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# Multiplication of Rational Expressions

## Multiply and reduce fractions with variables in the denominator

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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}{x-9}\end{align*} and the other expression were \begin{align*} \frac {x^2-10x+9}{x^2-4}\end{align*}, would you be able to multiply them together? Could the product be simplified?

### Multiplying Rational Expressions

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*}

#### Let's multiply the following rational expressions:

1. \begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}.

\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 sections.

\begin{align*}\frac{4ab^3}{80a^2 b^8}=\frac{1}{20ab^5}\end{align*}

1. \begin{align*}9c^2 \cdot \frac{4y^2}{21c^4}\end{align*}

\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*}

When rational expressions become complex, it is usually easier to factor and reduce them before attempting to multiply the expressions.

#### Now, let's multiply  \begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}\end{align*}:

Factor all pieces of these rational expressions and reduce before multiplying.

\begin{align*}\frac{4x+12}{3x^2} \cdot & \frac{x}{x^2-9} \rightarrow \frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\\ & \frac{4\cancel{(x+3)}}{3x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x-3)}\\ & \frac{4}{3x} \cdot \frac{1}{x-3} \rightarrow \frac{4}{3x^2-9x}\end{align*}

### Examples

#### Example 1

Earlier, you were asked about a cell phone game where you had to multiply \begin{align*} \frac {x^2+3x+2}{x-9}\end{align*} and \begin{align*} \frac {x^2-10x+9}{x^2-4}\end{align*}. Can you multiply and simplify these two rational expressions?

Yes, you can multiply these two rational expressions together. First, we will factor all pieces. Then we will reduce and multiply.

\begin{align*}\frac{x^2+3x+2}{x-9} &\cdot \frac{x^2-10x+9}{x^2-4} \\ \frac{(x+2)(x+1)}{(x+9)} &\cdot \frac{(x+9)(x-1)}{(x-2)(x+2)}\\ \frac{(x+1)\cancel{(x+2)}}{\cancel{(x-9)}} &\cdot \frac{\cancel{(x-9}(x-1)}{\cancel{(x+2)}(x-2)} \\ &=\frac{(x+1)(x-1)}{(x-2)}\\ &= \frac{x^2-1}{x-2}\end{align*}

#### Example 2

Multiply \begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}\end{align*}.

Factor all pieces, reduce, and then multiply.

\begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} & \rightarrow \frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\\ \frac{(3x+2)\cancel{(4x-3)}}{\cancel{(x+1)}(x-1)} \cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x-3)}(x-6)} & \rightarrow \frac{(3x+2)(x+6)}{(x-1)(x-6)}\\ \frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} &= \frac{3x^2+20x+12}{x^2-7x+6}\end{align*}

### Review

In 1–10, perform the indicated operation and reduce the answer to lowest terms

1. \begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
2. \begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
3. \begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
4. \begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}
5. \begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
6. \begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}
7. \begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}
8. \begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}
9. \begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}
10. \begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}

Mixed Review

1. 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?
2. Solve for \begin{align*}r\end{align*} and graph the solution on a number line: \begin{align*}-24 \ge |2r+3|\end{align*}.
3. What is true of any line parallel to \begin{align*}5x+9y=-36\end{align*}?
4. Solve for \begin{align*}d: 3+5d=-d-(3x-3)\end{align*}.
5. Graph and determine the domain and range: \begin{align*}y-9=-x^2-5x\end{align*}.

To see the Review answers, open this PDF file and look for section 12.5.

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### Vocabulary Language: English

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.