ALGEBRA. 



gether, as in common arithmetic; the sign 

 and the literal product being determined 

 by the preceding rules. 



Thus, Sax 5 b= I5ab ; because 3 XaX 

 5x6=3x5xax=15a6; 4a- X lly 

 = 44-ry; 9x 5c=-f45Ac ; 

 6d x 4m= 24 met. 



The powers' of the same quantity are 

 multiplied together by adding the indices; 

 thus o> x a> =a* ; for aa X aaa = aaaaa. 

 In the same manner, a m X a n =a m ' f '* ; 

 and So 1 .r3y5a a- y l = 153 a* 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 of all the pro- 

 ducts taken, for the whole product of the 

 two quantities. 



Ex. 1. 



Ans. a c-\-b c-\-or c-\-a d-\-b d-\-x d 



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

 c+d times, i. e. c times and d times. 



Ex. 2. Mult, a + b x 

 by c d 



Ans. a c-\-bc x c a d b d-\-x d 



Here a-f b 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+6 

 by a+b 



a b+b* 



Ans. 



Ex. 4. Mult, x-f y 

 by .r- y 



Ans. a- 1 * 



Ex. 5. Mult. 3 a 1 5 b d 

 by 5 a-f 4 6 <f 



15 a-H-25 a 1 b d 

 -\-V2a 1 bd- 



20 A d 1 



Ans. 15 aM-37 a* A </ 20 A* 



Ex. 6. Mult a+2 a 

 by a 1 



Ex. 7. Mult. 1 x-^-x- 

 by 1 -f x 



+ 



Ans. 1 



Ex. 8. Mult. x /> 

 by x + a 



px*-\-qx 



-\- ax 1 a p x-\-a g 



Ans. x? p 



Ans. a* 



Here the co-efficients of x 1 and x are 

 collected ; p a . x 1 = p x 1 a x? 

 and q a p. x=q x apx. 



To divide one quantity by another, is to de- 

 termine hone of ten the latter is contained in the 

 former, or -what quantity multiplied by the 

 latter will produce the former. 



Thus, to divide a b by a is to determine 

 how often a must be taken to make up 

 a k ; 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 

 + : but if their signs be unlike, the sign 

 of the quotient is . 



If abe divided by a, the quo- 

 tient is -f b ; because a X + * 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, a , C =c ; because a Amultipli- 

 a o 



ed by c gives a b c. 

 If" we first divide by a, and then by 



b, the result will be same ; for = b 



c, and =sc, as before. 



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* a' 1 fl 



Thus, =o 1 ;- = -=a- 3;= aW-*. 



as a' ai a. n 



If only a part of the product which 

 forms the divisor be contained in the divi- 



