8.2 Reducing rational expressions

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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: understand and be able to use the process of reducing rational expressions.</para>

Overview

• The Logic Behind The Process
• The Process

The logic behind the process

When working with rational expressions, it is often best to write them in the simplest possible form. For example, the rational expression
$\frac{{x}^{2}-4}{{x}^{2}-6x+8}$
can be reduced to the simpler expression $\frac{x+2}{x-4}$ for all $x$ except $x=2,4$ .

From our discussion of equality of fractions in Section [link] , we know that $\frac{a}{b}=\frac{c}{d}$ when $ad=bc$ . This fact allows us to deduce that, if $k\ne 0,\frac{ak}{bk}=\frac{a}{b},$ since $akb=abk$ (recall the commutative property of multiplication). But this fact means that if a factor (in this case, $k$ ) is common to both the numerator and denominator of a fraction, we may remove it without changing the value of the fraction.
$\frac{ak}{bk}=\frac{a\overline{)k}}{b\overline{)k}}=\frac{a}{b}$

Cancelling

The process of removing common factors is commonly called cancelling .

$\frac{16}{40}$ can be reduced to $\frac{2}{5}$ .   Process:

$\frac{16}{40}=\frac{2·2·2·2}{2·2·2·5}$

Remove the three factors of 1; $\frac{2}{2}·\frac{2}{2}·\frac{2}{2}.$

$\frac{\overline{)2}·\overline{)2}·\overline{)2}·2}{\overline{)2}·\overline{)2}·\overline{)2}·5}=\frac{2}{5}$

Notice that in $\frac{2}{5}$ , there is no factor common to the numerator and denominator.

$\frac{111}{148}$ can be reduced to $\frac{3}{4}$ .   Process:

$\frac{111}{148}=\frac{3·37}{4·37}$

Remove the factor of 1; $\frac{37}{37}$ .

$\frac{3·\overline{)37}}{4·\overline{)37}}$

$\frac{3}{4}$

Notice that in $\frac{3}{4}$ , there is no factor common to the numerator and denominator.

$\frac{3}{9}$ can be reduced to $\frac{1}{3}$ .   Process:

$\frac{3}{9}=\frac{3·1}{3·3}$

Remove the factor of 1; $\frac{3}{3}$ .

$\frac{\overline{)3}·1}{\overline{)3}·3}=\frac{1}{3}$

Notice that in $\frac{1}{3}$ there is no factor common to the numerator and denominator.

$\frac{5}{7}$ cannot be reduced since there are no factors common to the numerator and denominator.

Problems 1, 2, and 3 shown above could all be reduced. The process in each reduction included the following steps:

1. Both the numerator and denominator were factored.
2. Factors that were common to both the numerator and denominator were noted and removed by dividing them out.

We know that we can divide both sides of an equation by the same nonzero number, but why should we be able to divide both the numerator and denominator of a fraction by the same nonzero number? The reason is that any nonzero number divided by itself is 1, and that if a number is multiplied by 1, it is left unchanged.

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