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# Multiplying Rational Expressions

Objective Learn to multiply rational expressions.

In this lesson, you will learn to multiply rational expressions. The techniques are completely analogous to multiplying rational numbers (fractions). Keep this analogy in mind throughout the entire lesson.

## Multiplying Rational Numbers

Remember that to multiply rational numbers, multiply the numerators and then divide by the product of the denominators. Then simplify by dividing by the common factors of the numerator and the denominator.

Example 1

Find .

Solution

 The GCF of the numerator and the denominator is 6. Cancel the GCF.

An alternative way to approach this problem is to divide by the common factors before multiplying.

## Multiplying Rational Expressions

Multiplying rational expressions is done the same way. Multiply the numerators, and then divide the product by the product of the denominators. Then simplify by dividing the common factors of the numerator and the denominator.

Example 2

Find .

Solution

Method 1:

 Cancel the GCF of the numerator and the denominator, 6yz.

Now just like with multiplying fractions, there is an alternative method. Namely, divide by the common factors before multiplying.

Method 2:

There are some problems that require factoring the polynomials in order to find the GCF of the numerator and denominator.

Example 3

Find .

Solution

To simplify, factor the quadratic expressions in the numerator and then cancel the common factors of the numerator and the denominator.

For practice, do one more example.

Example 4

Find .

Solution

First, find the common factors. To do this, factor the quadratic expression in the denominator.

 Cancel the common factors.

We therefore conclude that

This technique is very important because it reinforces prior techniques like factoring and multiplication of rational numbers.