## Simplifying Rational Expressions

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Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is *simplified* if it has no common factors, other than 1, in its numerator and denominator.

### Simplified Rational Expression

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example:

- \(\frac{2}{3}\) is simplified because there are no common factors of 2 and 3.
- \(\frac{2x}{3x}\) is not simplified because
*x*is a common factor of 2*x*and 3*x*.

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify **rational expression**s.

### Equivalent Fractions Property

If *a*, *b*, and *c* are numbers where \(b\ne 0,c\ne 0\), then \(\frac{a}{b}=\frac{a·c}{b·c}\) and \(\frac{a·c}{b·c}=\frac{a}{b}\).

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see \(b\ne 0,c\ne 0\) clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Let’s start by reviewing how we simplify numerical fractions.

## Example

Simplify: \(-\frac{36}{63}.\)

### Solution

Rewrite the numerator and denominator showing the common factors. | |

Simplify using the Equivalent Fractions Property. |

Notice that the fraction \(-\frac{4}{7}\) is simplified because there are no more common factors.

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

Simplify: \(\frac{3xy}{18{x}^{2}{y}^{2}}.\)

### Solution

Rewrite the numerator and denominator showing the common factors. | |

Simplify using the Equivalent Fractions Property. |

Did you notice that these are the same steps we took when we divided monomials in the tutorial on polynomials?

To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

Note that removing the *x*’s from \(\frac{x+5}{x}\) would be like cancelling the 2’s in the fraction \(\frac{2+5}{2}\)!

### Example: How to Simplify Rational Binomials

Simplify: \(\frac{2x+8}{5x+20}.\)

### Solution

We now summarize the steps you should follow to simplify rational expressions.

### Simplify a Rational Expression.

- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!

We’ll use the methods we covered in the tutorial on factoring to factor the polynomials in the numerators and denominators in the following examples.

## Example

Simplify: \(\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+6\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor}\phantom{\rule{0.2em}{0ex}}x+2\phantom{\rule{0.2em}{0ex}}\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(x+2\right)}\left(x+3\right)}{\require{cancel}\cancel{\left(x+2\right)}\left(x+6\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{x+3}{x+6}\hfill \end{array}\)

Can you tell which values of *x* must be excluded in this example?

## Example

Simplify: \(\frac{{y}^{2}+y-42}{{y}^{2}-36}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}+y-42}{{y}^{2}-36}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y+7\right)\left(y-6\right)}{\left(y+6\right)\left(y-6\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor}\phantom{\rule{0.2em}{0ex}}y-6\phantom{\rule{0.2em}{0ex}}\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y+7\right)\require{cancel}\cancel{\left(y-6\right)}}{\left(y+6\right)\require{cancel}\cancel{\left(y-6\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+7}{y+6}\hfill \end{array}\)

## Example

Simplify: \(\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using grouping to factor the numerator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{2}\left(p-2\right)+2\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left({p}^{2}+2\right)\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ \\ \begin{array}{c}\text{Remove the common factor of}\phantom{\rule{0.2em}{0ex}}p-2\hfill \\ \text{from the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left({p}^{2}+2\right)\require{cancel}\cancel{\left(p-2\right)}}{\left(p-5\right)\require{cancel}\cancel{\left(p-2\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{2}+2}{p-5}\hfill \end{array}\)

## Example

Simplify: \(\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2n\left(n-7\right)}{4\left({n}^{2}-4n-12\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2n\left(n-7\right)}{4\left(n-6\right)\left(n+2\right)}\hfill \\ \\ \text{Remove the common factor, 2.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{2}n\left(n-7\right)}{\require{cancel}\cancel{2}·2\left(n-6\right)\left(n+2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)}{2\left(n-6\right)\left(n+2\right)}\hfill \end{array}\)

## Example

Simplify: \(\frac{3{b}^{2}-12b+12}{6{b}^{2}-24}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{3{b}^{2}-12b+12}{6{b}^{2}-24}\hfill \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3\left({b}^{2}-4b+4\right)}{6\left({b}^{2}-4\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3\left(b-2\right)\left(b-2\right)}{6\left(b+2\right)\left(b-2\right)}\hfill \\ \\ \text{Remove the common factors of}\phantom{\rule{0.2em}{0ex}}b-2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3.\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{3}\left(b-2\right)\require{cancel}\cancel{\left(b-2\right)}}{\require{cancel}\cancel{3}·2\left(b+2\right)\require{cancel}\cancel{\left(b-2\right)}}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{b-2}{2\left(b+2\right)}\hfill \end{array}\)

## Example

Simplify: \(\frac{{m}^{3}+8}{{m}^{2}-4}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{m}^{3}+8}{{m}^{2}-4}\hfill \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using the formulas for sum of cubes and}\hfill \\ \text{difference of squares.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(m+2\right)\left({m}^{2}-2m+4\right)}{\left(m+2\right)\left(m-2\right)}\hfill \\ \text{Remove the common factor of}\phantom{\rule{0.2em}{0ex}}m+2.\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(m+2\right)}\left({m}^{2}-2m+4\right)}{\require{cancel}\cancel{\left(m+2\right)}\left(m-2\right)}\hfill \\ & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{m}^{2}-2m+4}{m-2}\hfill \end{array}\)