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Rational Expressions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

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.

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.

Objectives of this module: be able to recognize a rational expression, be familiar with the equality and negative properties of fractions.

Overview

  • Rational Expressions
  • Zero-Factor Property
  • The Equality Property of Fractions
  • The Negative Property of Fractions

Rational Expressions

In arithmetic it is noted that a fraction is a quotient of two whole numbers. The expression abab, where aa and bb are any two whole numbers and b0b0, is called a fraction. The top number, aa, is called the numerator, and the bottom number, bb, is called the denominator.

Simple Algebraic Fraction

We define a simple algebraic fraction in a similar manner. Rather than restrict­ing ourselves only to numbers, we use polynomials for the numerator and denomi­nator. Another term for a simple algebraic fraction is a rational expression. A rational expression is an expression of the form PQPQ, where PP and QQ are both polyno­mials and QQ never represents the zero polynomial.

Rational Expression

A rational expression is an algebraic expression that can be written as the quotient of two polynomials.

Examples 1–4 are rational expressions:

Example 1

x+9x-7x+9x-7 is a rational expression: PP is x+9x+9 and QQ is x-7x-7 .

Example 2

x3+5x2-12x+1x4-10 x3+5x2-12x+1x4-10 is a rational expression: PP is x3+5x2-12x+1x3+5x2-12x+1 and QQ is x4-10x4-10 .

Example 3

3838 is a rational expression: PP is 3 and QQ is 8.

Example 4

4x-54x-5 is a rational expression since 4x-54x-5 can be written as 4x-51 4x-51 : PP is 4x-54x-5 and QQ is 1.

Example 5

5x2-8 2x-1 5x2-8 2x-1 is not a rational expression since 5x2-8 5x2-8 is not a polynomial.

In the rational expression PQPQ, PP is called the numerator and QQ is called the denominator.

Domain of a Rational Expression

Since division by zero is not defined, we must be careful to note the values for which the rational expression is valid. The collection of values for which the rational expression is defined is called the domain of the rational expression. (Recall our study of the domain of an equation in Section (Reference).)

Finding the Domain of a Rational Expression

To find the domain of a rational expression we must ask, "What values, if any, of the variable will make the denominator zero?" To find these values, we set the denominator equal to zero and solve. If any zero-producing values are obtained, they are not included in the domain. All other real numbers are included in the domain (unless some have been excluded for particular situational reasons).

Zero-Factor Property

Sometimes to find the domain of a rational expression, it is necessary to factor the denominator and use the zero-factor property of real numbers.

Zero-factor Property

If two real numbers aa and bb are multiplied together and the resulting product is 0, then at least one of the factors must be zero, that is, either a=0a=0 , b=0b=0 , or both a=0a=0 and b=0b=0 .

The following examples illustrate the use of the zero-factor property.

Example 6

What value will produce zero in the expression 4x4x ? By the zero-factor property, if 4x=04x=0 , then x=0x=0 .

Example 7

What value will produce zero in the expression 8(x-6)8(x-6)? By the zero-factor property, if 8(x-6)=08(x-6)=0, then

x-6=0x=6 x-6=0x=6
Thus, 8(x-6)=08(x-6)=0 when x=6x=6 .

Example 8

What value(s) will produce zero in the expression (x-3)(x+5)(x-3)(x+5)? By the zero-factor property, if (x-3)(x+5)=0(x-3)(x+5)=0, then

x-3=0orx+5=0x=3x=-5 x-3=0orx+5=0x=3x=-5
Thus, (x-3)(x+5)=0(x-3)(x+5)=0 when x=3x=3 or x=-5x=-5 .

Example 9

What value(s) will produce zero in the expression x2+6x+8x2+6x+8? We must factor x2+6x+8x2+6x+8 to put it into the zero-factor property form.

x2+6x+8=(x+2)(x+4)x2+6x+8=(x+2)(x+4)

Now, (x+2)(x+4)=0(x+2)(x+4)=0 when
x+2=0orx+4=0x=-2x=-4 x+2=0orx+4=0x=-2x=-4
Thus, x2+6x+8=0x2+6x+8=0 when x=-2x=-2 or x=-4x=-4 .

Example 10

What value(s) will produce zero in the expression 6x2-19x-76x2-19x-7? We must factor 6x2-19x-76x2-19x-7 to put it into the zero-factor property form.

6x2-19x-7=(3x+1)(2x-7)6x2-19x-7=(3x+1)(2x-7)

Now, (3x+1)(2x-7)=0(3x+1)(2x-7)=0 when
3x+1=0or2x-7=03x=-12x=7x=-13 x=72 3x+1=0or2x-7=03x=-12x=7x=-13 x=72
Thus, 6x2-19x-7=06x2-19x-7=0 when x=-13 x=-13 or 7272.

Sample Set A

Find the domain of the following expressions.

Example 11

5x-15x-1.

The domain is the collection of all real numbers except 1. One is not included, for if x=1x=1 , division by zero results.

Example 12

3a2a-83a2a-8.

If we set 2a-82a-8 equal to zero, we find that a=4a=4 .

2a-8=02a=8a=4 2a-8=02a=8a=4
Thus 4 must be excluded from the domain since it will produce division by zero. The domain is the collection of all real numbers except 4.

Example 13

5x-1(x+2)(x-6)5x-1(x+2)(x-6).

Setting ( x+2 )( x-6 )=0 ( x+2 )( x-6 )=0 , we find that x=-2 x=-2 and x=6 x=6 . Both these values produce division by zero and must be excluded from the domain. The domain is the collection of all real numbers except –2 and 6.

Example 14

9x2-2x-159x2-2x-15.

Setting x 2 -2x-15=0 x 2 -2x-15=0 , we get

( x+3 )( x-5 ) = 0 x = -3,5 ( x+3 )( x-5 ) = 0 x = -3,5
Thus, x=-3 x=-3 and x=5 x=5 produce division by zero and must be excluded from the domain. The domain is the collection of all real numbers except –3 and 5.

Example 15

2x2+x-7x(x-1)(x-3)(x+10) 2x2+x-7x(x-1)(x-3)(x+10) .

Setting x( x-1 )( x-3 )( x+10 )=0 x( x-1 )( x-3 )( x+10 )=0 , we get x=0,1,3,-10 x=0,1,3,-10 . These numbers must be excluded from the domain. The domain is the collection of all real numbers except 0, 1, 3, –10.

Example 16

8b+7(2b+1)(3b-2) 8b+7(2b+1)(3b-2) .

Setting ( 2b+1 )( 3b-2 )=0 ( 2b+1 )( 3b-2 )=0 , we get bb = -12-12, 2323. The domain is the collection of all real numbers except -12-12 and 2323.

Example 17

4x-5x2+14x-5x2+1.

No value of x x is excluded since for any choice of x x , the denominator is never zero. The domain is the collection of all real numbers.

Example 18

x-96x-96.

No value of x x is excluded since for any choice of x x , the denominator is never zero. The domain is the collection of all real numbers.

Practice Set A

Find the domain of each of the following rational expressions.

Exercise 1

Exercise 2

5x x( x+4 ) 5x x( x+4 )

Solution

0,4 0,4

Exercise 3

2x+1 ( x+2 )( 1x ) 2x+1 ( x+2 )( 1x )

Solution

2,1 2,1

Exercise 4

5a+2a2+6a+85a+2a2+6a+8

Solution

2,4 2,4

Exercise 5

12y3y2-2y-8 12y3y2-2y-8

Solution

-43, 2 -43, 2

Exercise 6

2m-5m2+32m-5m2+3

Solution

All real numbers comprise the domain.

Exercise 7

k2-45k2-45

Solution

All real numbers comprise the domain.

The Equality Property of Fractions

From our experience with arithmetic we may recall the equality property of fractions. Let aa, bb, cc, dd be real numbers such that b0b0 and d0d0.

Equality Property of Fractions

  1. If ab=cdab=cd, then ad=bc ad=bc .
  2. If ad=bc ad=bc , then ab=cdab=cd.

Two fractions are equal when their cross-products are equal.

We see this property in the following examples:

Example 19

23=81223=812, since 2·12=3·82·12=3·8.

Example 20

5y2=15y2 6y 5y2=15y2 6y , since 5y·6y=2·15y25y·6y=2·15y2 and 30 y 2 =30 y 2 30 y 2 =30 y 2 .

Example 21

Since 9a·4=18a·29a·4=18a·2, 9a18a=24 9a18a=24 .

The Negative Property of Fractions

A useful property of fractions is the negative property of fractions.

Negative Property of Fractions


The negative sign of a fraction may be placed

  1. in front of the fraction, -ab-ab,
  2. in the numerator of the fraction, -ab-ab,
  3. in the denominator of the fraction, a-ba-b.

    All three fractions will have the same value, that is,

    -ab=-ab=a-b -ab=-ab=a-b

  • The negative property of fractions is illustrated by the fractions
  • -34=-34=3-4 -34=-34=3-4

To see this, consider -34=-34 -34=-34 . Is this correct?

By the equality property of fractions, -3·4=-12-3·4=-12 and -3·4=-12-3·4=-12. Thus, -34=-34-34=-34. Convince yourself that the other two fractions are equal as well.

This same property holds for rational expressions and negative signs. This property is often quite helpful in simplifying a rational expression (as we shall need to do in subsequent sections).

If either the numerator or denominator of a fraction or a fraction itself is immediately preceded by a negative sign, it is usually most convenient to place the negative sign in the numerator for later operations.

Sample Set B

Example 22

x-4x-4 is best written as -x4-x4.

Example 23

-y9-y9 is best written as -y9-y9.

Example 24

-x-42x-5-x-42x-5 could be written as -x-42x-5-x-42x-5, which would then yield -x+42x-5-x+42x-5.

Example 25

-5 -10-x . Factor out -1 from the denominator. -5 -( 10+x ) A negative divided by a negative is a positive. 5 10+x -5 -10-x . Factor out -1 from the denominator. -5 -( 10+x ) A negative divided by a negative is a positive. 5 10+x

Example 26

- 3 7-x . Rewrite this. -3 7-x Factor out-1 from the denominator. -3 -( -7+x ) A negative divided by a negative is positive. 3 -7+x Rewrite. 3 x-7 - 3 7-x . Rewrite this. -3 7-x Factor out-1 from the denominator. -3 -( -7+x ) A negative divided by a negative is positive. 3 -7+x Rewrite. 3 x-7

This expression seems less cumbersome than does the original (fewer minus signs).

Practice Set B

Fill in the missing term.

Exercise 8

-5y-2=y-2 -5y-2=y-2

Solution

5 5

Exercise 9

-a+2-a+3=a-3 -a+2-a+3=a-3

Solution

a+2 a+2

Exercise 10

-85-y=y-5 -85-y=y-5

Solution

8

Exercises

For the following problems, find the domain of each of the rational expressions.

Exercise 11

6x-4 6x-4

Solution

x4 x4

Exercise 12

-3x-8 -3x-8

Exercise 13

-11xx+1 -11xx+1

Solution

x1 x1

Exercise 14

x+10x+4 x+10x+4

Exercise 15

x-1x 2 -4 x-1x 2 -4

Solution

x2,2 x2,2

Exercise 16

x+7x 2 -9 x+7x 2 -9

Exercise 17

-x+4x 2 -36 -x+4x 2 -36

Solution

x6,6 x6,6

Exercise 18

-a+5aa-5 -a+5aa-5

Exercise 19

2bb(b+6) 2bb(b+6)

Solution

b0,6 b0,6

Exercise 20

3b+1bb-4b+5 3b+1bb-4b+5

Exercise 21

3x+4xx-10x+1 3x+4xx-10x+1

Solution

x0,10,1 x0,10,1

Exercise 22

-2xx 2 4-x -2xx 2 4-x

Exercise 23

6a a 3 a-57-a 6a a 3 a-57-a

Solution

x0,5,7 x0,5,7

Exercise 24

-5a 2 +6a+8 -5a 2 +6a+8

Exercise 25

-8b 2 -4b+3 -8b 2 -4b+3

Solution

b1,3 b1,3

Exercise 26

x-1x 2 -9x+2 x-1x 2 -9x+2

Exercise 27

y-9y 2 -y-20 y-9y 2 -y-20

Solution

y5,4 y5,4

Exercise 28

y-62y2-3y-2 y-62y2-3y-2

Exercise 29

2x+76x3+x2-2x 2x+76x3+x2-2x

Solution

x0, 1 2 , 2 3 x0, 1 2 , 2 3

Exercise 30

-x+4x3-8x2+12x -x+4x3-8x2+12x

For the following problems, show that the fractions are equivalent.

Exercise 31

-35-35 and -35-35

Solution

( 3 )5=15,( 3·5 )=15 ( 3 )5=15,( 3·5 )=15

Exercise 32

-27-27 and -27-27

Exercise 33

1 4 and 1 4 1 4 and 1 4

Solution

( 1·4 )=4,4( 1 )=4 ( 1·4 )=4,4( 1 )=4

Exercise 34

-23-23 and -23-23

Exercise 35

-910-910 and 9-109-10

Solution

( 9 )( 10 )=90and( 9 )( 10 )=90 ( 9 )( 10 )=90and( 9 )( 10 )=90

For the following problems, fill in the missing term.

Exercise 36

-4x-1=x-1 -4x-1=x-1

Exercise 37

-2x+7=x+7 -2x+7=x+7

Solution

2 2

Exercise 38

-3 x+42 x-1=2 x-1 -3 x+42 x-1=2 x-1

Exercise 39

-2 x+75 x-1=5 x-1 -2 x+75 x-1=5 x-1

Solution

2x7 2x7

Exercise 40

-x-26 x-1=6 x-1 -x-26 x-1=6 x-1

Exercise 41

-x-42 x-3=2 x-3 -x-42 x-3=2 x-3

Solution

x+4 x+4

Exercise 42

-x+5-x-3=x+3 -x+5-x-3=x+3

Exercise 43

-a+1-a-6=a+6 -a+1-a-6=a+6

Solution

a+1 a+1

Exercise 44

x-7-x+2=x-2 x-7-x+2=x-2

Exercise 45

y+10-y-6=y+6 y+10-y-6=y+6

Solution

y10 y10

Exercises For Review

Exercise 46

((Reference)) Write (15x-3y4 5x2y-7 )-2 (15x-3y4 5x2y-7 )-2 so that only positive exponents appear.

Exercise 47

((Reference)) Solve the compound inequality 16x-5<13 16x-5<13 .

Solution

1x<3 1x<3

Exercise 48

((Reference)) Factor 8 x 2 -18x-5 8 x 2 -18x-5 .

Exercise 49

((Reference)) Factor x 2 -12x+36 x 2 -12x+36 .

Solution

( x6 ) 2 ( x6 ) 2

Exercise 50

((Reference)) Supply the missing word. The phrase "graphing an equation" is interpreted as meaning "geometrically locate the

          
to an equation."

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