# 12.4: Rational Expressions

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Simplify rational expressions.
• Find excluded values of rational expressions.
• Simplify rational models of real-world situations.

## Vocabulary

Terms introduced in this lesson:

lowest terms
canceling common factors
common terms
excluded value
removable zero

## Teaching Strategies and Tips

In this lesson, students learn what removable zeros are, how to find them, how they are related to excluded values, and what they look like graphically.

• Point out that removable zeros are “divisions by zero” removed by simplifying the rational expression.
• See Example 2.

Have students review factoring.

a. 16x2+12x4x=4x(4x+3)4x=4x+3, (x0)\begin{align*}\frac{16x^2 + 12x}{4x} = \frac{4x(4x+3)}{4x} = 4x+3, \ (x \neq 0)\end{align*}

b. 56x3+14x24x2+5x+1=14x2(4x+1)(x+1)(4x+1)=14x2x+1, (x14)\begin{align*}\frac{56x^3 + 14x^2}{4x^2+5x+1}=\frac{14x^2(4x+1)}{(x+1)(4x+1)}=\frac{14x^2}{x+1}, \ (x \neq -\frac{1}{4})\end{align*}

Optional: In some rational expressions, it may appear to students like nothing will cancel at first, despite the similar terms. Suggest that students try factoring out a negative and then rearranging the order of the terms.

c. 1xx1=(x1)x1=1, x1\begin{align*}\frac{1-x}{x-1} = \frac{-(x-1)}{x-1}= -1, \ x \neq 1\end{align*}

When factoring out the minus sign, suggest that students use brackets to keep the negative outside and so that it will be harder to lose it.

d. 1x2x2+x2=(1x)(1+x)(x1)(x+2)=[(x1)(x+1)(x1)(x+2)]=x+1x+2, x1\begin{align*}\frac{1-x^2}{x^2+x-2} = \frac{(1-x)(1+x)}{(x-1)(x+2)} = -\left [\frac{(x-1)(x+1)}{(x-1)(x+2)} \right ] = -\frac{x+1}{x+2}, \ x \neq 1\end{align*}

Point out that the rules for working with rational expressions are the same as those for working with ordinary fractions.

• Simplifying a rational expression means the same as simplifying a fraction – that the numerator and denominator of the rational expression have no common factors.
• Remind students after canceling all the terms of a numerator (or denominator) that a factor of 1\begin{align*}1\end{align*} remains.

In Example 3, have students use scientific notation.

## Error Troubleshooting

General Tip: Suggest that students check their work after simplifying a rational expression by substituting the variable with a number.

General Tip: Remind students that in order to cancel a factor, it must be common to the entire numerator and denominator.

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Feb 22, 2012
Aug 22, 2014
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CK.MAT.ENG.TE.1.Algebra-I.12.4