Loading...
Multiplying and Dividing Rational Expressions
A
B
C
D
E
. . . .

Division of Rational Expressions

Example 3

Divide and then simplify if possible.

a. ( 4x+3y )÷ 12 x + 9 y 2 x
b. ( x − 2 ) 2 x + 4 ÷ x 2 − x − 2 x 2 − 16


Solution
a. ( 4x+3y )÷ 12 x + 9 y 2 x
= 4 x + 3 y 1 ⋅ 2 x 12 x + 9 y
= ( 4 x + 3 y ) ( 2 x ) 12 x + 9 y

Factor 12x+9y in the denominator. Then cancel common factors.

( 4 x + 3 y ) ( 2 x ) 12 x + 9 y = ( 4 x + 3 y ) ( 2 x ) 3 ( 4 x + 3 y )
= 2 x 3
Thus, ( 4 x+3y )÷ 12 x + 9 y 2 x = 2 x 3 .

b. ( x − 2 ) 2 x + 4 ÷ x 2 − x − 2 x 2 − 16
= ( x − 2 ) 2 x + 4 ⋅ x 2 − 16 x 2 − x − 2
= ( x − 2 ) 2 ( x 2 − 16 ) ( x + 4 ) ( x 2 − x − 2 )

Factor the expressions in the numerator and the denominator. Then cancel common factors.

( x − 2 ) 2 ( x 2 − 16 ) ( x + 4 ) ( x 2 − x − 2 )
= ( x − 2 ) ( x − 2 ) ( x + 4 ) ( x − 4 ) ( x − 2 ) ( x + 4 ) ( x + 1 )
= ( x − 2 ) ( x − 4 ) x + 1
Thus, ( x − 2 ) 2 x + 4 ÷ x 2 − x − 2 x 2 − 16
= ( x − 2 ) ( x − 4 ) x + 1
or x 2 − 6 x + 8 x + 1 .