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Ex. 6. -55 tas-b -5s? Tasb 2559-5as8 +565 –5a5 ta's'-abs 565—abs

+62 25s*— 10as:—(106+a+)s2abs +6?

Ex. 7.

Ex. 8. 2-3ax+4.x2

5x+4ax:+3a*x+as 5a2_6ax-2x2

2x - 3axta 10a15ax+20a%2*

102°—Bax' +6a+r+2a*z* -120*x+18a*x?_24ax. -15ax4—12a*x+9a%22%3a*x

-4aRx: +6ax:_8x4 +5aRx8+4aRx++3a*rtas 10a —27a*x+34dor—18ax'_844 10x— Taxt—oʻr—3a*r* * ter Ex. 9.

Ex. 10. 2a2–3ax+4x2

5x +4ax+3aʻxtas ба? бах2°

2x-3axta 10a4—15aRx+20a

10x5 +8ax* +6a*x® +2a®z -12ax+180 x_24.az

-15ax*—-12aRx_9am_3a*x -4a'r +6ax_824

-5a2 +42*x+3a*x+as 100_27ar+34a*x?_18ar-8c* 10x6_7ax-ax343aom * tas

xty

Ex. 11.

Ex. 12. 32°—2x+5

x?ty 6x—7 18x_12x +302 co Frog

-21x7140—35 +xy+y? 1828–33x+44x—35 2+2x+y+yo

Ex. 13.
x2+xyty
y-ryty
x2 + xy + xy2
—x*y—~*uj248

tay?+xy+y*
x2 * txy

+y*

2

Ex. 14.

Ex. 15. 2-ax+bx-C

x2+xyty 2-date

X-Y -ax+6x3cx

28 ta'y+xy -dx +adx_bdx*+cdx tex--aex'+bex-ce

203 * *. y -a+d)x+6+ad+e)x_(c+od+ae)x++(cd be)x-ce

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Ex. 16. 114_^{1962—(22+24)?}or, 114^(1962*—*.*_4824-576) or mult. 114-» (148x*—**_576) by 114-^ (148x-2-576) 1142_1147 (148x576)

-114M (148x?_x4_576)+(148x21576) Prod. 12996–228/(1482_x4_576)+(148x*—3—576)

Ex. 17. 3x8+2x+y+3y 223–3x*y?+54 628 +4xRj+62%y% -9x®y? -6x*y_9xy

+15x®y®+:10x+y+15y® 62825x®y-6x*y* +21x*y*+xy+15yo

Ex. 18.
af+a+c+c*
a? -c2
ao+a*c*tac

a c4_6
as

а

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CX

ac

Note. When the quantities that are to be multiplied together have literal co- axtb.

a-bx efficients, pro

-d

c- dx ceed as before, acx++-bcx

-bcx putting the adx-bd

-adx+bd.x2 sum, or difference of the co

aca? +(bc-ad)24bd acm(bc+ad)x+bda efficients of the resulting terms, between brackets, as in the former rule. And if several compound quantities are to be multiplied together, multiply the first by the second, and then that product by the third, and so on to the last factor, as below. a+2x

За —x: -3x

2a +4.0 a*+2ax

6a_2ax -Зах ба?

+12ax_4.x2 -6x2

64*+10ax-4.22 a+46

40-2x as-ax-bax?

24a'+40a x-16ax* +4a'r-ax-24.*

-12aʻx—20az+848 a+3ax-10ax?_242

24a3+28a*2—36ax?_822

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3

To this we may add, that it is usual, in some cases, to write down the quantities that are to be multiplied together, between brackets, or under a vinculum, without performing the whole operation ; as 3ab(a+b)Xan (a?)

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Ex. 19. 28-> (202–10x) multiply add -2352 (203_10x) 28V (202-10x) by

and +(10x—20x®)» (20x?—10x)
(—10x sum +(102—20x*2352) (20x7-10x)

(
784—56(202?–10x)+202—10x product. Multiply

28% (202?–10x) again by this.
21952—15681 (20x410x)+5602*_280x

-7847 (20x2—10x)+56(202?_10x)—(20x+10x) (20x?_10x)
21952–23527 (20x2—10x)+1680x?_840.6+(10x—20x®) (2022—10x)

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5531

Ex. 20.
2352 +20x2—10x multiply
2352+20x2—10x by
5531904+47040x2_23520x
+47040x*-+-400.c -200.x®

--23520x—200x8+100.22
5531904—47040x+94180x2—400x+400x* sum.
Multiply 20x2–10c again.
110638080x?_940800x+1883600x4—8000x®+8000x®

455319040x-+-4704002-941800x+4000x4—40002 111108480x—55319040x—1882600x3+1887600x4—12000x3 +80002

8107-21952 by

Ex. 23.
72–16762° +840x21952 multiply
7003_1676
49.26 —117322 +5880x153664.ch
-11732x+280897624-140784028 +367915522*

+-5880d -1407840m+7056002–18439680.0

-153664x2+367915522?–18439680x+481890304 49.x23464.25 +2820736x31230082+7428870422-36879360x +481890304

Multiply ex+ma* +ndotrze

by ax+bx+ca+dx*
aeg tamtanza taras

bext +-bmx' +_bnw +brzo
+órten +dmz+(cr+dn)x+ +druk
fcex' +cmz'tcnct.crx"

Ex. 24.
dex+dmzo+dnx"+drzo

2

1+2

I
m

L
n

mtn

2*+1.

a

NOTE. The products of the powers of the same quantity are found by adding
their indices. Thus : a'xa'=a?, or a Xa=al+la?; a'xaal+=a; a*Xa=

*+*_12
; a" Xa'zanth; ao xaza

; a"xa"=am n; 2* X X=
Obs. 1. In multiplication, as well as in addition and subtraction, the order of
the letters is of no consequence. Thus, if abc be multiplied by d, the product is
abcd, bacd, cbda, &c., each of which is of the same value ; but it is usual to ar-
range them according to the order of the alphabet.

Obs. 2. In algebra it is customary to begin the multiplication on the left; but
because the steps are merely indicated, it is of no consequence where the opera-
tion commences,

3
Obs. 3. It may be useful to observe, that, according to Euclid, Lib. II. Prop.
V., the product of the sum and difference of any two quantities is equal to the
difference of their squares; thus, (1.) (a+b)(ab)=29-6?

’?? -Ba--
(2.) (a + b)(a)=a*_=(a + b)(a +b)(amb).
(3.) (a+b)(a —b)=a*—6=(a + b)(a + b) (

abo)=(a*+B)(d®+)(a+b)(-6). These compositions and decompositions of quantities are often found to be of great utility in the solution of equations.

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DIVISION.

41. Division in Algebra is the method of finding the quotient arising from the division of one algebraic quantity by another. Division is generally divided into three cases, namely, when the divisor and dividend are both simple quantities ; when the divisor is a simple quantity and the dividend a compound one; and when the divisor and dividend are both compound quantities.

Case I. When the divisor and dividend are both simple terms.

RULE. Place the divisor in the form of a denominator under the dividend ; cancel those letters which are common to both, and divide the coefficients by any number that will divide them without a remainder, and the result will be the quotient required.

Rule II. Divide the coefficients as in common arithmetic, and to the quotient annex those letters in the dividend which are not found in the divisor.

A general rule for the signs in all the cases of division : When the signs of the divisor and dividend are alike, (that is, both + or both — , the sign of the quotient will be + When they are unlike, (that is, the one + and the other — ,) the sign

. of the quotient will be — The above rule briefly expressed in one view, is as follows: Div'r. Div'd. Quo't. Div't. Div'd. Quo't. +

} + { 1. Plus

Both minus. 2

+ + } } Minus Minus Plus

AGAIN, THUS: tab

-ab

-ab +b;

tab b;

-b; ta

ta and these four are all the cases that can possibly happen with regard to the variation of the signs.

Powers and roots of the same quantity are divided by subtracting their idices, that is, subtract the index of the divisor from the index of the dividend.

a?
a-1

a’xas Thus,

a? a

+

plus { Plus 3 }

{

+ b;

a

a

-1

ca', or a;

=

a

-5

25-3;
at
at

= -2,

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