Main content

Course: Algebra (all content) > Unit 13

Lesson 3: Multiplying & dividing rational expressions

Multiplying rational expressions

Learn how to find the product of two rational expressions.

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. The domain of a rational expression includes all real numbers except those that make its denominator equal to zero.

We can simplify rational expressions by canceling common factors in the numerator and the denominator.

If this is not familiar to you, you'll want to check out the following articles first:

What you will learn in this lesson

In this lesson, you will learn how to multiply rational expressions.

Multiplying fractions

To start, let's recall how to multiply numerical fractions.

Consider this example:

\begin{aligned} \frac{3}{4} \cdot \frac{10}{9} \\ = \frac{3}{2 \cdot 2} \cdot \frac{2 \cdot 5}{3 \cdot 3} & Factor numerators and denominators \\ = \frac{3}{2 \cdot 2} \cdot \frac{2 \cdot 5}{3 \cdot 3} & Cancel common factors \\ = \frac{5}{6} & Multiply across \end{aligned}

In conclusion, to multiply two numerical fractions, we factored, canceled common factors, and multiplied across.

Example 1: $\frac{3 x^{2}}{2} \cdot \frac{2}{9 x}$ ‍

We can multiply rational expressions in much the same way as we multiply numerical fractions.

\begin{aligned} \frac{3 x^{2}}{2} \cdot \frac{2}{9 x} \\ = \frac{3 \cdot x \cdot x}{2} \cdot \frac{2}{3 \cdot 3 \cdot x} & Factor numerators and denominators \\ (Note x \neq 0) \\ = \frac{3 \cdot x \cdot x}{2} \cdot \frac{2}{3 \cdot 3 \cdot x} & Cancel common factors \\ = \frac{x}{3} & Multiply across \end{aligned}

Recall that the original expression is defined for

x \neq 0

. The simplified product must have the same restictions. Because of this, we must note that

x \neq 0

We write the simplified product as follows:

\frac{x}{3}

for

x \neq 0

Check your understanding

1) Multiply and simplify the result.

\frac{4 x^{6}}{5} \cdot \frac{1}{12 x^{3}} =

for

x \neq

Example 2: $\frac{x^{2} - x - 6}{5 x + 5} \cdot \frac{5}{x - 3}$ ‍

Once again, we factor, cancel any common factors, and then multiply across. Finally, we make sure to note all restricted values.

\begin{aligned} \frac{x^{2} - x - 6}{5 x + 5} \cdot \frac{5}{x - 3} \\ = \frac{(x - 3) (x + 2)}{5 \cdot (x + 1)} \cdot \frac{5}{x - 3} & Factor \\ (Note x \neq - 1, and x \neq 3) \\ = \frac{(x - 3) (x + 2)}{5 \cdot (x + 1)} \cdot \frac{5}{x - 3} & Cancel common factors \\ = \frac{x + 2}{x + 1} & Multiply across \end{aligned}

The original expression is defined for

x \neq - 1, 3

. The simplified product must have the same restrictions.

In general, the product of two rational expressions is undefined for any value that makes either of the original rational expressions undefined.

Check your understanding

2) Multiply and simplify the result.

\frac{5 x^{3}}{5 x + 10} \cdot \frac{x^{2} - 4}{x^{2}} =

What are all the restrictions on the domain of the resulting expression?

3) Multiply and simplify the result.

\frac{x^{2} - 9}{x^{2} - 2 x - 8} \cdot \frac{x - 4}{x - 3} =

What are all the restrictions on the domain of the resulting expression?

What's next?

If you feel good about your multiplication skills, you can move on to dividing rational expressions.

Want to join the conversation?

Sort by:

No posts yet.

Do you understand English? Click here to see more discussion happening on Khan Academy's English site.

Algebra (all content)

Course: Algebra (all content) > Unit 13

What you should be familiar with before taking this lesson

What you will learn in this lesson

Multiplying fractions

Example 1: 3x22⋅29x‍

Check your understanding

Example 2: x2−x−65x+5⋅5x−3‍

Check your understanding

What's next?

Want to join the conversation?

Example 1: $\frac{3 x^{2}}{2} \cdot \frac{2}{9 x}$ ‍

Example 2: $\frac{x^{2} - x - 6}{5 x + 5} \cdot \frac{5}{x - 3}$ ‍