## Multiplying Rational Expressions

Contents

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

### Multiplication of Rational Expressions

If \(p,q,r,s\) are polynomials where \(q\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}s\ne 0\), then

To multiply rational expressions, multiply the numerators and multiply the denominators.

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

## Example

Multiply: \(\frac{10}{28}·\frac{8}{15}.\)

### Solution

Multiply the numerators and denominators. | |

Look for common factors, and then remove them. | |

Simplify. |

Remember, throughout this tutorial, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x\ne 0\) and \(y\ne 0\).

## Example

Mulitply: \(\frac{2x}{3{y}^{2}}·\frac{6x{y}^{3}}{{x}^{2}y}.\)

### Solution

Multiply. | |

Factor the numerator and denominator completely, and then remove common factors. | |

Simplify. |

### Example: How to Multiply Rational Expressions

Mulitply: \(\frac{2x}{{x}^{2}+x+12}·\frac{{x}^{2}-9}{6{x}^{2}}.\)

### Solution

### Multiply a rational expression.

- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.

## Example

Multiply: \(\frac{{n}^{2}-7n}{{n}^{2}+2n+1}·\frac{n+1}{2n}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{n}^{2}-7n}{{n}^{2}+2n+1}·\frac{n+1}{2n}\hfill \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)}{\left(n+1\right)\left(n+1\right)}·\frac{n+1}{2n}\hfill \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)2n}\hfill \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{n}\left(n-7\right)\require{cancel}\cancel{\left(n+1\right)}}{\left(n+1\right)\require{cancel}\cancel{\left(n+1\right)}2\require{cancel}\cancel{n}}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n-7}{2\left(n+1\right)}\hfill \end{array}\)

## Example

Multiply: \(\frac{16-4x}{2x-12}·\frac{{x}^{2}-5x-6}{{x}^{2}-16}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{16-4x}{2x-12}·\frac{{x}^{2}-5x-6}{{x}^{2}-16}\hfill \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(4-x\right)}{2\left(x-6\right)}·\frac{\left(x-6\right)\left(x+1\right)}{\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(4-x\right)\left(x-6\right)\left(x+1\right)}{2\left(x-6\right)\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\left(-1\right)\frac{\require{cancel}\cancel{2}·2\require{cancel}\cancel{\left(4-x\right)}\require{cancel}\cancel{\left(x-6\right)}\left(x+1\right)}{\require{cancel}\cancel{2}\require{cancel}\cancel{\left(x-6\right)}\require{cancel}\cancel{\left(x-4\right)}\left(x+4\right)}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-\frac{2\left(x+1\right)}{\left(x+4\right)}\hfill \end{array}\)

## Example

Multiply: \(\frac{2x-6}{{x}^{2}-8x+15}·\frac{{x}^{2}-25}{2x+10}.\)

### Solution

Factor each numerator and denominator. | |

Multiply the numerators and denominators. | |

Remove common factors. | |

Simplify. |

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