ALGEBRA. 
give a b ; which we know is b. From this con- 
sideration are derived all the rules for the di- 
vision of algebraical quantities. 
If the divisor and dividend be affected with 
like signs, the sign of the quotient is — j— : but 
if their signs be unlike, the sign of the quo- 
tient is — . 
If — a b be divided by — a, the quotient is 
-f- 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 
he found in the dividend, the other part of the 
dividend, with the sign determined by the 
last rule, is the quotient. 
Thus, because n b multiplied by 
e gives a be. 
If we first divide by a, and then by b, the 
result will be the same ; for an( j Af 
’ a ’ 0 
~c, as before. 
Hence, any power of a quantity is divided 
by any other power of the same quantity, by 
subtracting the index of the divisor from the 
index of the dividend. 
rru a ” 2 a " 1 3 a m 
Thus, - = ft 2 — ~ a™-*, 
a' , a a a “ f 
If only a part of the product which forms 
the divisor be contained in the dividend, the 
quantities contained both in the divisor and 
dividend must be expunged. 
Thus, 15a 3 6 z c divided by — 3 a 2 b x, or 
15a 2 b 2 c — babe 
— 3 a 2 by ~~ y 
First, divide by — 3 a 2 b, and the quotient 
is — babe-, this quantity is still to be divided 
by y, and as y is not contained in it, the divi- 
sion can only be represented in the usual way ; 
, , . — babe. , 
that is, - — is the quotient. 
If the dividend consist of several terms, 
and the divisor be a simple quantity, every 
term of the dividend must be divided by it. 
a 3 x 2 — 5 a b .r 3 -}- 6 a r 4 
Thus, — ' = a 1 — 
a x‘ 
5 b x 6 
When the divisor also consists of several 
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 first 
term 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 remainder as many other terms of the di- 
vidend as the case may require, and repeat 
the operation till all the terms are brought 
down. 
Ex. 1. If a 1 — - 2 a b -f- b 2 be divided by 
a — b, the operation will be as follows : 
a — b) a 1 — 2 a b -}- 6 2 (a — b 
a 2 — a b 
— a b -)- b 2 
— ab + b 2 
* 
The rea on 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 2 , which 
gives a for the first term of the quotient, then 
multiplying the whole divisor by it, we have 
a 2 — a b to be subtracted from the dividend, 
and the remainder is — ab-\-b 2 , with which 
we are to proceed as before. 
The whole quantity a 2 — 2 a b -j- b 2 is in 
reality divided into two parts by the process, 
each of which is divided by a — b; therefore 
the true quotient is obtained. 
Ex. 2. a h) a c -|- a d -j- b c -j- b d (c d 
ac -\-b c 
ad-lfbd 
ad^bd 
* # 
Ex. 3. 
l—r) 1 ( 1 Remainder 
1 — r 1 — 
-j- X .T 2 
-j- X 2 
+ x 2 — x l 
+ .r 3 
-(- x z — X 4 
-\-x 4 &c. 
Ex. 4. y-l)y 3 — + l 
y 3 — y 2 
+r 
y 2 -y 
+y - 1 
v — i 
