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# 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} &\phantom{=}\dfrac{3}{4}\cdot\dfrac{10}{9}\\\\ &=\dfrac{\greenD3}{\blueD2\cdot 2}\cdot \dfrac{\blueD2\cdot 5}{\greenD3\cdot 3} &&\small{\gray{\text{Factor numerators and denominators}}} \\\\ &=\dfrac{\greenD{\cancel{3}}}{\blueD{\cancel{2}}\cdot 2}\cdot \dfrac{\blueD{\cancel{2}}\cdot 5}{\greenD{\cancel{3}}\cdot 3}&&\small{\gray{\text{Cancel common factors}}} \\\\ &=\dfrac{5}{6}&&\small{\gray{\text{Multiply across}}} \end{aligned}
In conclusion, to multiply two numerical fractions, we factored, canceled common factors, and multiplied across.

## Example 1: $\dfrac{3x^2}{2}\cdot \dfrac{2}{9x}$start fraction, 3, x, squared, divided by, 2, end fraction, dot, start fraction, 2, divided by, 9, x, end fraction

We can multiply rational expressions in much the same way as we multiply numerical fractions.
\begin{aligned} &\phantom{=}\dfrac{3x^2}{2}\cdot\dfrac{2}{9x}\\\\\\ &=\dfrac{3\cdot x\cdot x}{2}\cdot \dfrac{2}{3\cdot 3\cdot \goldD x}&& \small{\gray{\text{Factor numerators and denominators}}}\\ \\ &\quad \small{(\text{Note } \goldD{x\neq 0})}\\ \\ \\&=\dfrac{\blueD{\cancel{3}}\cdot \greenD{\cancel{ x}}\cdot x}{\purpleC{\cancel{2}}}\cdot \dfrac{\purpleC{\cancel{2}}}{\blueD{\cancel{ 3}}\cdot 3\cdot \greenD{\cancel{ x}}}&& \small{\gray{\text{Cancel common factors}}} \\ \\ &=\dfrac{x}{3}&&\small{\gray{\text{Multiply across}}} \end{aligned}
Recall that the original expression is defined for x, does not equal, 0. The simplified product must have the same restictions. Because of this, we must note that x, does not equal, 0.
We write the simplified product as follows:
start fraction, x, divided by, 3, end fraction for x, does not equal, 0

1) Multiply and simplify the result.
start fraction, 4, x, start superscript, 6, end superscript, divided by, 5, end fraction, dot, start fraction, 1, divided by, 12, x, cubed, end fraction, equals
for x, does not equal

## Example 2: $\dfrac{x^2-x-6}{5x+5}\cdot\dfrac {5}{x-3}$start fraction, x, squared, minus, x, minus, 6, divided by, 5, x, plus, 5, end fraction, dot, start fraction, 5, divided by, x, minus, 3, end fraction

Once again, we factor, cancel any common factors, and then multiply across. Finally, we make sure to note all restricted values.
\begin{aligned} &\phantom{=}\dfrac{x^2-x-6}{5x+5}\cdot\dfrac {5}{x-3}\\\\\\ &=\dfrac{(x-3)(x+2)}{5\cdot \goldD{(x+1)}}\cdot \dfrac{5}{\maroonD{x-3}}&&\small{\gray{\text{Factor}}}\\ \\ &\quad \small{(\text{Note }\goldD{x\neq -1}}, \text{ and }\maroonD{x\neq 3} )\\ \\ &=\dfrac{\blueD{\cancel{(x-3)}}{(x+2)}}{\greenD{\cancel{5}}\cdot({x+1})}\cdot \dfrac{{\greenD{\cancel{5}}}}{\blueD{\cancel{x-3}}}&&\small{\gray{\text{Cancel common factors}}}\\ \\ &=\dfrac{x+2}{x+1}&&\small{\gray{\text{Multiply across}}} \end{aligned}
The original expression is defined for x, does not equal, minus, 1, comma, 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.

2) Multiply and simplify the result.
start fraction, 5, x, cubed, divided by, 5, x, plus, 10, end fraction, dot, start fraction, x, squared, minus, 4, divided by, x, squared, end fraction, equals
What are all the restrictions on the domain of the resulting expression?