ALGEBRA. 



tlend, the quantities contained both in the 

 divisor and dividend must be expunged. 

 Thus, 15 a3 b 1 c divided by 3 a a b x, 

 15 q3 &* c 5 a be. 



r Sa^tT 3 ~~# 



First, divide by 3 a 1 6, and the quo- 

 tient is 5 a b c ,- this quantity is still to 

 be divided by y t and as y is not contained 

 in it, the division can only be represented 



in the usual way ; that is, -- is the 



quotient. 



If the dividend consists of several 

 terms, and the divisor be a simple quan- 

 tity, every term of the dividend must be 

 divided by it. 



a3 x l 5 a b x3 4- 6 a x* 

 Thus, --- -- =r a"' 

 ax 1 



5 b ar-f 6 * J . 



When the divisor also consists of seve- 

 ral terms, arrange both the divisor and 

 dividend according to the powers of some 

 one letter contained in them ; then find 

 how often the first term of the divisor is 

 contained in the firstterm of the dividend, 

 and write down this quantity for the first 

 term in the quotient ; multiply the whole 

 divisor by it, subtract the product from 

 the dividend, and bring down to the re- 

 mainder as many other terms of the divi- 

 dend as the case may require, and repeat 

 the operation till all the terms are brought 

 down. 



Ex. 1. If a 2 2 a b+b* be divided by 

 ^ b, the operation will be as follows : 

 a _ )a z _ 2 a b+b*(a b 



Ex. 2. a+b*)a c-f-a d+bc+b a(c+d 

 a c-\-b c 



'a'd+bd 

 a d+b d 



Ex. 3. 



Remainder 



1 jc 



Ex. 4. -lr/3 



Ex. 5. 



x ax> 



pa-\-qxl ax 1 

 a p.r 2 -\-qx 

 a p.x* u4 pa.x 



-^pa+q-xr 



r+a>- 



Remainder a3 pa%-\-qa r 



a 

 ab+b* 



THE THAXSFOIUrATION OF FRACTIONS 

 ' TO OTHERS OF F.a^AL VALUE. 



The reason of this, and the foregoing 

 rule, is, that as the whole dividend is made 

 up of all its parts, the divisor is contained 

 in the whole, as often as it is contained in 

 all the parts. In the preceding operation 

 we inquire, first, how often a is contained 

 in a 1 , which gives a for the first term of 

 the quotient, then multiplying the whole 

 divisor by it, we have a 1 a b to be sub- 

 tracted from the dividend, and the re- 

 mainder is a i-j-fi 1 , with which we are 

 to proceed as before. 



The whole quantity a 1 2 a b+b 1 is in 

 reality divided into two parts by the pro- 

 cess, each of which is divided by a b ; 

 therefore the true quotient is obtained. 



If the signs of all the terms both in the 

 numerator and denominator of a fraction 

 be changed, its value will not be altered. 

 For 



a b -\-ab , ab 



= -4- b = ; and. = b =* 



a -f- a 



a b. 



a 



If the numerator and denominator of a 

 fraction be both multiplied, or both divi- 

 ded, by the same quantity, its value is 

 not altered. For 



ac a _. 



' 



a .1 y z y 

 a b c z b c 



be b 



Hence, a fraction is reduced to its low- 

 est terms, by dividing both the numera- 



