ALSEBRA. 



gether, as in common arithmetic; the 

 sign and the literal product being deter- 

 mined by the preceding rules. 



Thus,'3x56= 15ab; because 3xc>X 

 x5Xx=15a,- 4 a:X 11 y 

 a-y ; 91) X 5c= +456c / 

 m= 24 m d. 



The powers of the same quantity are 

 multiplied together by adding the indices; 

 thus a^X^ 3 =rt 5 : for aa X <* tt = acmaa. 

 in the same manner, a"> X a" =+; 

 anc l _ 3 a - x^xS* x y== 15^3 ^ y . 



If the multiplier or multiplicand con- 

 sist of several terms, each term of the lat- 

 ter must be multiplied by every term of 

 the former, and the sum ot all the pro- 

 ducts taken, for the whole product of the 

 two quantities. 



Ex. 1. Mult, 



by c+d 



r Ans. a c-4-6 c+x c+<* d+b d+x d 



Here a +- -f x is to be added to itself 

 c-\-d times, i. e. c times and d times. 



Ex. 2. Mult, a 4- b x 



by c d ___ 

 Ans. a 



cx ca d b d-\-x d 



Here n-j-6 is to be taken c d times, 

 that is, c times wanting d times; or c times 

 positively and d times negatively. 



Ex. 3. Mult, a+b 



Ans. q3+2 a b+b* 



Ex. 4. Mult. or+# 

 by a: y 



x y y~ 



Ans. x 1 * a 1 



Ex. 5. Mult. 3 a2_ 5 b d 

 by 5 a*-f 4 b d 

 15 aM-25 a 1 b d 



Ans. 15 flH-37 a* 6 d 20 A* rf* 



Ex. 6. Mult, a^ 2 a 

 by a 1 2 a 



Ex. r. Mult. 1 x+x* 

 by 1 



x * 4. 3-3 



Ans. 



Ex. 8. Mult, .r 1 p x -j- q 

 by x + a 



-}- a x 1 ap x-\-a q __ 

 Ans. jrtp ax*-\-qap.x+att 



Here the co-efficients of x and x are 

 collected ; p a . x 1 = p 

 and q a p . 07=5 a- a /> x. 



a a >: 



Ans. a* * 



To divide one quantity by another, is to de- 

 termine how often the latter is contained in tJic 

 former, or ivhat quantity multiplied by tht 

 latter tvitt produce the former. 



Thus, to divide a b by a is to determine 

 how often a must be taken to make np 

 a b,- that is, what quantity multiplied by a 

 will give a b ,- which we know is b. From 

 this consideration are derived all the rules 

 for the division of algebraical quantities. 



If the divisor and dividend be affected 

 with like signs, the sign of the quotient is 

 -f : but if their signs be unlike, the sign 

 of the quotient is . 



If a b be divided by a, the quo- 

 tient is + b ; because a X -f- b gives 

 a b , and a similar proof may be given 

 in the other cases. 



In the division of simple quantities, if 

 the co-efficient and literal product of the 

 divisor be found in the dividend, the other 

 part of the dividend, with the sign deter- 

 mined by the last rule, is the quotient. 



Thus, f- =c; because a b multipli 

 a b 



ed by c gives a b c. 

 If we first divide by a, and then by 



b, the result will be the same; for = 



c, and =c, as before. 



a 



Hence, any power of a quantity is divi- 

 ded by any other power of the same quan- 

 tity, by subtracting the index of the divi- 

 sor from the index of the dividend. 



a$ <j5 1 a m 



* 6f3 fj3 @3 a^ 



If only a part of the product which 

 forms the divisor be contained in the divi- 



