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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 building rational expressions and know why it is often necessary to build them, be able to find the LCD of one or more expressions.</para>

Overview

  • The Process
  • The Reason For Building Rational Expressions
  • The Least Common Denominator (LCD)

The process

Recall, from Section [link] , the equality property of fractions.

Equality property of fractions

If a b = c d , then a d = b c .

Using the fact that 1 = b b , b 0 , and that 1 is the multiplicative identity, it follows that if P Q is a rational expression, then

P Q · b b = P b Q b , b 0

This equation asserts that a rational expression can be transformed into an equivalent rational expression by multiplying both the numerator and denominator by the same nonzero number.

Process of building rational expressions

This process is known as the process of building rational expressions and it is exactly the opposite of reducing rational expressions. The process is shown in these examples:

3 4 can be built to 12 16 since

3 4 · 1 = 3 4 · 4 4 = 3 · 4 4 · 4 = 12 16

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4 5 can be built to 8 10 since

4 5 · 1 = 4 5 · 2 2 = 4 · 2 5 · 2 = 8 10

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3 7 can be built to 3 x y 7 x y since

3 7 · 1 = 3 7 · x y x y = 3 x y 7 x y

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4 a 3 b can be built to 4 a 2 ( a + 1 ) 3 a b ( a + 1 ) since

4 a 3 b · 1 = 4 a 3 b · a ( a + 1 ) a ( a + 1 ) = 4 a 2 ( a + 1 ) 3 a b ( a + 1 )

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Suppose we're given a rational expression P Q and wish to build it into a rational expression with denominator Q b 2 , that is,

P Q ? Q b 2

Since we changed the denominator, we must certainly change the numerator in the same way. To determine how to change the numerator we need to know how the denominator was changed. Since one rational expression is built into another equivalent expression by multiplication by 1, the first denominator must have been multiplied by some quantity. Observation of

P Q ? Q b 2

tells us that Q was multiplied by b 2 . Hence, we must multiply the numerator P by b 2 . Thus,

P Q = P b 2 Q b 2

Quite often a simple comparison of the original denominator with the new denominator will tell us the factor being used. However, there will be times when the factor is unclear by simple observation. We need a method for finding the factor.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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