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### Course: Precalculus > Unit 4

Lesson 1: Reducing rational expressions to lowest terms# Reducing rational expressions to lowest terms

Learn what it means to reduce a rational expression to lowest terms, and how it's done!

#### 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**is all real numbers except those that make the denominator equal to zero.For example, the domain of the rational expression $\frac{x+2}{x+1}$ is $x\ne -1$ .

*all real numbers except*$\mathit{\text{-1}}$ , orIf this is new to you, we recommend that you check out our intro to rational expressions.

You should also know how to factor polynomials for this lesson.

#### What you will learn in this lesson

In this article, we will learn how to reduce rational expressions to lowest terms by looking at several examples.

## Introduction

A rational expression is

**reduced to lowest terms**if the numerator and denominator have no factors in common.We can reduce rational expressions to lowest terms in much the same way as we reduce numerical fractions to lowest terms.

For example, $\frac{6}{8}$ reduced to lowest terms is $\frac{3}{4}$ . Notice how we canceled a common factor of $2$ from the numerator and the denominator:

## Example 1: Reducing $\frac{{x}^{2}+3x}{{x}^{2}+5x}$ to lowest terms

**Step 1: Factor the numerator and denominator**

The only way to see if the numerator and denominator share common factors is to factor them!

**Step 2: List restricted values**

At this point, it is helpful to notice any restrictions on $x$ . These will carry over to the simplified expression.

Since division by $0$ is undefined, here we see that ${x\ne 0}$ and ${x\ne -5}$ .

**Step 3: Cancel common factors**

Now notice that the numerator and denominator share a common factor of $x$ . This can be canceled out.

**Step 4: Final answer**

Recall that the original expression is defined for $x\ne 0,-5$ . The reduced expression must have the same restrictions.

Because of this, we must note that $x\ne 0$ . We do not need to note that $x\ne -5$ , since this is understood from the expression.

In conclusion, the reduced form is written as follows:

### A note about equivalent expressions

Original expression | Reduced expression |
---|---|

The two expressions above are $x$ -values. The table below illustrates this for $x=2$ .

**equivalent**. This means their outputs are the same for all possibleOriginal expression | Reduced expression | ||
---|---|---|---|

Evaluation at | |||

Note | The result is reduced to lowest terms by canceling a common factor of | The result is already reduced to lowest terms because the factor of |

For this reason, the two expressions have the same value for the same input. However, values that make the original expression undefined often break this rule. Notice how this is the case with ${x=0}$ .

Original expression | Reduced expression (without restriction) | ||
---|---|---|---|

Evaluation at |

Because the two expressions must be equivalent for $x\ne 0$ for the reduced expression.

*all*possible inputs, we must require### Misconception alert

Note that we cannot cancel the $x$ 's in the expression below. This is because these are terms rather than factors of the polynomials!

This becomes clear when looking at a numerical example. For example, suppose ${x=2}$ .

As a rule, we can only cancel if the numerator and denominator are in factored form!

### Summary of the process for reducing to lowest terms

**Step 1:**Factor the numerator and the denominator.**Step 2:**List restricted values.**Step 3**: Cancel common factors.**Step 4**: Reduce to lowest terms and note any restricted values not implied by the expression.

### Check your understanding

## Example 2: Reducing $\frac{{x}^{2}-9}{{x}^{2}+5x+6}$ to lowest terms

**Step 1: Factor the numerator and denominator**

**Step 2: List restricted values**

Since division by $0$ is undefined, here we see that ${x\ne -2}$ and ${x\ne -3}$ .

**Step 3: Cancel common factors**

Notice that the numerator and denominator share a common factor of ${x+3}$ . This can be canceled out.

**Step 4: Final answer**

We write the reduced form as follows:

The original expression requires $x\ne -2,-3$ . We do not need to note that $x\ne -2$ , since this is understood from the expression.

### Check for understanding

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