Multiply and Divide Rational Expressions
Multiplying rational expressions is very similar to the process we use to
multiply fractions. It is essentially a big fraction which is reduced
(simplified) by dividing out the common factors. Dividing fractions is
accomplished by multiplying by the reciprocal of the second fraction.
Example 1: Multiply:
25x2 ∙ 24y4
9y8 ∙ 55x7
=
25x2
9y8
∙
24y4
55x7
25 · x2 ∙ 24 · y4
5 · x2 ∙ 8 · y4
3 · y8 ∙ 11 · x7
9 · y8 ∙ 55 · x7
Reduce the coefficients by dividing out common factors 3 and 5
5 · x2−7 ∙ 8 · y4−8
3 ∙ 11
Use the quotient rule for exponents
5 · x−5 ∙ 8 · y−4
3 ∙ 11
5∙8
3 ∙ 11 · x5 · y4
Move the negative exponent terms to the denominator
The negative exponents becomes positive
40
33x5 y4
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
x2 −9
Example 2: Divide:
x2 −9
x2 +x−20
·
x2 +x−20
x2 −8x+16
3x+9
÷
3x+9
x2 −8x+16
Multiply by the reciprocal of the second fraction
(x+3)(x−3) · (x−4)(x−4)
Factor the polynomials
(x−4)(x+5) · 3(x+3)
(x−3) · (x−4)
(x+5) · 3
(x−3)(x−4)
3(x+5)
Example 3: Multiply:
Divide out common factors (x + 3) and (x – 4)
Which can be multiplied and written as
3x2 +10x+8
x2 −3x−10
∙
(3x+4)(x+2) · (2x−1)(x−5)
(x−5)(x+2) · (2x−1)(x+3)
(3x+4) · 1
1 · (x+3)
x2 − 7x+12
3x+5
2x2 −11x+5
2x2 +5𝑥−3
Factor the polynomials
Divide out common factors (x + 2), (x – 5), and (2x – 1)
3x+4
x+3
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)