2ab+b3, Hence the result is

5a3+10a b+5ab2 a3+2ab+2ab2+b3

; and

the greatest common measure of this result is a+b, which is

found thus ;

a3+2ab+2ab2+b3)5a3+10a2b+5ab2 (5

5a3+10a2b+10ab2+563

a3+2ab+2ab2+b3

and

found to divide the numerator without a remainder; and consequently dividing both the numerator and denominator of the 5a3+10a2 b+5ab2

fraction

in its lowest terms ; that is,

a3+2a2b+2ab+b3
a+b

by a+b, we have the fraction

To reduce fractions to other equivalent ones, that shall have a common denominator.

RULE I.

151. Multiply each of the numerators separately, into all the denominators, except its own, for the new numerators, and all the denominators together for the common denominator.

It is necessary to remark, that, if there are whole or mixed quantities, they must be reduced to improper fractions, and then proceed according to the rule.

4 X c Xa=4ac common denominator;

Hence the fractions required are

2x+1 Ex. 2. Reduce 3b (2x+1) X x=2x2+x (

2a2

3a2c 20ab
4ac 4ac

and to a common denominator.

x

2a2 X 3b=6a2b new numerators;

3b Xx=3bx common denominator; Hence the fractions required are

2x2+x

6a2b

and
,
3bx

3bx

152. Find the least common multiple of all the denominators of the given fractions, (Art. 147), and it will be the common denominator required.

Divide the common denominator by the denominator of each fraction, separately, and multiply the quotient by the respective numerators, and the products will be the numerators of the fractions required.

Ex. 4. Reduce- and

3a2b
x2

5ab 4αx2

to the least common de

nominator.

Here 4ax2 is the least common multiple of x2 and 4ax2;

Or, as 4ax (the least common multiple) is the denominator of one of the fractions, it is only necessary to reduce the

3a2b
X2

fraction to an equivalent one, whose denominator shall

These rules appear evident from (Art. 118). For, let

α C e

adf cbf edb be the proposed fractions; then

bdf' bdf' bdf

are

b' d'ƒ' fractions of the same value with the former, having the com

mon denominator bdf. Since

adf

bdf

b'bdf d'

a cbf__c edb e ;; and = bdf f

3a2b Y

5x2

Ex. 5. Reduce

and

" 4cx2' 2x'

to the least common

8ac2

8ac2x2

denominator.

Here, the least common multiple of 4cx2, 2x, and 8ac2; (Art. 147), is Bac2x2; then,

X3c2b2ac X3a2b=6a2bc

4cx2

8ac2x2

-Xy=4 ac2 x Xy=4ac2xy

new numerators;

2x

§ V. ADDITION AND SUBTRACTION OF ALGEBRAIC Fractions. To add fractional quantities together.

RULE.

153. Reduce the fractions, if necessary, to a common denominator, by the rules in the last case, then add all the numera