6-6(4;j)4(6y)4-6(4j5)3(4>-) + 9(4;9)2(6(/)2 -
6(47^)2i-12(4j3)(6?)(4r)-2(6<?)3 + 6(6?)i + 3(4/-)2.
11. SqS^ — 4siS3+3s', SQ.Sg — Gs^Sg + lS.SgS^ — lO.Sg, where Sq, »j,
&c., are the coefficients of the terras (taken in order) of
the quotient in No. 10.
12. x^{4:x^ -2Sx + l)-^{x^-Ux^+x-S8); Sj =0, Sg = 28, s_,
= _3, 8^ = 544, s^=-10, S6=8683; 2(a-6)4 = 4526,
2(a-6)6=264122.
Exercise xxiv.
1. (8m + 2)2, (C--1)^ 2. (y3_.3)3, 4^2(2x+y)2.
3. (3a& + 2c)3, 4?/2(3a;-2/)2 4. (|a;2 _ 4^2)3, (^^^2 -1^2^2)2^
5. (« + 6+c)2,(3a;4 -1^2)2. 6. (z-;c+2/)2,Hy) -(-j f .
7. (x2-z2)3. 8. (a;-r/)4. 9. (a + i)^, (i^;3_4y3)3.
10. (a;-2/)2. 11. 4(x2+y2)2. 12. (.c+^)4.
,3. iw".-(±)"r 14. (.-j+c)^.
_ 6 / \ rt
15. (a2-/;3_c2)3. 16. (2rt-2c)2. 17. (2*2 _g5 + 4.)8.
Exercise xxv.
1. (7« + 2t)(7a-26). 2. {Ba + lb){3a-U).
3. (3a-2fc)(9a2+4i2)(3a + 26). 4. (lOx- 6?/) (10.^+6//).
5. 5i(a+2x!/)(a-2.r;/) 6. (3x3-4.y2)(3x3+47/2).
7. (3, + i)(|c-l). 8. (22/2-|x2)(22/2 + fx-j).
9. (3a-l)(3a + l)(9«2 + l). 10. {a-2b){a+2b){a2 ^U^).
11. (a-6)(a + fc)(a2+t2)(a4 + 64)(a«+&8).
ANSWERS. IX
12 (a-^h-c)(a-h-}-c). 13. {o + 2b-Sx + 4:y){a-j-2h-Sx-\-iy).
14. (x2_y2)3 15. (a;_|.,/_|_92)(a;+_,/_2z). 16. 1G(.(,- + 1)(1 -a;).
17. (x + y-\-z){x+y-z)'z — x-^y){z-\-x~y).
18. Axy{x + y){x y). 19. (x- s + ?/)(a;-z — </)(.r4-2+i/)(a;+2- ?/).
20. ^{a-Vc){h-^d). 21. 24a:(l + 2x2). 22. 8ai(a + i)2.
23. {a^h-\'C-\-d){a-irc-b-d){a-b-c-\-d){a + b-c — d).
24. (x+2/+z)(a;-?/— z)(j;+2/-z)(a:-t/-|-2).
25. 6a363^ae_3rt3;,3^^G). 26. (fl3 -i./>3)(,f3 _i3)3.
27. (x2 + j/2+52)(a.2+y2^22-2a:?/-2?/2-2z.r;)-
28. (x+22)(a;-2?/). 29. (« + fc-c)(«-i+c)(^ + c+«)(6 + c-a).
80. {x-y^z){x^y-z){x + y-\-z)[x-y-z).
Exercise xxvi.
1 (a:-7)(a-+2), (a;-7)(.r-2), (a:+4)(a; + 3).
2. (a;-3)(a;-6). (a;-7)(a;-12), («- 12)(a;+5).
3. 2(2x-5)(x+2), 3(3a:-20)(a;-10).
4. i(.K + 12)(ia--3), 5(a;+l)(5x + 3), (3a;3 -4)(3x3 -5).
5. (ia; + 4)(ix + 3). 4(4a:-5)(a;+l).
6. {x-a){:x + a)(x-b){x-\-b), {'2.{x+y)~U]{2{x+y) + ^].
7. (a;2+7/2-a2)(a;2+2/2+62). s. (a + 6-3c)(a4-6+c).
9. (a;+?/)(H-a;+?/){x+?/+(a;-?/)2}.
10. {(i-\-b){\-a-h){a-{-b+{a-bY).
11. (a;2+x-^+2/2 + 2x + 2/)x{A-2+x^+?/2-(a: + 2y)}.
12. (a-o6 + 36)(a + i-c). 13. (a;3+^3^^3j2 _ jg ^^g_
14. (A:2-10x-l2)(a;''-10a;+8).
15. (x2-14x+10)(a;-9)(a;-5). 16. (x^ -2^2)2.
17. (z + l)(2-l)(22-2;, (a;2-3)(x2 + l),
i ANSWERS.
(S:c*-\-5i/^){3x^-2y-^). 18. (c'»+2)(c'" - 1),
19. {x"'-ay''){x'" + by").
Exercise xxvii.
1. {x-bij)(bx-y). 2. 3{x+2;/){2x~y).
3. 4:[Ux-5i/){x-y). 4. 4.{Ux+5y){x-ij).
5. {14:x-y){x—20ij). 6. 4(7aj — oy)(2x — ;y).
7. 2(28a; + y/)(:«-10?/). 8. 'i{Ux~5y){x+y).
0. (8.c-5.y)(7x-4y). 10. {8x+5y){7x-4:y).
11. 2(3x+7/)(a;-3y). 12. (3x-2//)(2a; + 3?/).
13. 2i28x+y){x + 10y). 14. 2(28.«-07/)(a;- 2?/).
15. 2(28a;-f5?/((a;-2/y). 16. {5Qx-5y){x-Ay).
17. 2(4.^--?/)(7x-10.y). 18. A(Ux + y){x-5y).
19. 3(3./;+^j(4x-5i/). 20. (8x+5y){dx-8y).
Exercise xxviii.
1. (5.r-7)(2x-|-3). 2. (5.c + 3)(2x-7). 3. (5a;-3)(2x+7).
4. (2;c-5)(3x-ll). 5. (4a+l)(3a-2). 6. (3.s-7)(4x-3)-
7. (3x + 7)(4a;+3). 8 (Sa^ _462)(3a3+5i3).
9. (4x+l)(8x-l). 10. Sy^{x-y){-dx+2y).
11. (2«+3i/)(2a;+?/). 12. ic^(36+a:)(26-3a;),
13. (3x3 +72/2)(2a:3 -57/2). u. (2xa_9)(a;3 + 5),
15. (2x+2/)(2a;-j^)(a;~3?/)(a;+3?/).
16. {2.c+4-f.v)(2a;+4-7/)(.c+2-3.v)(a; + 2 + 3?/).
17. 169a;y. 18. (19!/3 + 60x//-6a;2)(35x--12a-?/+30?/3).
19. 2(4x7-3x2 -32/2)(61x2-49a;?/ + 61i/3).
20. 2(50^3 + 4x// + 10//2)(x2 + 10j;^+2(/3).
ANSWERS. Xt
Exercise xxix.
1. (7x+6// + B)(.r-2/-~). 2. (5a; - 52/-22)(4r+7/+4).
8. (Bz^+^i/^ + 13){x^-y^-l). 4. {4x+5y){5x-^)j+7):
5. (9x+8j/-20){8x-7/-l). 6. {x-\-Sy)ix-4!/-5).
7. (4.f + 3^-0)(2x + 3.y+z). 8. (3a:-2»/-22)(2rc- 3// + 42)!-
9. {Sx^—29/^+5z"){2x^+5!/^ -5).
10. (15a:2 + 8//2+5zn(a;=*-2//2+82-).
11. (2a-5i-7c)(2a + 3/;-f3c). ;.
12. (a - i + c)(rt + b- c){ci + b + c){a—h - c).
Exercise xxx.
1. -c'+'^+lv'S, 2a;2 + ^±^l/5. 2. ^r^ + 7,?/3^_3,/2 ^5^
^\{6x^-i-5y^±y^VlS). 3. i(4a;2 4-5±v/13),
TV{C(a:+2/)2+5z3+z3 ,/13}.
6. -rV(6-c24-5?/-')(6a;3+ll//2), {6A-2 + o)(6a;3 + llj.
7. i(5x2 4-10±3N/10), (2a3-f3±2,/2).
8. {2(x+y)2 + (3±2V2>2}; ^
^{10x^ + {l0±d\/10)y-}{10x''+(20-6y'10yj^}. \
9. -K9-*^^+7±yi3), A{2x■2 + (6±l/16)(2,•+2)n•
10. ■ ^(2^:2+6+ v/G), |(7u;2-f 20±,/85). i
11. |{4x2 4-(9±v/28)^2|. j
12. 4{7('t-6)2 + 8c2+t;v/29}, -^{3«2 -1-62.^3}. j
13. H^^' + l^+V^^)//'}. i{Ha-\-b)^ + {B±^/3){a-by}.
14. {7«2 + (6±v/14)i-}, (5«i3 4-9n2)(5;/i-+3/j2).
15. {7{vi+n)2 + {G± V U){m-7iy} .
XU ANSWERS,
ExEEciSE xxxi.
2. (x-±2xy + 2y^), {4:X- ±?xy-}'y^), {ix^±xy-\-y^).
3. (x- ±1/2x4-1), (iK- + ^/6x(/ + 3?/2), (ld=2!/-4y2).
4. (x2 + 3a;+l), cc2 + x/6j,- + 3, ia:2±2x?/+2/^.
6. (2^3+//2±|a:/y), {x^ +y^-±\xyVSd), (2a;2 + l±2x).
7. (a:2'" + 8?/2"' + 4a;'";/'"), x^"' + 2y^"'zt'2'x"'y"'),
(|a;2 — ^y^±:xy V5).
8. (2a;2_l + 2x), -(^a;2 -6//2 +a;//)(^.*-2 - 6?/2 -a;?/),
(ic- +a2?/3 -ffla-?/y'2).
9. mx'^-ny'^±xy^p), a;3'" + 2"'-»7/2'"-+-2"'^'»,
10. 4x2 -3±x, 2x2_2±2a;v/2,
- (3a;2 - 2//2 +xy){^x^ - 2.7/2 -xy).
11. 2x2±4.r^-3?/2, a;2 + 2x + 5.
12. 2(a2+„6-j-/;2)2^ (2rt2_^a_,_l)2.
13. {(ir+2/)- + 3(a;+?/)2+z2||(^ + ^j2_3(3.+y)^425|
14. (a + fc)2+|c2±|c2v/5.
15. {4a2 + 5a(6-c) + 2(6-c)2}{4r»2-5fl(6-c) + 2(^>-c)2}.
16. 4(a2+5ai-2i2)(i2 + 5f,i„2a2).
17. {(x2+y2_a,,/)2_j.3(a;2_|.^2_;,.^)^.,. + ^)
18. [i^a^^nb-\rh^)-\-l{a-li]^±^{a-bY^^].
19. (4a2+2rt + l), a;2 + 7^44.
20. (a;2±9x?/ + 9!/2), (IdzSz+Sz^).
21. 4(3a;2-2a;+l)(x2-2a; + 3).
Exercise xxxii.
1. (a,-2 + 3Xa:+3)(a;-l). 2. 20i-2 + 3)(.T3 + .r-3)
3. («2 + 4)(a;+4)(a;-l). 4. (a;+2)(a;-2j(3x2 +a: + 12).
ANSWERS.
5. ix'-3){5x^+4:x+15). 6. {x^ + 6){10x^+5x-G0).
7. (ia;2 + TV)U-«'+^0-e-TTj)- 8. (5a;2 - l)(5x2 _8a;4-l).
9. (5j;2-8)(7i^2_6^_12),
10. (3x2-4j(2U-2-13^-28). 11. (18.«2_|_i)(45^3 4.9^^s),
12. {nx'^ + l){22x^--Bx-2). 13. (U-^ -|j(^x-'+ix+|).
14. 8{x^-2y^){10x^-4:xy + 20y^).
15. (2a;2-5y^)(12*2-6a;(/ + 30//2).
16. (a;2-16?/2)(2x2 + .^xy + 32//2).
17. (a;2_ 6)^11x2 +10x+^/). 18. 10(^-2 +2)(4x2 + 3:c- 8).
19. (a;2_6//)2(l.Sx2-12.cy + 78^2).
20. (x2+4//2)(3.r2+3a;y-12^2|,
21. (.c2-3?/2)(5a:=+4x^ + 15y=).
22. 2(x2-2?/2)(2a-2 -7a;?/+2//2|.
23. (x2+ii/2)(/;2+80x2/-^i/2). 24. (x^- -Gy^){2x^-xy ^12y^-).
Exercise xxxiii.
1. (.i;^+3.c + 27)(x2-9x + 27). 2. a;2 +a;(l±|/3) + 4.
3. {.i-2+l+i(l±\/5)x}. 4. a;2 + l-a;(2±:,/5).
5. 2x2 + 2- 3j;±xv/23. 6. (x2_|_i5_g_ 5)(^2_a._5)^
7. (4x2-2)(4x2-6x-2). 8. (a;2+8x+4)(x2 -3x+4).
9. (x2+7x-2)(x2-x-2).
10. (x2+5x//+3;/2)(x2-xi/+3y2).
11. (x2+l0x-l)(x2 + 2x-l).
12. (x2+7xy + 2/2)(x2-3xi/+2/2).
13. 2x2 + J-?/ -5^2+ ^^^46. 14. (x2+7x//-J/2)(x2-x^-y2).
15. x2+2?/2+3x2/±xy|/3,
IG. {3x2 + 10xy-2y2)(3x2 -4X//-2//2).
17. ^\{nx-+22y'^ + 5xy±^^xyi/n}.
xiY ANSWERS.
Exercise xxxiv.
1. {y-z){v/^-y). 2. {hy + c){ax-^hy—c).
3. {2-'-{-a){x-\-a){x-a). 4. (2a;-«)(a;-2/;).
5. (.T+3rt)(a; + 2/>). 6. (x-b-^){x-a){x + a).
7. (x-6)(a; + M(a.— a)(a;2+rt.x-H-a2). 8. (2x + 3«)(4.t+5?)).
9. {a-\-hx){a-hx^cx^) 10. (a-&c)(a + &x+c.r2).
11. {ax-d){hx'^+rx-f). 12. (p.r-9)(x3 -ar-1).
lo_ («_j_c)(« + 26 + 3cl. 14. (« + fl)(.'c-+a:-fl).
15. (wx-w)(pa;2+?ic-r). 16. {x-a){x-h){x-c).
17. (cc + rO(a:-J)i,'c-c). 18. (a; + a)(a; + i)(a;-c).
19, (rt3^z)(x-«v/)(a:3-7/). 20. («6a;4-^^?/-<7/2)(«ic + %).
21. (^ax-\-c){ax'i -hx-^c). 22. (a;-?/)(xH-//)(w?a;-«//+7-z).
23. (wa;-w/)(«a;4-% + ^'z)- 24. {?«x + r?.)(aa--k"a4-a).
25. (c2-xz)('^2_y2)(a2_a;_y). 26. {x^-m^-x^-a){x^ -n + n'').
27. (l + x-a;2)(l-rta:+/^a;3-ca;3)-
28. («a--rf//)(««— t^)(«*'+ci/)- 29. {wx+q){2rx+n){>n^x-n).
30. (»ia;+n^)('»:c-n?/)(/^2a;3+22y2)(a. + l).
Exercise xxxv.
I (^a,JrOc){a-h). 2. («a:+%)(/w -«'?/).
3. (x-«)(a;+«)(a:^ +«»; + «-)• 4. .r(a+a:)(«2 + rta;+a;2).
5, {ax-h){cx + d). 6. (5a:2 - l)(5cc3 -a: + l).
7, («,_6)(« + i+.x--c). 8. (rt2 + />)(«+/,).
9
(a:-^/)(«+?/)^. 1^- {x-y + l){x''+xy+y^-).
11. (fc-2.v)(2 + i4 12. (a;-l)(a; + 2)2.
13. (/?-?)(;j^-2?2). 14. (a-l)(a3 + 2a+2).
15. («^3_i)(3at2 + l). 16. (2/-l)2(y+2).
ANSWERS. XV
17. {a-\-b){2a^ -Bcib + 2b2). 18. {b"" - l){b^'^ + 2b"' +2),
19. (^'^2")(ya.._3y.2..^,3»j. 20. {a-b){a^-\-ab-2h^).
21. (a"'-c")(a"' — '2c"). 22. (aa;-i)(»- -aa-i).
23. (5u;"-3«-)(7x" + 8(t2). 2-4. («6+6c-ca)(rt6 — 6c + ca).
25. (?rt-6)(OT, + /;)(a-m). 26. (^ -3a2)(l _3a)(l + 3a).
27. {x-y-z)(x^ -2x1/+!/^ +z). 28. (6m- 7w)(4m2 +«3).
29. (.c'" + */")(•*;" +y'')- 30. (a;2+.cy+«a;+!/2)(a;2+x(/-ax-.y)2.
Exercise xxxvi.
1. {x-7/){x-[-y){x^+xyi-tj^){x''-xij+y^-), (x-l){x^-+x+l),
{x + 2){x-^ -2a; + 4), i2a-dx){4:a' +Qax+9x-),
{2 + (ix){4:-2ax + a'-^x^).
2. {x-a^){x^+x^a^ -{-x^a^-^xa^ +a»),
(3rt-4)(9a3-hl2« + 16), {a^ -b2){a^ +b^){a^ + b^),
{x^-2u){£^+2x\i/+4:X^y'^ + Sx-!/^ + 16>j'^).
3. {a-b). 4. :« + 42/. 6. (.«+7/)(a:2 ^?/2)(u:4 + ,y4)
6. 5(//3-x2)(7x*-ll.<;3^3 4.7_y4), («2-2/>)(a2+26)(a* + 4i2).
9. (<< + /;)(m+a)(m2-«?rt+a2).
10. {x^+xy+y^){x^-xy+y^){x^-+2xy-y^-).
11. (a3+6c)(a4-4a26c+762c3).
12. (a:-a+i){(a;-a)2-(x-a)ft + i2}.
13. (x^-2xy + Ay^){x+2y-hix7j).
14. (2:c+3//)(2x-3»/)2. 15. H -2a;)(l+4a;3).
16. {a'^ + ubc+b^c^){a-\-bc)(a^-abc + b^c^).
Exercise xxxvii.
1. S(x+y){y+z)(z+x). 2. {a-b){b-c){a-c).
3. 3(a3-/.2)(i2-c3)(c2-a3). 4. (x + y){y-\-z){z-^x).
XVI ANSWERS.
5. 3{a + b)(b+c)(c + a). 6. (a-^h-\-c){a-h)(b-c)(c-a).
7. (a-i-h){b + c)(e-}-a). 8. (a^ -b)h^ -c)(c^ -a).
9. {a + b)(b + c)(c+a). 10. (a-i)(6-c)(c-a).
12. (rt2 4.62 + c2_rt6-6c-m)(a-6)(i-c)(c-rt)-
13. (a2 + &2^c2)(« + ;; + r).
14. (c-i3)(a_c3)(6-a3). 15. (.c2-?/^)(7/2_22)(^2_22).
16. (x + y-^z){x-7j+z){y-z+x){z^y-x).
17. (a-6)(/j-c)(a-c). 18. 8(a + i-fc)3.
24. (a-i)(^-c)(«-c)(«'-' + 63 + c2+ai + /;c+ca).
Exercise xxxviii.
1. (a-2)(«'--7fl + 2). 2. (x-2)(a;-3)(x-4).
8. (a;-3)(a;-2)2. 4. (a:_2)2(a; + 4).
5. {x-}-l){x^+2x + d). 6. {x^-+2x + S){x-' + 2x-\-S).
7. (.c + 2)(a;-l)2. 8. (x2 4-2a: + 3)(a;2 -2x-+3).
9. (/n — n)(»j2_2wm — 2)^2). 10. None.
11. {m-n){m-2n)^. 12. (&+3c)(62 _26c + 13c2).
13. ^(^yn-n)^{m^~vin + n). 14. (rf+26)(a— 26)(a2— 7a/j+462).
15. (a;- 5) (a: -3)2. 16. (a:+2)(a;2H-3a; + l).
17. («-l)(a2_2a-195). 18. {iJ + 2){p-l){p + A).
19. (a-l)'^(«4-2)(a+3). 20. (a2»_l)(a2»-2)(a2n_3).
21. «2_|.452 + 7rtj. 22. (./-i)2(a2 + 2a6 + 2i2).
23. {p-2){p^-2p^2). 24. (a:"-l)(a;2» + 5a;" +5).
25. (?/-2)(i/3-3i/2+2j/+4). 26. None.
27. {a-b){a'i + 2ab-\-db^). 28. (a"+l)(2r/2"-3^" + 2).
29. (a;-2)(a:-3)(a;-6)(x--7). 30. ix-y){x-2y){x-Syy.
ANSWERS. XVU
Exercise xxxix.
1. 2{x-l)ix''-9.r+10), {x-2yy'{z-3y).
2. (^x + 3y){3z^-xy + y'), (a;-l)(4a;-2)(2.c + 3).
3. {x-5a){3x^+a-^), {2x + 3y){x^+3x!/-y^).
4. (6+c)(ft-4c-)(2/>2_6c+c3), (oa+4:/*)(3a2+7a6-353).
5. {2p+q){2iJ-\-3q){p^-+q--').
6. {10x-9y){15x+lQy){x^-5xy+Sy^).
7. (2jr-3</)(2.c4-3//)(3.c+4yj(3x-5?/).
8. (5a;-2z)(2^3_3^.2^_j.8^y2 + 12y3).
Exercise xl.
4. (a;+2//)(a;2+8!/2). 5. l-2:K+3a;2.
6. {a-x){a+x)^. 7. x- +y- -Jrz^ +xy+^z-zx.
8. (a4-/')(3a + 6). 9. (a;-.y)(2^+3^).
10. a^'-b^+c^. 11. 7a3_3rti-[-2i2,
12. rt-7. 13. (a-fe)(6-c)(a-c),
14. (x-a)2-J(x— rt)+/>2. 15. x^+y2-\-z^ + l.
16. a:(a;2 -«./•+ 6j. 17. .t^ + z/S.
18. {x-y){x^+y''). 19. aS _/;-2_|.c2 4.i_
20. a3-63-c3. 21. a-\-x. 22. (e— fc)(a + 6 + r).
23. ab-ca-bc. 24. a;2+?/a + l _a;?,-f a; + ?/.
25. (a;3-2)(a;+l). 26. a3H-5a+3..
27. (2.£-i/ja2-(.c + ;/)a.c+:e3. 28. a(x2+a;4-l)-(a; + l).
Exercise xli.
1. x^-3. 2. x+r>. 3. a:2__^_^-L 4.. ax'-'-i-bx + c^
5. None. 6. c^+c". 7. {a-6)(x4-«). 8. %+z/).
9. (a-b){h~c)(c-a), 10. o.Z"' + \. ^
12. 5{a-b){b-c)(c-a). 13. {y-l)(x-l).
15. {x+l)(x" + l){x-l)^. 16. {x+l){x + ^){x-\-'d)(x''r4:). 17.5
18. Same as given quantity. 25. {a — l){b — c){c— a).
21). «4+.t;-'+2x-l-l.
Exercise xlii.
1. (x-l)--(.'r2 +4^+16), ic(3.v-7)-=-r/(72/-3).
2. (.?;2_«.,;-|-a2)H-(a.-^-«2), (^4.4)-^(aj_.l)3.
8. (a;-l)0T+2)-r-(a;= + 5.r + 5), (./;3 + 2a; + 3)--(x3 -2.T-3).
4. l-(i-2.c), l-^(a;3_2x-+2).
5. 5fl3(ft, + .r)^a;(a2+a.7T + a;2), (43:24.1)^(53.3 j_2; + l).
G. (a;-2/)-^(.c+y). 7. (S^^-o^ + l)--(4«3^4_|.2rta;2 _1),
(f7j; + i)/)-f-(rta;— %). 8. —1-i-abc.
9. -(a + ^ + c)-i-(«--6)(6-r)(c-a). 11. 5 - 7(^-'^ +a;^+?/2)
Exercise xliii.
1. (4~.r)-f-(5-a:), (a'' +b-)^2ah.
2. a-, 2«-f (a2 + l). 3. ^(l-f r/)--(l + 2rr + 3a2), x.
4. />2-=-a2, (/) + l)-^r762. 5. (rt(._i(/)^(rtc+M), A-^a.
8. 1. 9. -(a'i+«2^3_|.^,4^_=_,.i(«_ij2^
10. («. + /> + c)2-=-2&c. 11. 1^1 , 4:a^x^^{a^+x^).
12. (x + ?/)-f-(.t— ?/). 13. (a -^) »--(« + 6) 3.
14. {x+y)^{:x-y) 15. l-=-a;3.
lb. l^n. lY. ±(l-i)-^(l + c^ la. 1-^0.
ANSWERS. Xll
Exercise xliv.
1. (x9a)-L.5. 2. a-^b. 3. IGa'a;-^ (a* -a;*)'.
4. 0. 5. l^(;c+2). 6. l-f-(rt4-a;4).
7. 12^?/-4-(9.r2-4v3). 8. (43;2+2) -4-a:(16x4-l).
,9. l^{x + l}{x-[-2){x + 8). 10, 4(u;4 4-4.'c-i/2+2/4)H-(.c4-?/4V
11. (a-i)3_^(a;-frt)2(a;-f6)2. 12. 2a ---X.
13. (236-77x)H-18(lla;-8). 14. l--(a-6).
15. 15a(3a-.'c)-^(9«+2u:)(« + 3:r).
IG. (10x-7)^(a;-l)(2./;-o)-l-4-(2a;-7)(.«-4). 17. Z.
18. ?/"(«/"-«")■ 19. (a -6)^" + 2.
20. 0. 21. 4a;-^-^(a;i2_i),
22. - (a2 +o2)(a2 _ ab'\-b-^) -4- (aS -^s^^^a ^. <,6 + 63).
Exercise xlv.
1. x-y. 2. a + h. 3. 0. 4. 0. 5. 0,
C. {{a+b){c+a)x^ + 2(ah + ic+ca)ax-2a^bc} h-
{a-\-h){a+c){x + a)x{z-}-b){x+c.) 7. 1.
8. a+6 + c. 9. 1. 10. a;3_,/3 n. q.
12. (rt-i)(6-c)(a-c)-r-(« + ^')(6+c)(c + a).
13. a;3-=-(a:-a)(a;-6)(x-c). 14. 1. 15. 0.
16. {b{x + a-b)-{-ax} -^- {(7t'4-(o -«)(.«— ^)}.
Exercise xlvii.
1. (fl_fc)2+4c-^=0. 2 8. '3. 10. 4. «2+/;«,
5. 7J? = 2, «=1. 6. 2a;2, or 6. 7. 7/?.= — 5, w = 6.
8. ±12. 9. (a2+&2j(c2+d2). 11. _36c-46-2 4-i2c2-463.
12. (a;2-4a;+3)(a;2-4), sl]bo (x^ -Sx-\-2)lv^ -x -6).
13. i(-l±i/5). 15. a-=-c = d2 H-e-', rt-f-fe=/2 -r-e',
XX ANSWERS.
h-^c = d- -^f-. 17. ac'^^b^d &\ididad = bc.
19. i^;3_|_27g = o. 24. _p = 2?-/i32±2?7i^>/(m3+f).
25. 4.{p-2,)^q.
ExERCtsB»-- xlix.
1. 6, 31, a, -3. 2. -41, -a, 2, 10.
3. a + ^ 6--^/, 6-c, 3. 4. -2,6, -5,12.
5. -14, a -36, 2a- 36, 5i-3a.. 6. 7, 4, a, 6.
7. ic, 5-=- a, 0, 1. 8. -1, {{a + hy^ -a} -^b; a+h.
9. (/)-«), a + i. 10. 1-^a-b, l^{a~b), 1 ^ («'-'+ ^/2).
11. 2i, a. 12. a + 6, c-i-(a + 6), i^(a-c).
13. (i_c) -=-(«- 6), /^ + c. 14. ,-< + /;, a^+ab + b^.
15. fl2-«& + /^^ 1. IG. -1, {a + b)-^{a-b).
17. (<; + &)(e-?>), 2-f-15, 3^14. 18. -1-12, b^ac, a~b.
19. {a''-{-b-)-^a-b2)^a^b--', a{b^ -{- c^) ^ be.
20. 10, 12, 4, \. 21. 1000, |, |. 22. 9^^^^, ai, bc^a.
23. 63_^rtc, c(fl + &), ft(«+6)-=-«.
24. a-^h, {a-b)~{a + b), -{a+b)^-^{n-J))^.
25. -1, -1. 26. (a2_c3)--(a + 5)2, 2, 3^.
27. rt/^ i-^«, rtCH-^ 12. 28. 12, -ac^b.
29. 9, 2. 30. 12, 1. 31. 3, 1. 32. (2fl-l)(2« + 2), 0.
38. iH-m. ' 34. 1. 35. {ah + bc+ca)-^(a2 + bc+c^).
3(3. (^^^2^l,2J^c-)^{<th-^bc+ca). 37. a + b+c. 38. 1.
39. 1. 40. 1. 41. 1. 42. 15. 43. 16j. 44. 6.
45. 5. 46. {>i])cja+pqb + qc + d)-i-mnpg. 47. —^.
48. 0. 49. -25^136. 50. 1.
answers. xxi
Exercise 1.
1. 2, 3* 2. i, i. 3. ±2, 1:^. 4. 1, 1^. 5. +f,
±{a+b), a. 6. 4, 5, 2, 2^. 7. -3 or 2; 4, -3;
2i, — li. 8. 1 ; f or | ; i or 3. 9. — f or f, ^ or 6 ;
for -|. 10. -1, 2-, -i, 1. 11. 0, -b, dh.
12. <f, ±«V-1. 13. 1; i(-l+v5). 14. ±a.
15. +6c, — (i + c). 16. a + 2b. 17. // or +a.
18. -2ab, ^ab{lztV7). 19. a, ^/, -{a-^b). 20. a, 6.
21. rtorl-«. 22. -«, -6, a -26.
23. «, ^6(l-/y)H-(H-rt-^). 24. a3-6.c2-37.<; + 210.
25. a;4-4<fx3-13«-'a;:i+64fl3^-48c<4.
26. a;(a;-l)(x-T2j(u;-4)==0. 27. a;* -4.1-3 +.c-2 4-6x- + 2 = 0.
Exercise li.
1. 4. 2. -7f 3. -107. 4. 8. 5. 3a. 6. ^Vf-
7. 50-1^, 17. 8. 22, 46A. 0. 7, 3. 10, 10, 10, 11.
11. 0 or 11; 33. 12.3956^3971. 13. |(15±v/190).
14. 3. 15. 3. 16. 4. 17. If 18. U. 19. 3^.
20 4. 21. ±3. 22. 11. 23. 2and-l±i/-3.
24. 2^. 25. 0. 26. 3a. 27. |. 28. Jf. 29. 3.
80. 10. 31. 0, 1, or (-5±a/-23)-=-8. 82. 102f.
33. (-ll±:y'4681)H-20. 34. 2, i, |. 35. -4
86. Qov dz\/{a"+b^).
Exercise lii.
1. (l-<i)^(l+./), a(m4-l)-H-(w-l), b{m + l)-^a(m-l).
2. a — b, 0, 0. 3. fc, ijia-i-b, b ~ ca. 4. 1, —1, 0.
5. -^or-1. 6. {c-/v)(t-+c2)^2aic. 8. 14, 4^.
9. 2, 6^295. 10. 73-- 210, (./ f 6+r + t/) -r- (w+7i).
ANSWERS.
SI
11. b~a. 12. h-^a. 13. « or 0.
14. rhi/rt'+l-^ 2. 15. h 16. \^. 17. 0 or 4.
18. c-^ab. 19. 83i/(2a;-l) = 100A/(3x-3).
20. 75 --52. 21. 8. 22. 34/^^.
23. l^»(n — 1). 24. a6-^(/>-a). 25. 4, 3f or 13|.
20. a^h^ -~{a-hy~,'d. 27. 4a3-^(l+a)2, i(ft + i)3_j.(a-i)2.
28. (1+62) -=-2^6. 29. ^^(l-a;) = 2 H- (a + i)2.
30. ~a±as/ {{l+h+h^)-^-lb].
.r+li ' _ Z^^+l'
-1/ " U-li
ExEBGiSE liii.
1. 8. 2. 0. 3. 3. 4. {Vm + y'nY.
5. ab^il-'l-Jb). 6. 4^7. 7. l--(«.-2).
8. 18962-^-12393. 9. Vo -^ ( v/'/ + 2).
10. (c4-2Z)c2)^(2c3-26). 11. i. 12. 18a.
13. a;2=80-f-81. 14. zh's^v/ar- l^- +Av/-ll-
16. ±v/ {«'- ^^'-^^l^}- 17- 0. 18. A«-
19. (^c-a-hy^^'2nahc. 20. a:2 = a2(7i-l)2 -f. (2n- 1)
21. 16..7/=0i-4.^'-^)2. 22. 0, -|i.
23. /_^-l)^0. 24. 2v/(l-7«2)-=-mv'(4-m2).
25. (rt2_i)|a3-f-2+ V(«--|-lj}H-rt2.
26. {cn-an + cY~h{n-iy. 27. ±5.
28. 2\/(3x2 + 10) = (17;/17-3|/3)h-7. 29. ±5.
80. ±,/(36-2a). 31 ^^l{a^-h^).
ANSWERS. XXUl
32. (2// + 2z-2x-)3+216.tv/2=0. 33. fa/G.
34. a(rt2-47i-f-8) - (2/i-4). 85. a- +2a.
36. ±y'{'3a^+b^)-i^y"d.
Exercise liv.
1. - 1^1(2 ^lj2^^a. 2. {2a2+b-)^2a.
3. {(a- 6)a3 - 2c{a- +ab + b'')}^{a-' - 2c{a^-b'^)}.
4. - b. 5. a + 6+6-. 6. ab-7-[b — a).
7. a;--3«:K-«-=0, &c. 8. a. 9. ^{a + b + c) 10. lH-a6c.
11. l±{a + b-\-c). 12. (rt-6)(«c-26)-^(«+6)ac. 13. -c.
14. {fn + '^h)3. 15. ±2. 16. c^{a-b).
17. (a-6)-^(« + />). 18. |a. 19. ±2. 20. +2, &c.
21. ^(«+<)-^i"-0- 22. «, (3rti-3/^2_^,)^(l_,_3^j_3i)
23. ^/. 24. a, b, 'lb. 25. a, (c-* + 6«6)^6/>. 26. ^(c + 6a).
27. \a. 28. a + Z'. 29. (a/; + 6c+c«)^(a + 6+c).
30. ±6, ±a. 31. i/{l^(^ !)}•
32 {6(rt-5)-4c(c-6)}--{lc-3/>-a).
33. ((•2_a6)^(„+i^2c) 34. ^{-29±V37).
35. (x-)-«)2 = 262-a2. 36. ^ [b'^ - ^ab). 37. !(?>-«).
38. 3^, f 39. a;--6.'c = a. 40. lit:i/19. 41. b, b-a.
42. (r(2+6--)^(a + ^). 43. a;=-5H-2, 44. J (5 ± a/3).
45. -2a, 4^, la. 46. -3c/.
Exercise Iv.
1. bc^{a + c). 2. («2 + ^,_2a/.)-f(a + /^-).
3. (ad-ir)--(«-i). . 5. -^--^. 6. c. 7. .^(a + i)
8. a-\-h. 9. 0. 10. 0. 11. abc.
12 («2+/>2_^e-3)-=-(a + 6 + c). 13. (a+Z/ + c)H-(a2 + fc2-}-6-3).
14. {a^+h^^c^\^{ah+hcJrca) 15. —7. 16. -^•
17. U. 19. 4. 20. -140. 21. 17. 22. 10. 23. a.
^■^- aC+c^-M: 25. Si, 0. 2G. S^. 27. (r^fc-c2)-(« + 6).
28. -&, rt, 29. 0, 0. 30. ^(fl-fi-c). 31. -^.
32. (f. 33. ah~{a^-h^). 34. -3f. H.5. |. 36. -3|.
37. lufinity. 38. 10. 39. abc-^{ab + bc + ca).
40. {ab-{-hc + ca - ad - 5 J - cd)-^(a +b-\-c - 3(/).
41. aib + c)^ -^(b'^ + c^ - ab+bc - ca).
42. ic(rf-fl) + (({-6)(5-c)(c-J)H-(fl/)-fif4r^/-flc^-&^-c2).
43. bc^ —b^c-ac--\-b-d-ah(l + acd~-{ab-\-hc-ac-b^).
44. -(r/-f5 + r). 45. a-\-b-i-r. 40. («& + &(• + ra) -^ "Jr.
47. -i(/'+c). 48. {ab + c)~2a. 49. 9. 50. 2. 51. 7.
52.4. 63. ^V(5+v/785). rA. A, (am-nb)^{n-m + a--b).
55. T-V, i(« + c)--(^/2 4.,,/,_^i3). 50. 0, -I, +. . 67. 10.
Sn. «^,^(6-r), c{a"--ir{b-c)a-bc]+a{b^-c^-)^
{a"- +b^ -c^ +ah-bG-ac). 60. 6^{rt + 5).
61. vipcq-^apnq-^iapn'^ — cqm~).
62. {(')?/?(a — c) + 0-/(6— ff) 4-r/^j(c - b)] -^
{m((rt-c)+/t(6-a)+7-(c- /;)}. G3. {u^ ■]-b^)~ab, 0, ^J.
dhi-o) -q(h-d)
^^ ^ ^ ' a{n- g) — m{b — d} *' '
66. 100. 67. 13, 111. 68. 11, 7.
a2 +2ac+ad+2bc + 2ab
69. (a + ^-m-7z). 70.
fl — (i
XXV
71. U'i+b)±i/{i(a-b)3-lc2},aorb. 72.0. IS. a + b + c.
7-i.
UW-f "C-t-C(l
75. a + i-j-c.
76. « + 6+c.
obc
77.
(ab + hc + cu) -.
■-(«+/> + (•)• 78.
i2+a3_c3.
79.
c — a — b.
80. 0. 81.
0 or 11.
Exercise Ivi.
1. .4 = 0, or 7^" = 0. 2. .^ = 0, or i? = 0, or C's* 0.
8. « = 0, ora — Z^ = 0. 4. x = 0, or?/ = 0.
5. In the first case either « — 5^ = 0, or x— 4^+3 = 0, in the
second case both conditions hold. 6. s; = 0, or a;:=a.
7. x — O, or x= —b. 8. x = a, or x^c-~h.
9. .f=-0, or a; = 3. 10. sj = 0, or;7; = a4-6.
11. j; = 0, orjj=+a*. 12. a; = 0, ora; = i^-Ha.
13. x^O, oru; = a. 14. a; = 0, or a.
15. x = Q, or x^a + h. 16. a; = 0, or « + />.
17. -{2ah)-^{a+b). 18. a; = a, or6.
19. x = a, orb, ore. 20. 5. 21. 1. 2^. 21.
23. x=\, x^'S. 24. x = 9, .r = 4.
m
25. a;=l, orB. 26. («&) h- (ff + 6).
27. .r = a, or 6. 28. :c = {a^ +b^) ^(a+b), x = b+a.
''.9. {2nh)-i-(a + h). 30. a; = a, orA.
1. ?;=1, or (l4-a)-^(l -«)• 32. a; = ff.
34. .i- = a-6, or i(i4-c)- <^5. x = a + b, or ^{a + c).
36. a; = " , or 1. 37. a-{-b-c.
^-^b+c
89, a; = a, or 4 (46 -a). 39. /;s -c, or a + fe + e.
XXVI ANSWERS.
4f\ 1 m — n .^ nc — vh
40. a; = l, or 41. x =
n — p mc — ap
^2_ p(a-b)-c{m-n)^ ^3 ^^i^^^^,)^ ^r i(6- a).
vi(c ~b) — a[7i —p)
44. x = 2a — b, ox Bb — 2a. '45. x = a-\-c — b, ora; =
46. x = a-\-b, or . '—
6 8
47. x = 4n+b, or a + b. 48. x = , or __
b — c c
49. {a-b){b-c)x^-{a^-\-b^+c--ab-bc-ca)x +
{a-c){a.-b) = 0.
50. x=±'S, or +2, 51. x=±Q, or +2.
a+h a—b a h
7, or — |— r, 54. X = -r-, or - — •
a — b a-{-u 0 a
2ri + 3b Sa + 26
55. x~b — 2a, or a — 26. 56. x= — ^ , or — ~^ — ^
57. x = {mb+na)-^{m-irn), ov {ma-nb)-i-(m+n).
68. x=V{{m + 2n)- \/{m-2n)}^]/ {{)n-\-2n)+ ^{m -2n)}.
17(6+1)- A/(c-l)f
60. «{v/(36--2) + /(2-c)}h-{v/(3c-2)--/(2-c)}.
61. a{N/(2c-l) + l}-^{l-A/(2c-l)}.
62. i(a + 26). 63. i(a + 6)-(a -6) \/(m-2;i)-f-N/(m4-2n).
64. -^-, or -^. 65. 2a6~(a+6).
8a + 5b 36 - 5a
66. « = — g — , or g — •
67. x = 2a{y'(c + 4)-v(c--4)}--{v/(c + 4)+ v/(c-4)}.
ANSWEKS, XXVU
68. .r = 4, or 3.
69. ^{«±|/a3-4w) where m^a^ ±:^ ^^tc-^a"^) ; 3 or 1.
70. a, b. 71. x = i[a + b±y/{{a + bp—4.{ab-\-t)}],
where t = l{a-b)^± ly' {{a-by+ ir}.
72. x = 0, or a, or ^a(l±i/-8); a; = 4, or 2.
73. x = 0, ov a + b, ov :i^{{a-\-b)±^V{a~b)2 -4:ab}.
74. z--{a-b)x+ab = &c. 75. x^ l{Sa-b). ov l{3b-a).
76. a; = 3rt-26, or 36-2rt.
77. y^ —vi- =0, where y — vi—x and 2m = a-\-b. See Key.
78. 2/2-m-'=0. 79. 2/--m2=0. 80. y^-w^^O.
81. 2/--«i^=0. 82. j/2_„,2=o. 83. ;?/2-«t3=0.
84. (2/=^ -^2K5y2+7/.2)^0, (where also A- = |(rt-i).
85. A:4-Y/4=c. 86. A5+10^-3y2^ 5^7/4 = c(^^4_y4) &c.
87. «?/±/cv/(A;-3c±:r) = 0, where .s-2 =3/t + c, and r- = {k-^c)^
+(A;-c)(3A;+c-). 88. -3± v/(9±12/24).
89 — 102. Work with a variable to such that Mu; = a;- +1.
89. r(; = (rt±s)-4-6, where .s = rt3 + 262.
90. if = (3rt+2i±s)-^2(a-6) where .s= ±5,/(a2 +2a6 + 46'-').
91. w = (3±&-)^(l±:.s) where s = (i- 4a) -=-/>.
92. (u' + lj2=rt^(«_i). 93. w2^2a~-[b-a).
94. (a;-)-l)-4-(a,--l) = a^(fl-8?>).
95. (if + 2)--(Mj-2)=.-^(l±6') where s=r (16'/ + i)-^6.
96. ii'2(4a-6)^(rt-6). 97. iv^ = {^-la-'db)~[^a-b).
98. w = (7)±s)-=-2a where s^ = /;2 + le^s.
09. u; = (fl + 6±«)-i-2(a — 6) where s = («-)- 6) 2 -i-8(a-6).
XXVIU \NSWEES.
100. w — {a-l-b±s)^2{a-b) where s'^~{a — b)» =
{(a + 5)2+4(a-i)2}-f(a-&)2.
101. (vf+2)±(w-2)= +s-=-(4 + 3s) where s^ =2a-i-{a-\-b).
102. {ic + 2)~-iv = ±i/{5a^{a-\-U)}.
103. i{2a + b), i(a + 26). 104. 2a-b, |(«+5), &c.
105. 1, 2, 4, 5. lOG. ±1, 2, 4. 107. 1, 2, 3, 4.
108. -i, -1 1, f. 109. -1, 8, 4. 110. -a, 5a, 5a.
111. 15, 20. 112. 2i«. 113. 4, -1.
114. 7, —1. 115. |(6f^rt + ca-4-A + r/7)^c). IIG. ±a^m, &c.
117. 2s(s-ft)(s-6)(s-c)-f-|/{.s'--rt2)(s'3_ 0^(5/0 _^2j^ wliere
2s = a + ft + c, 2.s'i=a3+62^c.
118. (2a6+2ac2+26c3-rt^-6-2-c4)-f-4c2. 119. a, b, ^{a + b).
120. ±rt or ij'jfv'S. 121. a, b. ^{a+b).
Exercise Ivii.
2. x,2; y, 1. 3. x, S ; y, 1.
5. X, -lOi; 2/, 5*. G. a;, -2; ?/, i
8. .r, -2; ?/, -3. 9. x, -|; ?/, a,
11. X, 12; ?/, 8. 12. a;, 8; ?/, -9.
14. «, 12; y, 15. 15. a:, 18; y, 13.
17. a:, 7 ; y, .9. ' 18. a;, 7 ; ?/, -3.
20. X, 2 ; 7/, 3. 21. x, 3 ; ?/, 4.
23. a;, -3; ?y, |. 24. x, 12; v/, 15
20. a;, 8; y, 9. 27. x, 3; ?/, 1.
29. a;, 11; //, 7. 30. a;, 17; y, 13.
32. a:, -4^; y, -\^. 33. a:, 13; y, 10.
35. x, 11; ?/, 6. 36. a;, 7*?/, 5.
1.
X,
7; ?/, 9.
4.
X,
9 ; y, 5.
7.
X,
-1; 2/, 1.
10.
X,
-i; y,h
13.
X,
10; y, 12.
16.
X,
•3 ; y, -2.
19.
X,
7: y, 3.
22.
X,
25.
X,
7 • 1; 3
28.
X,
7; 2/, 8.
81.
X,
6; y, -4.
S4.
X,
4|; 2/,3A
ANSWERS.
XXIX
37.
x,2; y, 3.
88.
X, 5 ; y, 3.
89. Equations
40.
x, 3 ; y, 1.
41.
X, 7 ; y, 5.
not independent.
42.
x = 0 = y = Q.
43.
0, 0. 44.
X
= 0 or 13 ; y = Q or f f-
45.
X, 17 ; y, 20 ;
z, 5.
46.
X
23 '
', 234 -, 247
> 2/' T30' ^^ rSTT-
47.
11, 7, 9.
48.
21, 22, 23.
49.
-15, -6, -8.
50.
3, 4, 5.
51.
12, 15, 10.
52.
5, 3, 1.
53.
h H. f •
54.
3, 5, 7.
55.
11, 13, 17.
56.
5, 3, 1.
57.
9, 7, 3.
58.
7i, 8i, 9i.
59.
31, 2i, 14..
60.
2-3, 3-4, 4-5.
61.
30, 20, 70.
62.
88 -=-59, 109^
}^51
[), 1004 --59.
63.
30, 12, 70.
64.
6, 12, 20.
65.
5, 2, 0.
%Q.
1, 1, 1.
67.
11, 9, 7.
68.
5, 3, 1.
69.
2, 3, 1.
70.
3, 4, 5.
71.
XXX
.S' 3' f
72.
5, 4, 3.
73.
7, 3, 1.
74.
2, 3, 1.
75.
1, 3, 5.
76.
0, 1, 2.
77.
1755-698, 360
^349
, —15705^698.
78.
I h 1-
79.
5, 4, 1, 3.
80.
4|,3A.2A,U.
81.
31, 41, 51, 21.
82. 7,
41
,4,8
h
83.
20, 10, 0, 30.
84. 11
-^
24, :
\, 1-^24,1
85.
270 -- 117, -
52--
■117, 15 H- 117,
-
-126
^117.
86.
Each 210.
Exercise Iviii.
1. {alc—ac')M^'b-<tb'). 2. h{cn- dm)~{nd -he).
3. h{d-c){d-a)-^d{h-c){h-a), c{d-a){d-b)~d{c-a){o-b).
4. j/ = c2-j-ff?<. -f ^'f' + f'^;, z = <hi-\-ew-{-ax-\-hy,
u = ew+ax-{-by + cz, ic = ax-\-by-^cz-\-d\i.
5. « = -^»!(a-^4-c), &c. Q. x={p{a^-b)-m{ab-l) +
XXX ANSWERi5.
7. x={l- arn + abn — abc2:)-}-(ibcdr)~-{l-\-abcd€), &c.
8. 1 = n-^{l-^a) + b-T-{l +b) + c-^[l +c).
9. 1 = ai + Zw + ca + 2abc.
Exercise lix.
1. (nc — hd)^(na — hm), (vie — ad)-^(mh — na).
2. (no-\-bd)-i-{a?i + bm.), (mc — ad)-i-{bm-\-an).
8. c{n — b)-^ {an — 771 b), c{m~a) -^ [bm— am).
4. {b — c)a-i-{b—a), b{a — c)-7-(a — b). 5. ab-i-(a-\-h), y, same.
6. ab"-i-{a^ + b"), a~b^{a-+b^). 7. ac-^ia + b), icH-^a + ft)
8. (a- — i-)-J-(an( — 5m), (/;2 _(:j3j-^^5„i_ ^^),
9. a + b — c, c + a — b. 10. a + c. 5-fc.
11. a(cn— rf»?)^-(iici — ac), b{cn — dm)-^{ad—bc).
12. i/={93(a2_c2)-6(6 + 2a)}-^{(«-i)''*-c34-4fcc}.
13. a + 6 — c. a-5 + c. 14. rt + 6-c, c+a— 6.
15. (m-rv)(w— a)-f-(Z)-c), &c. 16. i-r-^a-i)(a — t), &c.
17. (?»-ic)(Z-a)-^(c — a)(a — 6), &c.
18. a;=J:?-j-(;?^+«^5+w) + '^ so y and f.
19. p[l — (Ja-\-vib -\'nc)\-r-{}d-\-mq-{-nr)-\-n . &c.
20. (m'-^ + 2a3-62-c2)-=-3<f, &c. 21. y = a-Z,-fr. &c.
22. X = (rt.Z^+6c4-ca)(6+c - 2a)(26- a - c)-=-{(a - c)(6 +c - 2a) +
(t_c)(26 — a — c)|. Corrected equation, a; = i(i+c), &c.
23. ma-^(a + 64-c), &c. 24. )ipi~-{aup-+bmp-Vcmg).
25. l---(6-c), &c. 20. i(i+c-a), &c.
27. «M-4-(i, &c. 28. 2=l-f-(a+o-c).
29. a + b, &c. SO. 1^2a, &c.
ANSWERS. XXXI
31. (m8+»5-^2)-^-2w7^, &c. S2. ^(a + c-b), &c.
33. l{m-+n--')-^''lmn, &c Si. 1h-(6+c- a), &c.,
35. Iic^^b+c), &o. 86.h + c~a,&o.
37. a, h, c. 3b. h'^ -c^, &c.
39. i(a + 2/'-c + 3rf), &c. 40. i(-ia+6 + 3c-2d + 5e).
Exercise Ix.
1. aJfh, a-b. 2. K*2_|_5)^ ^(a^-S),
4, a^{a—h), b^{a + b). 5. l-=-(rt— 6), l^(a-f5),
8- a-{-b — c, a — b+c. 7. a + i — <;, a — b+c.
9. {ab~l)^{a-l){b-l), {a-b)^{a-l){b -1).
10. (l+a)-^-(«i-l), (l-t-/>)-^(«6-l).
11. {a + l)(b-\-i)---{ab-l), {a-b)^{ab-l).
12. fl(a+6), b{a-b). 13. a{i(«+C')-c(a-c)}^(a»-bc),
a{/;(a-6) + c(a+c)}H-(«2_6c). 14, ~-(a + b). ab.
15. i(&+c), &c. 16. («-2/^+3c)-9-38, &c.
17. 2-r-(6+c), &c. 18. rt + 6, &c. (by symmetry).
19. b^-c-,&c. 20. i3-c3, &c.
21. ^abc, {l-a)(l--h){l-c), (2-a)(2-5)f2 -c),
22. 2fl6c-4-(a6 + 6c-ca). 23. 1, i, 1.
24. ar — [ma + iib-[-pc+qd), &,c.
25. « = 0, or(--l)-^(^-^)^
26. (6+c-rt)H-(a + ^+cJ, y = {b -c -aj-^ia - b - e),
27. ^(a — 6 + 7?i — n), &o.
XXXU ANSWEB8.
28. (4a + 2c — <i— 36), y-^z by symmetry-
29. —{a-b + c-{-d), {ab-^bc, &c.), -{abd-{-&c.), abed.
30. ^(a — b-irc — d+e), others by symmetry.
31. x = {a — lb-\- Imc — Imnd + Imnpc) — ( 1 + Imnpq), the others bj
symmetry. 32. x-=b-\-c — e, &c.
84. ?/ = (a+5i + 3c-7tH9e)-j-22, &c.
35. 0 = |(a-f-c), then symmetry. 36. 2 = (;4-f^ + «, &c.
S7. a; = a— 26+3c — 2cZ+e, then by symmetry.
Exercise Ixi.
1. x={1al + a-Jrh-^r)^2{a-b) where r = 4a(62+6-(-l) _j-
(3a-Z/)(36-rt).
2. a; = (ar + l)^(«r-l) wliere r^ = (J2 - 1) -e-3(a3 -62).
3. a;={i/(l + a)(l + ^)-l/(l-«)(l-6)}^
{|/(l+rt.)(l + /;)+v^(l-a){l-6)}.
4. ^/76-a/3)H-(a6 + a^), (rt|3 + 6a)H-(rt/J-6a).
6. x={i/{a + h-irc){a+b-c)+->/ (b->rC—a){u-\-c-b)} ^
{ ^/[a + b + c){a + h-c) - 1/... }.
6. ■{a + b)-^{l-ah). 7. x= (a/3-a6) -i- (fl;3+6a)
la; — 1/ " (a—ni^ib — n)*
10. x={b-^G-a) -4- UV^ + c-a)(c-f ft-6)|^a + 6-c).}
11. a; = (6 + c)-5-^(a+6)(/; + cXc+a).
12. a;= ^/(a + i + c) -j- «, &c.
13. a;={62+c2-a(6+c)} -^ i/2(a3 + ?,s ^c» - 3fl6r).
14. (6+^-a}, &o. 16. a or (a--6)-i- {l-a6).
1
ANSWERS. XXXUl
16. x+y=s/(a-hb){a+2h) ~ y{a-b),
x — y= y/(a + h){a — 1h) -=- \/(a — 6), &c.
18. {x-\-y)-^{x-y)= -/(a-f-Si) — |/(a.-^) = »i suppose.
19. y^ =m-T- (am^ — vi+l) where m =
.1 -=- a+6± v/(aa -6s +1) -^ (a-^b).
20. a, 6. 21. «={v^(a-c) + |.-c}?/-=-{i,/(a-c)-iA}.
22. x=(a-{-c)y -=- (a — c), &c.
23. x-\-y = {ab-l)-i-ia-b), &G.
24. a;i/={v/(6 + 2)-i/(&-2)}-i/{(6 + 2)--,/(ft-2)) =
p suppose, x^y- {|/« + 2) + |/'(a- 2)}-^
{N/(« + 2)-i/(a-2)}.
25. a;2y2j,2 = ^(a + i._c)(fc-j-c-a)(c+«-6\ (&c.
26. x=(a6 — fee — ca)-j-2N/rt6c. 27. a — x"^ = rfc:wi, where m is
the value of v in the equation 4crt — 4(c+aj)-f-ii;'' =
(ca — ah — be) ^ -{-^bica — ab — bc)v -f ih^v^ .
28. X = ii/{abc) i-r- + — 1,7/ and z by symmetry,
29. a(i3+C=)-T-(fc='+c2)^(/>2_|_c3)^ &c.
30. c{V{a + b) + ^{a-b)}^V{a + b}.
81. 0 ora(6 + c)-=-26c, &c.
32. »= -1 or a|/(a^ — l)-=-v'(a-* — 6=*), &e.
xxxjv answers.
Examination Papers,
I.
1. {a+b+c){x+7j+z).
4. 2a;2'»-ix3'", 4-- — - -—• 5. -!-14i/-19-=-27, 0 or f
•^ ^ 0 c a •
6. 16, 7. 199, 8, AB 37, CA 52, 56' 45.
II.
1 (a+h)^. 2. (a+b + c){a^ + hs+r^ -dahc). 8. l-^a;^.
4. (a2'»4.2a'" + 2)H-(«'" + 2), (a + /> + c)^(a-6- c).
7. — -^j 4 or 6. 8. p-q-^(r-pq).
9. 684. 10. (aw.+6/i)-r(a + />), (by common rule).
III.
1. -117, a^{z-x) + (x-y)ah + {y-z)b2.
2. a+bx+cx^, 3 — 4a;-i-7x--^-10a;3
3. 21 ; 6, 9, 12. 4. 160 eggs.
5. 40,35. 6. VU'^+a)+ym-a).
7. 5ori;4. 9. {a + b+c){a—b){b-c)[a-c.
IV.
1. (^a + b+c)^. 2. rt^&. 3. (4.^3 - 9?/2)(43;3— 47/5). 1 + Va;.
4. ii/3-^i/6, 17 5, 4. ■ 6. 1. 7. Oor4;8, 1.
8. 7t, 12. 9. 4 or 6. 10. *(-3± -/ -39).
V.
1. 8x3 + 1-^125. 3. 16a'-' /^2. 4. ft-f-8rt. 5. 15, 12.
6. ia, ia; 4 or -9, 7. 6, 7, 8, or -6, -7, -8.
ANSWERS. XXXV
YI.
1. (,,24./,D). 2. a- +h^ + c^ +ah+ac-bc.
3. x-+i/-'±U!/, i7x + G!/-9){x-y + A). 4. -20, 0.
5. b or 1^1. 7. x = 4c or 9, y = d or 4 ; 1, 2, 3 ;
a; = M-7 + i/33). 8. -1, 0, 1, or 5, 6, 7, or — V", -4, 3.
9. x(x-+3)(x2-2a;-l) = 0. 10. a;-7-2/-3-=-4.
VII.
2. --01. 4. a;2+a;- 1.
0. (36a;2 + 18.c + 9)H-(16j;4_81), (^3 -a/>)^(j;2 + a6).
6. d; 3; x^i{a-b).
VIII.
3. {x--xij + y^y^ ; a^ + b'i-aVj^.
J 6 7
5. «^3 +ix + ca;~^ ; -^a; +^x — ix
6. a;24.^,^+^/^ ; 50(.-«-f-5)(a;-4)(a;-6).
7. a = 0 = /vorrt=l, 6 = 2.
8. 3 or —43-^7 satisfies the equation 2— y . . , &c.
IX.
4. axy+b. 7. +\/«/!'; V'(«+6)-f-|/(rt — 6), and t/ = reciprocal
of this. 8. P(22a-216)--20a(a-6). 9. 7, 15, 48.
X.
1. ax^ +hy^ +2cxy. 3. x-2. 4. 24.
.5. 5.c--3«u;+4rt-^;-^4-^-l/i. 7.3;^. 8.4,6.
If X
XI.
1. 2. 2. G, 8. 3. (4a;*-9/)3, V^x4+/j^4.
7. 0 or -2J ; ?/=±3 or ±^-9, &c.
XXXVi ANSWERS.
xn.
6. w- —m=p, q = 0.
7. a;3 = (a-2)-^(a+4), -,/rc= -J/^(w- 1) - en} -^ |/a(w— 1)
8. a6-{(a-l))6-l)-l}. 9. 37.
XIII.
1. 2. 3. 1+r ^x + c„x- +&C., vfhere c^, ffg^ *^c., represent the
combiui^tions of a^, a^, . • ■ taken one, two, &c. at a times
4. $4000. 5. {m{b'c — bc'} — n{ca' — c'a)}-i-{ab'-a'b),
'^^J^^'c^' 7 (« + /3)'+(a-^)'-2/ = 0. 9.65.
XIV.
1, l^iihr. 4. Smiles. 5. rtm-=-?i ; .'c+z/ = ±5 or ±1,
x-y=±l or ±5. 6. 20.
XY.
2. A. 3, ^3(r3-ifn+c/2(?;2-fl'c)+ar(nr,7-ftc). 4. 2-=-(w-n).
If 7?i— « is uegative x is neg. which shows that they were
together before noon. If ?«-n = 0, a; is infinite, i.e., they
are never together.
5. a;2(a;2_„2)(a;-2«); (a:^ -r?2)(-x2-?/3).
6. {a + b+c-{-dd)^{a + b + c + d) : {x + y-\-z) ^ {x-y -hz);
XVI.
4. 2a;4 -3.r3+4.r4-3. ' 5. See paper XIX., prob- 4.
IX T
a |{l±i/(a" + 4t")--\/4J". 8. {b,c,-b,c^)-^{a,b^-a^b,)
^Q>^c^-b.,c^)-^{a.,b^-a^b^). 9. acH-(a-6),
ANSWERS. XXXvii
XVII.
1. {2ah-(a + h)}^ab. 2. x* -2z''{a-\-b^) + {a-b^)^ ]
8a»-2a6-10rtC-3/)2+2c2 + o6c. 6. m^-n=4:.
B. (m 4-n)(^hq—iic)—-{inq—pn}, {p-\-(i)(mc— bn)-T-{niq—pn).
xvin.
1. l-a;'": 2.1. 4. x-a-fc. 6. (a;- 7/)(a:-2).
7. 1. 8. 9a; ±1^ \/2 ov zt^VG. 9. 10,11.
XIX.
1. K4x-'' + ri3. 3. 5a.2_l; (;r2+a;.i/-|-2/2)2.
4. {a-c)(a-d){b-c){b-d)^(a + b-c-dy-.
6. (a + 2)2H-4(a2+«). 7. (T^)^ -a2)^12rt. 8.18,22,50.
XX.
1. (l + m'^x+(l-n)ij. 3. a;2_i.
4. (a;+2/--s.){//+z— a:)(2 + ar-?y)-=-(a; + ?/+2)2;
(a26-2a62 -a3+rt63 -,.^3)^(^4 _54).
5. 1. 6. a''{cb'-hc') + b''{ac'-a'c) + c'\a'b-ah') = 0.
8. a(6-c)H-(6-a), b{c-a)-^{b-a). 9. 8. 10. 2000.
XXI.
1, 8. 2. 24a7)c; 2a» +4:?j'i^x^ + 62a^.c'i'+Ua^x^ + 18x^.
4. 0; ^7_|_i6. 5. 16; x+2ri x -2a. 7. 3?.77 oz. of gold,
783 oz. of silver. 9. - (8±4i/3)H-(3zh2 \/3 ;
a;=±2or V — 1, 7/==Fl or =p2i/-l; 8, 4.
10- y = cost of 2nd bale = 60 ;i: 20 ^7.
XXiVlll ANSWERS.
XXII.
1. '02997, a^+ag+p^. 2. ah'' -b^ -hc = 0,
{a).{a-b){8a-Sb). {b).x{x~ l){a—b){b-c){a-c).
3. b^=Uc. (a), (d+b)^. 5. (a) {ax-by)^{ax-{-hij)
7. («) i(a-Hi + 6-), {b), I, f, 2.^
(c). b — c,c — a,a — h. (d). — 1±V''2.
XXIII.
1. ^x^+^x^-x^ + kx- + ^l + {95x^+^lx^-A0x+A2)^
(3x4 - 21x3 -f 9a; -6); -882. 4. x = a + 2c, y = b-\-Sc.
5. (2). ^V 7. (1). a+i + c. (2), 1, 2, 3, 4. (3), 0 or j\.
12. Coll. to Newmarket 63 miles.
XXIV.
1. \^^(x-y){y-~z](z-x). 2. S-m-v, 0. 8. 5, 3. 4^.
4. (1). A = bc-^2a, B = ac-i-2b, C = ab^2c; (2). a^.^fc^^c'-'.
11. 7y=+2, a;=+5, &c. ; a;«-10x=-19or -16,
— ^ . — ^ 4P«
.-. ic^B or 3, &c.; {b ±a )p-«
XXV.
1. (cr--Bx)M^^-x^); 1. 2. — -— -/.2(.T+fl)-«2x2.
5. a;2-2x = 2, a; = 3, ?/ = 2; ?/ = 53-f-24, &c. 6. 4 miles, 3 do.
ANSWESS. SXXIX
111
10 1+ -j-+r724-^-72:o + &c. = 2-71828 approximately;
{(5x):h-3''"^^}{1-6-11 . .. (5r-4)}^|ji. 11. -| and -^,
XXVI.
2. a-v/(a6-a2); 4. 3. 2, i, or ^-(-3± V^o) ; a;+7/+2 =
v/(a2+2i/2), .•.2 = c+i/(a3+262), &c. ; i(-4±,/76).
4. 3. 7, p2^.i^r^_^27 = 0. 10. (l+.r)--(l -x)2 -
a:"-i{3x-l + 2«(l-x)}--(l-x)M vi--(15n + 9), ^V
XXYII.
1. {hn-n7(fi. 2. («+ /,+c)(«, -/>)(/>- c)(rt-c).
3. (a-6)(/j-c) + (6-c)(c-a) + (c-a)(a-&); {a-b)^ +
(6-c)24-(c-a)2. 4. a;=:a^(^,2+^3_}.c2), &c.
5. Vab; 18 or -2; i + 3j. 7. i{a + h)&c.;
x={l+a^-b^±-l^{l-a-b){l-a + h){l-{-a-b){l+a+b)\
H-2a; a;-f-y=:(l+rt)(l— ^>)-H(l-rtcj, &c.
8. —8; {p^-Ap^-q+8pr)-^{p^-Aq).
9. (m + «)-^-2»m, (n — m) -i-2wn.
I.
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I know of. They will soon, I have no doubt, supersede texr-
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TTAMBIilN SMITH'S ST A.TICS.-Elementary
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Examination Piimor in Canadian
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T3»ox T Contains the words in comtnon use in daily life, to-
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fchol early, he should at least know how to spell the words
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