12.5: Multiplication of Rational Expressions
Suppose you were playing a game on your cell phone in which you were randomly given two rational expressions and were asked to identify the product of the two expressions. If one of the expressions were \begin{align*} \frac {x^2+3x+2}{x9}\end{align*}
Guidance
Because a rational expression is really a fraction, two (or more) rational expressions can be combined through multiplication and/or division in the same manner as numerical fractions. A reminder of how to multiply fractions is below.
For any rational expressions \begin{align*}a \neq 0, b \neq 0, c \neq 0, d \neq 0\end{align*}
\begin{align*}\frac{a}{b} \cdot \frac{c}{d}= \frac{ac}{bd}\\
\frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}\end{align*}
Example A
Multiply the following: \begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}
Solution:
\begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2} \rightarrow \frac{4ab^3}{80a^2 b^8}\end{align*}
Simplify exponents using methods learned in previous Concepts.
\begin{align*}\frac{4ab^3}{80a^2 b^8}=\frac{1}{20ab^5}\end{align*}
Example B
Simplify \begin{align*}9c^2 \cdot \frac{4y^2}{21c^4}\end{align*}
Solution:
\begin{align*}9c^2 \cdot \frac{4y^2}{21c^4} \rightarrow \frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}\\
\frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}=\frac{36c^2 y^2}{21c^4}\\
\frac{36c^2 y^2}{21c^4}=\frac{12y^2}{7c^2}\end{align*}
Multiplying Rational Expressions Involving Polynomials
When rational expressions become complex, it is usually easier to factor and reduce them before attempting to multiply the expressions.
Example C
Multiply \begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^29}\end{align*}
Solution:
Factor all pieces of these rational expressions and reduce before multiplying.
\begin{align*}\frac{4x+12}{3x^2} \cdot & \frac{x}{x^29} \rightarrow \frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x3)}\\
& \frac{4\cancel{(x+3)}}{3x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x3)}\\
& \frac{4}{3x} \cdot \frac{1}{x3} \rightarrow \frac{4}{3x^29x}\end{align*}
Video Review
Guided Practice
Multiply \begin{align*}\frac{12x^2x6}{x^21} \cdot \frac{x^2+7x+6}{4x^227x+18}\end{align*}
Solution: Factor all pieces, reduce, and then multiply.
\begin{align*}\frac{12x^2x6}{x^21} \cdot \frac{x^2+7x+6}{4x^227x+18} & \rightarrow \frac{(3x+2)(4x3)}{(x+1)(x1)} \cdot \frac{(x+1)(x+6)}{(4x3)(x6)}\\
\frac{(3x+2)\cancel{(4x3)}}{\cancel{(x+1)}(x1)} \cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x3)}(x6)} & \rightarrow \frac{(3x+2)(x+6)}{(x1)(x6)}\\
\frac{12x^2x6}{x^21} \cdot \frac{x^2+7x+6}{4x^227x+18} &= \frac{3x^2+20x+12}{x^27x+6}\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Multiplying and Dividing Rational Expressions (9:19)
In 1–10, perform the indicated operation and reduce the answer to lowest terms

\begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
x32y3⋅2y2x 
\begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
2xy2⋅4y5x 
\begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
2xy⋅2y2x3 
\begin{align*}\frac{4y^21}{y^29} \cdot \frac{y3}{2y1}\end{align*}
4y2−1y2−9⋅y−32y−1 
\begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
6aba2⋅a3b3b2 
\begin{align*}\frac{33a^2}{5} \cdot \frac{20}{11a^3}\end{align*}
33a2−5⋅2011a3 
\begin{align*}\frac{2x^2+2x24}{x^2+3x} \cdot \frac{x^2+x6}{x+4}\end{align*}
2x2+2x−24x2+3x⋅x2+x−6x+4 
\begin{align*}\frac{x}{x5} \cdot \frac{x^28x+15}{x^23x}\end{align*}
xx−5⋅x2−8x+15x2−3x 
\begin{align*}\frac{5x^2+16x+3}{36x^225} \cdot (6x^2+5x)\end{align*}
5x2+16x+336x2−25⋅(6x2+5x) 
\begin{align*}\frac{x^2+7x+10}{x^29} \cdot \frac{x^23x}{3x^2+4x4}\end{align*}
x2+7x+10x2−9⋅x2−3x3x2+4x−4
Mixed Review
 The time it takes to reach a destination varies inversely as the speed in which you travel. It takes 3.6 hours to reach your destination traveling 65 miles per hour. How long would it take to reach your destination traveling 78 miles per hour?
 A local nursery makes two types of fall arrangements. One arrangement uses eight mums and five blackeyed susans. The other arrangement uses six mums and 9 blackeyed susans. The nursery can use no more than 144 mums and 135 blackeyed susans. The first arrangement sells for $49.99 and the second arrangement sells for 38.95. How many of each type should be sold to maximize revenue?
 Solve for \begin{align*}r\end{align*}
r and graph the solution on a number line: \begin{align*}24 \ge 2r+3\end{align*}−24≥2r+3 .  What is true of any line parallel to \begin{align*}5x+9y=36\end{align*}
5x+9y=−36 ?  Solve for \begin{align*}d: 3+5d=d(3x3)\end{align*}
d:3+5d=−d−(3x−3) .  Graph and determine the domain and range: \begin{align*}y9=x^25x\end{align*}
y−9=−x2−5x .  Rewrite in vertex form by completing the square. Identify the vertex: \begin{align*}y^216y+3=4\end{align*}
y2−16y+3=4 .
Image Attributions
Here you'll learn how to find the product of two rational expressions.