College Algebra & Trigonometry, 2018a

1.6. MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 57
Multiplying and Dividing Rational Expressions
In multiplying and dividing rational expressions, it is often easier to identify and
cancel out common factors before multiplying rather than afterwards. Multiplying
rational expressions works the same way that multiplying numerical fractions
does - multiply straight across the top and straight across the bottom. As a
result, any factor in either numerator of the problem will end up in the numerator
of the answer. Likewise, any factor in either denominator of the problem will
end up in the denominator of the answer. Thus, any factor in either numerator
can be cancelled with any factor in either denominator.
Example
Multiply. Express your answer in simplest form.
x 2 +5x +6
∗ x2 − 2x − 15
25 − x 2 x 2 +6x +9
x 2 +5x +6
∗ x2 − 2x − 15
25 − x 2 x 2 +6x +9
(x +2)(x +3) (x − 5)(x +3)
= ∗
(5 + x)(5 − x) (x +3)(x +3)
= (x +2) ✘ ✘✘ ✘
(x +3)
(5 + x) ✘ ✘ ✘ ✘
(5 − x)(−1) ∗ ✘ ✘ ✘ ✘
(x − 5) ✘ ✘ ✘ ✘
(x +3)
✘(x ✘ +3) ✘✘ ✘ (x ✘✘✘
+3)
= − x +2
x +5
Dividing rational expressions works in much the same way that dividing numerical
fractions does. We multiply by the reciprocal. There are several ways to
demonstrate that this is a valid definition for dividing. First, it is important to
understand that the fraction bar is the same as a “divided by” symbol:
8
2
=8÷ 2=4.