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# Multiplying Rational Expressions

A rational number is any number that can be written as a fraction with integers. An example is $\frac{1}{2}$ or $\frac{4}{15}$ .

A rational expression is one that can be expressed as a quotient of polynomials. That is, a/b where a and b are polynomials and $b\ne 0$ .

## How to multiply a fraction by a fraction

To multiply a fraction by another fraction, you simply multiply the two numerators and multiply the two denominators and then simplify as needed.

$\frac{a}{b}×\frac{c}{d}=\frac{a×c}{b×d}$

An example of this would be:

$\frac{3}{7}×\frac{4}{5}$

$=\frac{3×4}{7×5}$

$=\frac{12}{35}$

There are no common factors, so this cannot be reduced any further. Therefore, $\frac{3}{7}×\frac{4}{5}=\frac{12}{35}$

## How to multiply rational expressions

Multiplying rational expressions is basically the same as multiplying fractions. That is to say, you multiply the numerators to get the numerator of the product and you multiply the denominators to get the denominator of the product.

The rule that guides the multiplication of rational expressions is:

For all rational expressions $\frac{a}{b}$ and $\frac{c}{d}$ with $b\ne 0$ and $d\ne 0$ , $\frac{a}{b}×\frac{c}{d}=\frac{ac}{bd}$ .

The steps to multiply rational expressions are as follows:

1. Factor all of the polynomials.
2. Cancel any common factors.
3. Multiply the resulting numerator and denominator.

Example 1

Multiply the following rational expression.

$\frac{7{x}^{2}}{3}×\frac{9}{14x}$

Note the variable in the denominator of the second fraction. It cannot be 0 because that would lead us to divide by 0, which is forbidden.

There is no factoring to be done, so our first step is to cancel duplicate factors.

$\frac{7{x}^{2}}{14x}=\frac{x}{2}$

$\frac{9}{3}=3$ .

$\frac{x}{1}×\frac{3}{2}=\frac{3x}{2}$

$\frac{3x}{2}$ for $x\ne$

We need to add the $x\ne 0$ because $\frac{3x}{2}$ makes it possible for x to be 0, but the original expression does not allow for x to be 0 because x is in the second fraction's denominator.

Example 2

Multiply the following rational expression.

$\frac{2x+1}{{x}^{2}-1}×\frac{x+1}{2{x}^{2}+x}$

First, we see that we can factor each of the denominators. The first denominator is a case of the difference of two squares. The second denominator is easy because we can pull out a factor of x.

$\frac{2x+1}{{x}^{2}-1}=\frac{2x+1}{\left(x+1\right)\left(x-1\right)}×\frac{x+1}{x\left(2x+1\right)}$

Now we can multiply the numerators and denominators by placing them side by side.

$\frac{\left(2x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)\left(x\right)\left(2x+1\right)}$

Now we can factor out common factors. We can factor out $2x+1$ and $x+1$ . That leaves us with

$\frac{1}{\left(x-1\right)\left(x\right)}$

Which can be simplified to

$\frac{1}{x\left(x-1\right)}$

## Flashcards covering the Multiplying Rational Expressions

Precalculus Flashcards

## Get help learning how to multiply rational expressions

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