J. A. (James Alexander) McLellan.

The teacher's hand-book of algebra; containing methods, solutions and exercises illustrating the latest and best treatment of the elements of algebra online

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EACHER'S HAND-BOOK




LIBRARY

OF THE

University of California.



Gl FT OF



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M. 3 Cage's ^atljematital Scries.

THE .TEACHER'S

Hand-Book of Algebra ;



CONTAINING

/

METHODS, SOLUTIONS AND EXERCISES



ILLUSTRATING

THE liATEST AND BEST TREATMENT OF THE ELEMENTS

OF ALGEBRA.



BY



J. A. McLELLAN, M.A., LL.l).,

M
HIGH SCHOOL INSPECTOE FOR ONTABIO.



The object of pirre Mafliemati'r<>.ivhicli is mwther vanipfor Alfiehra,istheunfoldivo
of tlie laws of the human intelligence.'' — SyiiViiHTEB.



FIFTH EDITION-REV ^SEO AND ENLARGED.



W. -T. GAGE & COMPANY,

TOKONTO AND WINNIPEG.




Entered according to Act of Pt^rliament of Canada in the year
1880 by W. J. Gage & Coitpany. in the office or tne iviinisier
of Agriculture.



PREFACE.



This boot — embodyincr the substance of Lectures at Teachers'
A.ssociations — has been prepared at the almost unanimous request
o^ the teachers of Ontario, who have long felt the need of a work
to supplement the elementary text-books in common use. The
following are some of its special features :

It gives a large number of solutions in illustration of the best
methods of algebraic resolution and reduction, some of which are
not found in any text-book.

It gives, classified under proper heads and preceded by type-
solutions, a exeat number of exercises, many of them illustrating
methods and principles which ai'e unaccountably ignored in
elementary Algebras.

It presents these solutions and exercises in such a way that
the student not only sees how Algebraic transformations are
effected, but also perceives how to form for himself as many
additional examples as he may desire.

It shows the student how simple principles with which he is
quite familiar, may be applied to the solution of questions which
he has thought beyond their reach.

It gives complete explanations and illustrations of important
topics which are strangely omitted or barely touched upon in the
ordinary books, such as the Principle of Symmetry, Theory of
Divisors, Factoring, Applications of Horner's Division, &c.

A few of the exercises are chiefly supplementary to those pro-
posed in the text-books, but the intelligent student will find that
even these examples have not been selected in the usual appar-
ently aimless fashion; he. will recognise that they are really
expressions of certain laws ; they are in fact proposed with a view

l(w220



IV PREFACE.

to lead liim fr, investigate these laws for himself as soon as he
has sufficiently advanced in his course. Nos. 8, 9, 10 and 11
afford instances of such exercises.

Others of the questions proposed are preparatory or interpreta-
tion exercises. These might well have been omitted, were it not
that they are generally omitted from the text-books and too often
neglected by teachers. Practice in the interpretation of a new
notation and in expression bv means of it, should always precede
Nits use as a symbolism itself subject to operations. Nos. 23 to
36 of Ex. iii., and nearly the whole of Ex. xv. may serve for
instances.

By far the greater number of the exercises, are intended for
practice in the methods exhibited in the solved examples. As
many, as possible of these have been selected for their intrinsic
value. They have been gathered from the works of the great
masters of analysis, and the student who proceeds to the higher
branches of mathematics will meet again with these examples
and exercises, and he will find his progress aided by his familiar-
ity with them, and will not have to inten-upt his advanced
studies to learn processes properly belonging to elementary
Algebra. In making this selection, it has been found that the
most widely useful transformations are, at the same time, those
that best exhibit the methods of reduction here explained, so that
they have thus a double advantage. A great part of the exercises
have, of necessity, been prepared specially for this work.

Articles and exercises have been prepared on the theory of
substitutions, on Elimination, &c., but it has finally been decided
to hold these over for Pt. ii,, which will probably appear if the
prSsent work be favorably received.



CONTENTS.



Chapter I. — Sobstitution, Horner's Division. &c.

PAGE.

Sect. 1. — Nnmeiical and Literal Substitution 1

Sect 2.— Fundamental Formulas and their Applications 10

Sect. 3. — Horner's Methods of Multiplication and Division, and their

Applications 21

Chapter II. — Principle of Symmetry, &c.

Sect. 1.— The Principle of Symmetry and its Applications 33

Sect. 2. — The Theory of Divisors and its AppUeations..- 39

Chapter in. — Factoring.

Sect. 1. — Direct Application of the Fundamental Formulas 62

Sect. 2. — Extended Application of the Formulas 71

Sect. 3.— Factoring by Parts 79

Sect. 4. — Application of the Theory of Divisors 83

Sbct. 0. — Factoring a Polynome by Trial Divisors 90

Chapter IV. — Measures and Multiples, &c.

Sect. 1. — Division, Measures and Multiples 101

Sect. 2. — Fractions IO9

Sect. d. — Ratios 122

Sect. 4. — Complete Squares, &o 130

Chapter V.
Simple Equations op One Unknown Quantity 138

Preliminary Equations. Resolution by Factors. Fractional Equa-
tions. Application of Ratios. Equations involving Suids,
Higher Equations, Sec.

Chapter VI.
Simultaneous Equations I70

Equations of Two Unknown Quantities. Systems of Equations.
Application of Symmetry. Equations of Three Unknowns.
Systems of Equations.

Chapter VQ.
Examination Papers 207




:ir.R^R>t^



CHAPTER I.
Bection I. — Substitution.



Exercise i.
1. If a = 1, 6 = 2, c = 3, rf = 4, a; = 9, ?/ = 8, find the
value of the following expressions : —



l_a_(l-l-a;)}.
a-{x-y)-{b-c){d-a)-{y-b){x + c).
x - !j^y - {y — a)\d + c(h - c)v']■
{x+d){y+b+c)+{x-d){a—b-(^)+(y+d](a-x -dy
{d-x)^ + {c + v^*

la-b){c^ -b-'x) -{c-d){b^ -a'-'x) + {d-b-c){d^ -f'*'^
d — ad + c (\d + b
d + a d—G d — b

2. If a = 3, 6 = - 4, c = - 9, and 2s' = a + b + c, find the
value of the following expressions : —

s{s — a){s — b){s-c).
««H-(5-a)2+(s-5)^+(s-c)2.
«« - (s-a){s — h) - {s — b){s -c) — (s — c)(s— a).
2{s — a)[s — b){s — c)-j-a{s-b){s-c) -\-b{s-c){s~a)+c{s- a){s-b).

8. If ffi = 2, 6 = - 3, c = 1, a; = 4^, find the value of the
following expressions : —

a2_62 „2+/,2 (a-^,)2 (a-h)^ *

c^^Tb^' ~a^b^' (a + by' (a + by'

a^^ab + b^ a^-b^ a; (2x-3 3«-l ).x-l



a'^-ab^b^ a^-b^ 2
{a + b)\{a + by-c^\ ^

Ab^c^-{a^-b^-c^)^'
a'{b - g) Vb''{c-a)^ ",^{a-b) ^
- Ja-b)(b-G){c-a)



2 SUBSTITUTION.



\



4. If a = G. ^ = 5, e == - 4, rf = — 3, find tlie value of tin
following expressions : —

y{b-' +ac)+ y(c2— 2ac), yib^+ac+ ^(c^— 2«c)}.

5. If a; = 3, i/ = 4, 2; = 0, find the value of: —

{3a;-v/(a;3+?/2)}2{2a;+v/fa:2+2/2+2)}.

G. Calculate the values of {x+y+z)^-^{x^+y^ + z^, ^^^^

xyz

(a) 9^=1, y = 2, z = H.

(b) x = %y = 3,z = 4:.

(c) a; = 3, 2/ = 4, 2 = 5.
(J) a: =10, .v = ll, 2=12.

7. Givcu x=S, y = 4:, z= —5, calculate the values of

{x+y+x}^ -S{x+y+z) (xy+yz+zx).
x^{y+z)-{-y^{z+x)+z^{x+y) + 2xyz.
x'^[y-z)-{:ij^z-x}+z^x-y).
(5«-4z)2+9(4a;-2)2-(18a;-5«)2.

{Sx + Ay + 5z)^ + {Ax + 8y + 12z)Z-(5x+5y + lSz)*,

8. If s = a+ 5 +c, find the value of

(2s_a)2 + (2s-ft)2-(2s+c)2, given
(1) a = S, 6 = 4, c = o, (2) a = 21, 6 = 20, c = 29,

(3) a=119, 6 = 120, c = 169, (4) a = S, b= -4, c = 5,
(5) ^=5, 6 = 12, c=-13.

9. If a = l, 6 = 3, c = 5, f? = 7, f = 9,,/"=ll, prove that



<i + 6 + c + (i + e+/=



2



1+1+1+1+1=1(1-1

ab be cd cle ef 2 \ a /

A+ V + l_+±=l/l-l

abc bed cd^ def 4 i fl6 et

J_+J^+_l.=l(l- M.

o6';i 6cde cdef 6 \a6p t/(//



SUBSTITUTION.

a2^h2.\.c2_ab — bc — ca = h^+c^+fl^ — bc — ed — db =
t.2 4. ,f2 _|_ ^2 _ cd - de -ec = d^+e^ -\-P -de- ef-fd.
10. lia=l,h = % 6 = 3, d = A, e=o,f=&,(j = l, prove that
a+b + c = U-d, a-{-b + c-\-d = ^de,
a^b+c+cl+e= W, a + b+c+d-\-e+f= ^fg,

aa^fc3+c.3= ^4l+^), a2_{.b3 4.^2+^3=:

de{d±^)^ «3+^,3 +,3+^3 +,3 = ^0
«3 +i2 +,2 +,/3 ^.,2 + /^ = M+'/J.

a3_|_i3 4.^3 4-^3 = (« + J4.(.+ rf)8,

a^-\.b^4.rS + d^+e^-{-p = {a+b+c + d-\-t+f)»
A , lA . A. cd(c+d)(c^d—l)
6c(o+c)



a4 4-64 4.^4 4_J4 =



6ie((i+e)(crfe— 1)



^44.,44.,4+J4 + ,4 = €Ai-^i^),

,4 + ,4+,44.d4+,44./4= ^(Z+^) (^^^tI).
c2+t^2=e3^ C3+J3 4.,3=y3

11. Assume any numerical values for x, y, and z, and calculate
the values of the following expressions : —

(a;5 - 10:c3 4. 5a;j 2 + (5a;4 - 10a;2 + 1)3 - (a;3 + 1) 5 .

(a,4.i)3_2(x+5)3-(a;+0)34-2(a;+ll)3 + (x+12)3-(a;+16)3.

(^2 _^3;3 4.(2x^)2 _ ^^2 4.^2)2
(a;3_3x^3)34.(^);c32/-^3)3_(a;3 4.^2)3.

(3x3 +4a;i/+2/2)3 + (4a;3 +2a;(/)2 - (5a;S+4a;.'/+i/2)2
(a;_2/)34-(v-5)34-(5-a;)3-3ia;-2/)(2/-5;(«-aj).
Art. I. If X = any number, aS, for example, 3, then x-
(which = x.a;) - 3*, x^ (which = x.x'^) = ?..(;2, a;* (which = x.x^) =
Bx-3, &c. Oi- 3 = x, 3j; = a;2, 3a;3 =0;^, '6x'^ = x^, &c. Hence prob-



4 SUBSTITUTION.

lems like the follomng may be solved like ordinary arithmetical
problems in " Keduction Descending."

Examples.

1. Find the value oi x^ — '2.x — 9 when x^H.

x^-2x-9
5

5x

-2x


5

15 Explanation,

-9 x^^6x,

.-. a?2-2a; = 8a;=15, and

6 .-. a;2-2a;-9=:15-9 = 6.

2. Find the value of a;* — *^ — ^x"^ —dx—b when x~'^

x'^—x^ — 4.x'^—Sx — 5
3

p^ 3x3

Tj 2a;3

3

Pa ^*^

-4a;3

r„ 2ar2

3 .-. x4-a;3-4j;«-3a:-6 = 4

— if ic = 3.

7>s ^-^

-3a;

Tj 3a!

3

P4 9

-5

% *•



SUBSTITUTION,

Explanation.

:. x^—x'^ = 2x^ = 6x^ ,
,•. x^-x^ — ix'^ —2x^ — 6x,
:. x'^-x'^-ix^-Sx = 3x^-9,
:. cc^ — x^ — 4a;2 — 3a; — 5 = 4.
8. Find the value oi 2x^ + V2x^ 4-6a;3 -12x+ 10,
Using coefficients only, we have

2+12+6-12+10
-5



Pi ••


. -10

+ 12


r,


+ 2

- 5


Va ...


-10


^2 ■••


+ 6


Vm. ...


- 4


'9 ••'


- 5


1t>, ...


20


r3


-12


fg ...


8




-6


Pa ...


-40


r4


+10


r. ...


-30



the quantity = —30 if ar= -5.
Art. II. If the coefficients, and aiso the values of x are small
numbers, much of the above may be done mentally, and the work
will then be very compact. Thus, performing mentally the mul-
tiplications and additions (or subtractions) of the coefficients,
and merely recording the partial reductions r^, r^, r^, and the,
result r^, the last example would appear as follows : —



6



SUBSTITUTION.



-5)2 4-12 +6 -12 +10
2
-4
8

Art. III. In the above examples, the coefficients are "brought
iown" and written below the wodncts p^, p^, p^, p^, and are
added or subtracted, as the case may require, to get the partial
reductions r^, r^, r^, and the result r^. Instead of thus " bring-
ing down " the coefficients, we may " carry up " the products j9^,
P2' Ps' Pv writing tliem beneath their corresponding coefficients,
and thus get r^, r^, r^, r^ in a third (horizontal) line. Arranged
in this way Ex. 2 will appear

1 -1 _4 _3 _5
+ 3 +6 +6 +9



1 +2 +2 +3; 4;
and Ex. 3 will appear

2 +12 +6 -12 +10
-10 -10 +20 -40



-5



2 +2 -4 +8; -30
Comparing these arrangements with those first given (Ex. 2
and 3), it will be seen that they are figure for figure the same,
except that the multiplier is not repeated.

Art. IV. When there are several figures in the value of a;,
they may be arranged in a column, and each figure used sepa-
rately, as in common multiplication. Where only approximate
values are required, *• contracted multipHcation " may be used.

4. Find the value of 3a;5-lG0a;4 + 344a;3_^700a;3-1910a;+
1200, given a;= 51.

3 -160 +844 +700 -1910 +1200

1 3-7-18 37 -23

50 150 -350 -650 1850 -1150



•7 -13 +37
result is 27.



-23; +27



SUBSTITUTION.



5. G'ven »;= 1-1S3, find the value of CAx*-lUx+4:5 correct to
three decimal places.

64 -144 +45

64 75-712 89-5673 -38-0419

6-4 7-5712 8-9567 -3-8042

5-12 6-0570 7-1654 -3.0434

•192 -2271 -2687 - 1141



1
1

8
8



64, 75-712, 89-5673, -88-0419,
.'. result is —-004.



•0036.



1.

2.

3.

4.
5.



Exercise ii.
Find the value of

xi-Ux^-Ux^-lSx+n, fora;=12.
xi + r)0x^-lGx^-16x-61, for ic= -17.'
2x4 + 249a;3-125a;2-|.l00, fora:= -125.
2.^3 _478a;3_ 234a;- 711, for a; = 200.
x'^—iix'^-8, for a; = 4.

6. a;6-515a;S-3127a;4+525a;3-2090a;2+315Ga;- 15792, for a:
= 521.

7. 2a;5+401a;4-199a;3 + 390a;3_602a:+211, for «= -201.

8. 1000x4 - 81a;, for a; = •!.

9. 99a;4 + 117a;3-257a;2-325a;-50,- fora;=lf.

10. 5a;^+497a;4 + 200a;3 + 19Sa;3- 218a;- 2000, fora;=-99.

11. 5a;5-620a;4-1030a;3 + 1045a;2-4120a:+9000, fora; = 205.
Calculate, correct to three places of decimals, —

12. a;3 4-3a;2-18a;-38 for a; = 3-58443, for a; =- 3-77931, and
for a; = -2-80512.

13. 7/4- 142/2 +1/+ 38 for t/ = 3-13131, for y= -1-84813, and
for ?/= -3-28319.

Exercise iii.
What do the following expressions become (1) when x = a, (2)

when x= -a?

1. a;4 -4ra3 + 6fl2a;3_4fl3a;-l-rt,4,

2. y\x^-ax+a^). 3. y(x^ + 2ax+a^).
4. (a;2-|-aa; + rt3)3-(a;2-ax- + r/2)3.

If a; = 1/ = 3 = «,^nd the value of the following expressions r



8 SUBSTITUTION.

5. {x-y) (y-z) (z-x).

6. (x-i-y)^ {y + z-a) (x+z-a).

7. x{y + z) (y2 +z2 -x^) +^ ^^, + x] {z^ +x^ -y^-)+z{x ^y) (x^ -^

8. -^ + JL + ^_.
y+z x+z x+y

Find the value of

n ■^ , X 1 abc

9. — + — .wtienjc=

a b a-{-b

10. + — + — -, when x= — (a-b+c),

a[b — x) b[c — x) a{x — c) a

11. ^+ -^-, wheua.= ^(Ll«)_.

a b — a b[b+a)

12. (a + x) {b+x)-n{b + c)+x^, when a; = —.

b

13. bx-\-cy-\-az, when x = b-\-c — a, y = c + a — b, z = 'i-^-b -c.

14. <l±^l±b^ - __1^ _, when a:= -a.

a(l+6) —bx a-2bx

15. — — - —! —, when a; =*(/;- ^0-

\x+bj x — a — 2,b

16. (p — q) {x+2r) + {r — x) (p+q), when a; = ^ ^ ^^ ~ ^ ■ ' ■.

17. ffl3(6-c)-f62(c-a) + c2(a-6), when a- 6 = 0.

18. (a-\-b + c) (6c+m+rt6) - (rt+6) (6-f-c) (e+a), when rt= — 6.

19. (a+6 + c)3-(a3+63_^c3), whena+6 = 0.

20. {x+y +z)'^ - (x+y)'^ - {y+z^ - {z+x}'^ -\-x'^ +y* +z*, when
x-i-y-r-z = 0.

21. a3(c-62) + 53(a_c2)_}.c3(j_a2)+a6c(a&c-l), when6-a»
= 0.

22. aW«_!+!iV + 6^/^^l±lT. when a^ +6^=0.

23. Express in words the fact that

(a-&)2=a2_2rt6+62.

24. Express algebraically the fact " that the snra of two quan-
tities multiplied by their difference is equal to tjie difference of
the squares of the numbers."



SUBSTITUTIOSI. 9

• 25. The firea of the walls of a room is equal to the height mul-
tiplied by twice the sum of the length and breadth : what are the
areas of the walls in the following cases :

(1) leuglh I, height h, breadth h.

(2) height x, length b feet more than the height, and breadth
6 feet less than the height.

26. Express in words the statement that

{x-^-a) {x+h)=x^-^{a+b)z+ab.

27. Express in symbols the statement that " the square of the
sum of two quantities exceeds the sum of their squares by twice
their product."

28. Express in words the algebraic statement,

(x+y)^ =x^ +y^ + Sxij{x-\-y).

29. Express algebraically the fact that "the cube of the differ-
ence of two quantities is equal to the difference of the cubes of
the quantities diminished by three times the product of the
quantities multiplied by their difference,"

30. If the sum of the cubes of two quantities be divided by
the sum of the quantities, the quotient is equal to tl: ^ -square of
their difference increased by their product ; express this algebrai-
cally.

81. Express in words the following algebraic statement:

""Lzyl^ix+yy-xy.
x-y

32. The square on the diagonal of a cube is equal to three
times the square on the edge ; express this in symbols, using
I for length of the edge, and d for length of the diagonal.

83. Express in symbols that " the length of the edge of the
greatest cube that can be cut from a sphere is equal to the square
root of one-third the square of the diameter."

34. Express in symbols that any "rectangle is half the rectan-
gle eoutained by the diagonals of the squares upon two adjacent
sides." [The square on the diagonal of a square is double the
square on a side.]

85. The area of a ckcle is equal to x multiplied into the square



10 SUBSTITUTION.

of the radius ; express this in symbols. Also express in symbols
the area of the ring between two concentric circles.

36. The volume of a cylinder is equal to product of its height
into the area of the base, that of a cone is one-third of this, and
that of a sphere is two-thiids of the volume of the cii-cumscribing
cylinder ; express these facts in symbols, using h for the height
of the cylinder, and r for the radius of its base.

Exercise iv.
Perform the additions in the following cases :

1. {b-a)x+{c-b)y, and {a+b)z+{h+c)y.

2. ax-lnj, {a — b)x-{a+b)y, and {a + b)x-{b—a)y,

3. (y~z)a^+iz-x)ab + {x-y)b^, and {x-y)a^-{z -y)ab-{x

4. ax+by + rz, bx+cy+nz, an."c.>t4-.''.'y + &z.

5. (a+b)x^+{b+c)y^+{a+c)z^, {b + c^x^ +{a+c)y^ + {a + b)z9,
{a+c)x^ + {a + b)y^ + {b+c)^^, and-(a+/;+c) {x^+y^+z^).

6. x(a-b)2 +y{b-c)2-\.z{c-a)*, y(^a-b)^ +z{b-c)^+z{c-
^)^, and z{a - b)^+x{b - c)3 +y(c-a)^,

7. {a-b)x^+{b-c)y^+{c-a)z^,{b-c)x^ + {c-a)y^+{a-b)z^,
and [c-a)x^ + in-b)y^+(b-c)z^.

8. {a-^b)x + {b+c)y-{c + a)z, {b + r,)z + {G + a)x-{n + b)y, and
(a+c)y+(a + b)z-{b + c)x.

9. rt3-3rt6-^^/;2, 263-363+C3. ab-^,b^+b^, and 2a&-^?;».

10. ax^-nbx'', -Qaaf+lbaf, and - 8bx" + I0ax'' .

11. What will {ax-by + cz)-\-{bx + cy-((z) -{cx + mj+bz) be-
come when X - y - z = l ?



Section II. — Funovmental Formulas and theib Appcication.

4. By Multiph cation we get

{x + r) (x + s)=x^ + {r + s) x + rs A.

(x-hr){x-ts){x + t) = x'' + {r + s + t)x'' + {rs-tst + tr)x + rst B,

From A we immediately get
(a; -}-j/) 2=^2 +2x^+2/2 [1]



FUNDAMENTAL FORMULAS. 11

{x+7j + z)^=x^+2xy + 2xz + y^ + 2yz t z^ [2]

(2«)^ = 2rt2 + 2 nab [3]

{x+y){x — y)^x'-^—y^ [4]

From B we derive

{x±y)^ ^x^±8x^y + Sxy^±ir [6]

= x^±/j^±3xij {x±y) [6]

{x + y4 »)^=a;2 +y'-^+z^ +'Sx-{y+z) + 8y^{z i- x) i-Sz^{x + y)

+ 6xyz • [7J

= x^+y^ + z^ + 3 {x + y) (y^-z) (z + x) [8]

= x^ +y^+z^ +8 {x+y + z) {xy -\- yz + zx) — Sxyz... [9]

(2a)3^2a3 + 8^a^b -\- Q^ahc [10]

[The symbol £ means the sum of all such terms as]
Formula [1] . — Examples.

1. We have at once {x -\- y)'^ + (^ — y)'^ = 2(^2 _j_ ^2^^ a^^j
{x + yY —{x — yY=4.xy.

2. (a + 6 + c + d) '^ +{a — h — c 4- d) '^ may be written

{{n + d) + (6 + c)}-^ -h {{a + d) — (6 + c)}2, which (Ex. 1; ==
. 2{('< + J)2i-(i + f)^} ; similarly

l^a — h -f c— (/)2+ [a + b—c — d)'^ = {{a — d) — {b-c)}-^-^

.-. (a+ 6 + C+ tf)2 + [a — h — G -ir-dY -f {o. — h + c — d^Jr
(a + 6-c-(/)3 = 2{(a + J)-^ + (6+c)s+(a-c/)2 + (/, _c)a} ^
(again by Ex. 1) 4(«3+63+,.2_i.,i2).

3. Simplify (« + ?j-fc)s - 2{a+b+c)c + c^ ;

This is the square of a binomial of which the first term is
^aJrb-^c) and the second -c; the given quantity .•. =
{(«+6+c)-c[- = (a



12 FUNDAMENTAL FORMULAS.

4. Simplify {a+b)* -^a'-i + b^) [a-i-b)^ + 2{a* + h^).

By Ex. 1. 2[aA + b't)=:{ct2^b^)^ + {a» -b^)^ ; :. given quan-

tity = (« + 6)4- 2[U^ + //2) (« + 6)2 4- („2 +02)2 + („2 _ ^>2^a =
{{a + 6)2 _^a3+62j}2 + ^,i2_63)2=rt4+2«2y3^ /;i = (a2 +62j2.

Exercise v.

1. (a;+3//2)2 + (a; -32/3)3, |i,,,3 + 3J2)2 _ ^,,2 _ 37,2)2.

2. Shew that {mx-{-n;j)2 + {nx-my)^ = {m^-^n^) [x^ + i/^).
B. " " {mx — ny)'^—{Hx — iny)^ = \^m''^—n^){x^-i/^).
4. Simplify ;./ + 3ij2_^2(«H-36) ((i-^jH-(a-&)3) {a-b\^.

6. " (a;+ 3)3^ (x -1-4)2 -(a;+ 6)3, and (4^3-2^2^2 -

(i(/2+2a;3)2.

6. Simplify (a + 6+c)2 + (6 + (;)»-2(6+c) {a + b+c)

7. Shew that ['ix + by)^ + {cx-\-dyY-^{ay - bxY -\- [cy - dx)^ =
(rt2 + i2 + c3+(Z3j (^3.34.^3^.

8." Simplify (a;-3y2)3^.(3^2 _^)3 _ 2(3a;3-^) (a:-3v/2).

9. " (a;3-i-a;^-^2)3_(^3_a.^„^3^3^and(l + 2a;+4a;3)2
-|-(l-2a;+4x2)«.

10. If « + />= - Jc, shew that (2(t-6)2 + (26 -c)3 + (2c-a)2 +
2(2«-6) (26-c) + 2(2Z>-c) (2c-«)-|-2(2c-a) (2«-?>) = /^jc2.

11. Simplify- 2 (ff- 6) 2 -(a -26) 2; (a^+^ah-^b^)^ -{a^-\-b^y.

12. " ((/ + M2-(6-H6-)3 + (c + t/)2-(^+«)2.

13. " (^a;-2/)3+ai/-«)2+a^-a;)2+2(ia:-2/) (Az-cr^

+ 2(A2/-^)(i-^-^) + 2Q^-2/)(i2/-=)-

14. Prove that [x- yf + {y -zf +{z-xY = '2{x-y) {z- y) +

^{y-x) {z-x) + ''l{z-y) {z-x).

15. Simplify (l + o;)* -2(1 +a;2) (1+^)3 + 2(1 +x4).

16. " (a;-l-.(/+,^)2-(.c+?/-z)3-(^+2-a;)2-(z+a;-y)2.

17. " (a;-2y+3z)3 + (3z-27/)3+2(a;-2?/ + 3z)i2?/-32).

18. " (r/2 4.62-,,. 3)3 _f.(c2_ 62)2 +2(/;2_^.2)(rt2_(.62_c2),

19. " {x+yy + {x-y)^-\x-y}-[x + y)».



FUNDAMENTAL FORMTJIiAS. 18

20. " {5a+3b)^ + 16{da+by^-{lSa + 5b)9.

21. Shew that (3a-ft)2+(3fe-6)34-(3c-a)3-2(i-3fl)(36-c)
+ 2{'db-c){Sc-a)-2{a-Si.){3u-b)-i{a + b + c)2=0.

22. If z2= 2x7/, prove that (2x3 -2/2)2 + (22 -2^ 3)2 +(a;3-2z2)3

-2(2x3-2/2)(22-2^2)^_2(x2-2z2)(23_2j/3)_
2(x3 - 222) (2x2 -1/2) = (x+2/)*.

23. Simplify (l+x+x3+x3)2 + (l-x-x2+x3)2 +

(1-X + X3-X3)2 + (1+X-X2-X3)2.

24. Simplify {ax-{-by)^-2{a^x^ + b^y'^) (ax+by)^ +

2{a^x^ + bhj^).

Formulas [2} and [3] . — Examples.

1. (l-2x + 3x3;2 = l_4x+6x2

+ 4x3-12x3

+9x*



= l-4x+l(ix - 12x3 +9x4.

2. (ah + bc + ra)^=a^h^-\-2ab^e + 2a^bc-{-b^c^-{-^abc^ + c^a^ =
a2b^+b^c^+c^a^+2abc{a + b + c).

3. \{x + y)^+x^ + y^}^^{x+ij)^ + 2(xi-y)^{x2+l/^)-\-x^ + 2x^

y2^^4 = (a. + j^)4 + (^|.y)3|(a;-H^)2 ^ (x-y)^} t X-^ + 2x^7/3 + z/4
= 2,X + ^)4+(x2 - 7/'3, 3 +^4+2x2^/3 +yi = ^{ix+y)^ + X* + 7/4}.

4. (x3+X?/ + /y3j3=x*+i^.X-'^ + 2x2^/3 +x3^2 -f 2x7/3 + ^/^ =
(x+7/)2x2+a;'//3+7/2(x + 7/)3.

5. In Ex. 3, substitute 5 -c for x, f -a forj/, and consequently
b — a for x+7/, then since (b — a)^ = {a — b)^, Ex. 3 gives

|(rt_/,)2 + (6_c)2+(c-a)2}2 = 2{(a-Z>}4 + (5-c)4 + (c-fl)4}.

6. Making the same substitutions in Ex. 4, we have
(a2+63^c2 -ah-bc -caY = {a —b)^{b - c)2 + (6-c)2(c- a)2 +
(c — a)2(a — fi)2, or, multiplying both sides by 4,

{(a_6)3^(5_c)3 + (c-a)3}2=4(rt_6)2(J_c)2 + 4(fe_c)2 X

(c-a)2+4(c-rt)2(a-6)2. and .-. from Ex. 5, (a-i)4 + (^6-o)4 +
(c-a)4 = 2(«-6)2(i>-c)2+2(fe-c)8(c-a)2 + 2(c-a)3>(a-6)3.



14 FUNDAMENTAL PORMULAS.

Exercise vi.

1. (l-2.'c+3a;3 -4x3)3, {l-x-tx^-x^)».

2. {l~2x+2x^-3x^-x^)^, (l + 3a;+3a;3+a;3)2.

3. (2a-6-c2_l)2, (l_;^ + y + 2)3, (la;- 1?/ + 03) 2.

4. (:c3-x2^+a;^3_^3)3, („a;+te2+cx3 + (/a;4)3.

o. Shew that (a^ ^b'^ +c^) (.c^ i- y2 ^z2 ^ _ (^^x + bij + •z)^ =
{a/j - 6a;) 3 + (ex - flz) 3 + (65 - cy ) 3 .

6. Prove that {a + b)x + (6 + c)y + (c + ^/)3 multiplied by {a — b)x
-^[b — c)y + (c — a)z, is equal to the difference of the squares of
two trinomials. .

7. Shew that (a-b) (a-c) + {b-c) (b-a) + {c-a) (c-b) -
i-{(a-6)2 + (6-c)34-(c-a)3}=0.

8. Simplify {a-(6-c)}3 +{6-(c- a)}» + {c-(a-6)}».

9. Shew that {a^+b^ -x^- )^ +{ai + bl-x^)^ + 2{aa^ +bb,y

^(,,2 ^ ,t2_a;3)2+(t2_l_i,2_^2)2+2(a6 + ai6i)3.

10. ^rovethsii{{a-b){b-c) + {b-c){c-a)-]-{c-a){a-b)}^ =
(a-bY(b-GY + {b-cY (c-a)3-h(c-rt)3 (a-6)2.

1 1 . Square 2(« — \bx —^cx-\-2dx.

12. If a; + ?/ + 2 = 0, shew that x^ + //* + 2* = (^2 -7/2)3 +

(^3_22)3+(22_^2)2.

13. Provethata3(6 + c)2-}.//2(c + rt)3-fc3(rt + 6)3 + 2a6c(a + 6 + c)
= 2(rt6 + 6c+crt)2.

Art. V. To apply formula [4] to obtain the product of two
factors which differ only in the signs of some of their terms : —
group together all the terms whose signs are the same in one fac-
tor as they are in the other, and then form into a second group
all the other terms.

Examples.

1. Multiply a + 6 — c-f t^ by a-6 — c — d; here the first group is
a — c, the second b-\-Ll\ :. we have

{(a -c) + (6 + <0} {{a-c)-{b+d)}={a-cy-{b-\-d)^.



PUNDAMENTAIi FORMULAS. 15

2. (1 J- Sx-^Sx^ + x^) (1 - nx + Bx^ - x^.) = {(1.+ ?>x"} +
{Sx-^x-)} {(1+3x3) - ('3a;+«3)}={l+3a;3)2 - (3a;+a-)3 = 1-
3x^ +8x4 -a;«.

3. Find the continued product of a +/'+c. h+c — a, c+n —b and
n + h — c.

The first pair of factors gives {{b + c)-\-a} {{h-\-c) — <-i\ ={6+c)^
-o8 = 62+26c+c2-«2.

The second pair gives {a — {h — c)] {a + {h — <:)] =a^ — h^-\-1hc
— c2 ; the only term whose sign is the same in both these results
is 26c ; hence, grouping the other terms, we have

{26c + (/y2+22_a2)} |2k-(63+c3 - a^)] =
(263)2 -(63+c2-a2)3 = 2a262 + 2^2,•3+2c3»2 _„4_ft4_c4.

4. Prove (^i.^-^ah^h)^ -aU^ = {a'^+ahY + {ah + b^)^ .

The expression ^{>t^-\-h^) {n^ + 2ah+h^) = {a'^ +b^) (^+6)2 =
a2(a+6)2+63(« + 6)2 = (a2_f.a5)2_}-(a6+63)2.

Exercise vii.

1. (a2+2a6 + 62) („2_2a6+62).

2. {:L;c^-xy+y^){hx^-\-y^+xy).


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