# 12.4: Rational Expressions

## Learning Objectives

- Simplify rational expressions.
- Find excluded values of rational expressions.

## Introduction

A simplified rational expression is one where the numerator and denominator have no common factors. In order to simplify an expression to **lowest terms**, we factor the numerator and denominator as much as we can and cancel common factors from the numerator and the denominator.

## Simplify Rational Expressions

**Example 1**

*Reduce each rational expression to simplest terms.*

a)

b)

c)

**Solution**

a)

b)

c)

When reducing fractions, you are only allowed to cancel common **factors** from the denominator but NOT common terms. For example, in the expression , we can cross out the factor because . But in the expression we can’t just cross out the terms.

Why can’t we do that? When we cross out terms that are part of a sum or a difference, we’re violating the order of operations (PEMDAS). Remember, the fraction bar means division. When we perform the operation , we’re really performing the division — and the order of operations says that we must perform the operations inside the parentheses before we can perform the division.

Using numbers instead of variables makes it more obvious that canceling individual terms doesn’t work. You can see that — but if we canceled out the 9’s first, we’d get , or -0.2, instead.

For more examples of how to simplify rational expressions, watch the video at

.

## Find Excluded Values of Rational Expressions

Whenever there’s a variable expression in the denominator of a fraction, we must remember that the denominator could be zero when the independent variable takes on certain values. Those values, corresponding to the vertical asymptotes of the function, are called **excluded** values. To find the excluded values, we simply set the denominator equal to zero and solve the resulting equation.

**Example 2**

*Find the excluded values of the following expressions.*

a)

b)

c)

**Solution**

a)

b)

c)

## Removable Zeros

Notice that in the expressions in Example 1, we removed a division by zero when we simplified the problem. For instance, we rewrote as . The denominator of this expression is zero when or when .

However, when we cancel common factors, we simplify the expression to . This reduced form allows the value , so is its only excluded value.

Technically the original expression and the simplified expression are not the same. When we reduce a radical expression to its simplest form, we should specify the removed excluded value. In other words, we should write our final answer as .

Similarly, we should write the answer from Example 1, part as and the answer from Example 1, part *c* as .

## Review Questions

Reduce each fraction to lowest terms.

Find the excluded values for each rational expression.

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