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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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