Reduce the following fractions to their lowest common denominator.

37.

4

5

(x-1)' (x−1)2' x+1' (x+1)2' x2-1'

39.

1

a2

40.

x2-ax+a2' x2+ax+α2, xa+a2x2+a1°

XVI. Addition or Subtraction of Fractions.

140. Rule for the 'Addition or Subtraction of fractions. Reduce the fractions to a common denominator, then add or subtract the numerators and retain the common denominator.

a-c
b

Here the fractions have already a common denominator, and therefore do not require reducing;

The student is recommended to put down the work at full, as we have done in this example, in order to ensure accuracy.

Here the common denominator will be the product of a+b and a-b, that is a2-b2.

By Art. 107 the L. C. M. of the denominators is

(x − 1) (x − 3) (4x2 — x−6);

4x3-92-15x+18(x-1)(x-3) (4x2-x-6)'

=

4x2-3x+2

4x3-9x2-15x+18

(x+1)(4x2-x-6)-(4x2-3x+2) (x−1)
(x-1) (x −3) (4x2 — x −6)

141: We have sometimes to reduce a mixed quantity

to a fraction; this is a simple case of addition or subtraction of fractions.

_5+12−(−2) _ _5+12−x+2

x3-6x+14

=

x2-3x+4

x2-3x+4

==

x2-3x+4

142. Expressions may occur involving both addition and subtraction. Thus, for example, simplify

The L. C. M. of the denominators is (a2-b2) (a2 + b2), that is a1-b4.

α a (a−b) (a2 + b3) _ aa — a3b+ a2b2 — ab3

=

a2-b2

a2 (a2-b2)
a1-b1

a

=

a+b a2-b2 a2+b2

a1 — a3b + a2b2 — ab3 + a3b + ab3 — (aa — a2b2)

a1 — a3b + a2b3 — ab3 + a3b+ab3 — aa + a2b2

at - b$

2a2b2