Skip to content Skip to navigation

Connexions

You are here: Home » Content » Rational Expressions: Reducing Rational Expressions

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • College Open Textbooks display tagshide tags

    This module is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook CollaborativeAs a part of collection: "Elementary Algebra"

    Comments:

    "Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Elementary Algebra"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Elementary Algebra"

    Comments:

    "Elementary Algebra covers traditional topics studied in a modern elementary algebra course. Written by Denny Burzynski and Wade Ellis, it is intended for both first-time students and those […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Rational Expressions: Reducing 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: understand and be able to use the process of reducing rational expressions.

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
 x 2 -4 x 2 -6x+8 x 2 -4 x 2 -6x+8
can be reduced to the simpler expression x+2 x-4 x+2 x-4 for all x x except x=2,4 x=2,4 .

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

Cancelling

The process of removing common factors is commonly called cancelling.

Example 1

16 40 16 40 can be reduced to 2 5 2 5 .   Process:

 16 40 = 2·2·2·2 2·2·2·5 16 40 = 2·2·2·2 2·2·2·5

Remove the three factors of 1; 2 2 · 2 2 · 2 2 . 2 2 · 2 2 · 2 2 .

2 · 2 · 2 ·2 2 · 2 · 2 ·5 = 2 5 2 · 2 · 2 ·2 2 · 2 · 2 ·5 = 2 5

Notice that in 2 5 2 5 , there is no factor common to the numerator and denominator.

Example 2

111 148 111 148 can be reduced to 3 4 3 4 .   Process:

 111 148 = 3·37 4·37 111 148 = 3·37 4·37

Remove the factor of 1; 37 37 37 37 .

 3· 37 4· 37 3· 37 4· 37

3 4 3 4

Notice that in 3 4 3 4 , there is no factor common to the numerator and denominator.

Example 3

3 9 3 9 can be reduced to 1 3 1 3 .   Process:

 3 9 = 3·1 3·3 3 9 = 3·1 3·3

Remove the factor of 1; 3 3 3 3 .

 3 ·1 3 ·3 = 1 3 3 ·1 3 ·3 = 1 3

Notice that in 1 3 1 3 there is no factor common to the numerator and denominator.

Example 4

5 7 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.

Consider the fraction 6 24 6 24 . Multiply this fraction by 1. This is written 6 24 ·1 6 24 ·1 . But 1 can be rewritten as 1 6 1 6 1 6 1 6 .

6 24 ⋅ 1 6 1 6 = 6⋅ 1 6 24⋅ 1 6 = 1 4 6 24 ⋅ 1 6 1 6 = 6⋅ 1 6 24⋅ 1 6 = 1 4

The answer, 1 4 1 4 , is the reduced form. Notice that in 1 4 1 4 there is no factor common to both the numerator and denominator. This reasoning provides justification for the following rule.

Cancelling

Multiplying or dividing the numerator and denominator by the same nonzero number does not change the value of a fraction.

The Process

We can now state a process for reducing a rational expression.

Reducing a Rational Expression

  1. Factor the numerator and denominator completely.
  2. Divide the numerator and denominator by all factors they have in common, that is, remove all factors of 1.

Reduced to Lowest Terms

  1. A rational expression is said to be reduced to lowest terms when the numerator and denominator have no factors in common.

Sample Set A

Reduce the following rational expressions.

Example 5

15x 20x . Factor. 15x 20x = 5·3·x 5·2·2·x The factors that are common to both the numerator and denominator are 5 and  x. Divide each by  5x. 5 ·3· x 5 ·2·2· x = 3 4 , x≠0   It is helpful to draw a line through the divided-out factors. 15x 20x . Factor. 15x 20x = 5·3·x 5·2·2·x The factors that are common to both the numerator and denominator are 5 and  x. Divide each by  5x. 5 ·3· x 5 ·2·2· x = 3 4 , x≠0   It is helpful to draw a line through the divided-out factors.

Example 6

x 2 −4 x 2 −6x+8 . Factor. ( x+2 )( x−2 ) ( x−2 )( x−4 ) The factor that is common to both the numerator and denominator is  x−2. Divide each by x−2. ( x+2 ) ( x−2 ) ( x−2 ) ( x−4 ) = x+2 x−4 , x≠2, 4 x 2 −4 x 2 −6x+8 . Factor. ( x+2 )( x−2 ) ( x−2 )( x−4 ) The factor that is common to both the numerator and denominator is  x−2. Divide each by x−2. ( x+2 ) ( x−2 ) ( x−2 ) ( x−4 ) = x+2 x−4 , x≠2, 4

The expression x−2 x−4 x−2 x−4 is the reduced form since there are no factors common to both the numerator and denominator. Although there is an x x in both, it is a common term, not a common factor, and therefore cannot be divided out.

CAUTION — This is a common error: x−2 x−4 = x −2 x −4 = 2 4 x−2 x−4 = x −2 x −4 = 2 4 is incorrect!

Example 7

a+2b 6a+12b . Factor. a+2b 6( a+2b ) = a+2b 6 ( a+2b ) = 1 6 , a ≠ -2b a+2b 6a+12b . Factor. a+2b 6( a+2b ) = a+2b 6 ( a+2b ) = 1 6 , a ≠ -2b
Since a+2b a+2b is a common factor to both the numerator and denominator, we divide both by a+2b a+2b . Since ( a+2b ) ( a+2b ) =1 ( a+2b ) ( a+2b ) =1 , we get 1 in the numerator.

Sometimes we may reduce a rational expression by using the division rule of exponents.

Example 8

8 x 2 y 5 4x y 2 . Factor and use the rule  a n a m = a n−m . 8 x 2 y 5 4x y 2 = 2⋅2⋅2 2⋅2 x 2−1 y 5−2 = 2x y 3 , x≠0, y≠0 8 x 2 y 5 4x y 2 . Factor and use the rule  a n a m = a n−m . 8 x 2 y 5 4x y 2 = 2⋅2⋅2 2⋅2 x 2−1 y 5−2 = 2x y 3 , x≠0, y≠0

Example 9

−10 x 3 a( x 2 −36 ) 2 x 3 −10 x 2 −12x . Factor. −10 x 3 a( x 2 −36 ) 2 x 3 −10 x 2 −12x = −5⋅2 x 3 a( x+6 )( x−6 ) 2x( x 2 −5x−6 ) = −5⋅2 x 3 a( x+6 )( x−6 ) 2x( x−6 )( x+1 ) = −5⋅ 2 x 2 3 a( x+6 ) ( x−6 ) 2 x ( x−6 ) ( x+1 ) = −5 x 2 a( x+6 ) x−1 , x≠−1, 6 −10 x 3 a( x 2 −36 ) 2 x 3 −10 x 2 −12x . Factor. −10 x 3 a( x 2 −36 ) 2 x 3 −10 x 2 −12x = −5⋅2 x 3 a( x+6 )( x−6 ) 2x( x 2 −5x−6 ) = −5⋅2 x 3 a( x+6 )( x−6 ) 2x( x−6 )( x+1 ) = −5⋅ 2 x 2 3 a( x+6 ) ( x−6 ) 2 x ( x−6 ) ( x+1 ) = −5 x 2 a( x+6 ) x−1 , x≠−1, 6

Example 10

x 2 −x−12 − x 2 +2x+8 . Since it is most convenient to have the leading terms of a polynomial positive, factor out −1 from the denominator. x 2 −x−12 −( x 2 −2x−8 ) Rewrite this. − x 2 −x−12 x 2 −2x−8 Factor. − ( x−4 ) ( x+3 ) ( x−4 ) ( x+2 ) − x+3 x+2 = −( x+3 ) x+2 = −x−3 x+2 , x≠−2, 4 x 2 −x−12 − x 2 +2x+8 . Since it is most convenient to have the leading terms of a polynomial positive, factor out −1 from the denominator. x 2 −x−12 −( x 2 −2x−8 ) Rewrite this. − x 2 −x−12 x 2 −2x−8 Factor. − ( x−4 ) ( x+3 ) ( x−4 ) ( x+2 ) − x+3 x+2 = −( x+3 ) x+2 = −x−3 x+2 , x≠−2, 4

Example 11

a−b b−a . The numerator and denominator have the same terms but they occur with opposite signs. Factor −1 from the denominator. a−b −( −b+a ) = a−b −( a−b ) =− a−b a−b =−1, a≠b a−b b−a . The numerator and denominator have the same terms but they occur with opposite signs. Factor −1 from the denominator. a−b −( −b+a ) = a−b −( a−b ) =− a−b a−b =−1, a≠b

Practice Set A

Reduce each of the following fractions to lowest terms.

Exercise 1

30y 35y 30y 35y

Solution

6 7 6 7

Exercise 2

x 2 −9 x 2 +5x+6 x 2 −9 x 2 +5x+6

Solution

x−3 x+2 x−3 x+2

Exercise 3

x+2b 4x+8b x+2b 4x+8b

Solution

1 4 1 4

Exercise 4

18 a 3 b 5 c 7 3a b 3 c 5 18 a 3 b 5 c 7 3a b 3 c 5

Solution

6 a 2 b 2 c 2 6 a 2 b 2 c 2

Exercise 5

−3 a 4 +75 a 2 2 a 3 −16 a 2 +30a −3 a 4 +75 a 2 2 a 3 −16 a 2 +30a

Solution

−3a( a+5 ) 2( a−3 ) −3a( a+5 ) 2( a−3 )

Exercise 6

x 2 −5x+4 − x 2 +12x−32 x 2 −5x+4 − x 2 +12x−32

Solution

−x+1 x−8 −x+1 x−8

Exercise 7

2x−y y−2x 2x−y y−2x

Solution

−1

Excercises

For the following problems, reduce each rational expression to lowest terms.

Exercise 8

6 3x-12 6 3x-12

Solution

2 ( x−4 ) 2 ( x−4 )

Exercise 9

8 4a-16 8 4a-16

Exercise 10

9 3y-21 9 3y-21

Solution

3 ( y−7 ) 3 ( y−7 )

Exercise 11

10 5x-5 10 5x-5

Exercise 12

7 7x-14 7 7x-14

Solution

1 ( x−2 ) 1 ( x−2 )

Exercise 13

6 6x-18 6 6x-18

Exercise 14

2 y 2 8y 2 y 2 8y

Solution

1 4 y 1 4 y

Exercise 15

4 x 3 2x 4 x 3 2x

Exercise 16

16 a 2 b 3 2a b 2 16 a 2 b 3 2a b 2

Solution

8ab 8ab

Exercise 17

20 a 4 b 4 4a b 2 20 a 4 b 4 4a b 2

Exercise 18

( x+3 )( x-2 ) ( x+3 )( x+5 ) ( x+3 )( x-2 ) ( x+3 )( x+5 )

Solution

x−2 x+5 x−2 x+5

Exercise 19

( y-1 )( y-7 ) ( y-1 )( y+6 ) ( y-1 )( y-7 ) ( y-1 )( y+6 )

Exercise 20

( a+6 )( a-5 ) ( a-5 )( a+2 ) ( a+6 )( a-5 ) ( a-5 )( a+2 )

Solution

a+6 a+2 a+6 a+2

Exercise 21

( m-3 )( m-1 ) ( m-1 )( m+4 ) ( m-3 )( m-1 ) ( m-1 )( m+4 )

Exercise 22

( y-2 )( y-3 ) ( y-3 )( y-2 ) ( y-2 )( y-3 ) ( y-3 )( y-2 )

Solution

1

Exercise 23

( x+7 )( x+8 ) ( x+8 )( x+7 ) ( x+7 )( x+8 ) ( x+8 )( x+7 )

Exercise 24

-12 x 2 ( x+4 ) 4x -12 x 2 ( x+4 ) 4x

Solution

−3x( x+4 ) −3x( x+4 )

Exercise 25

-3 a 4 ( a-1 )( a+5 ) -2 a 3 ( a-1 )( a+9 ) -3 a 4 ( a-1 )( a+5 ) -2 a 3 ( a-1 )( a+9 )

Exercise 26

6 x 2 y 5 ( x-1 )( x+4 ) -2xy( x+4 ) 6 x 2 y 5 ( x-1 )( x+4 ) -2xy( x+4 )

Solution

−3x y 4 ( x−1 ) −3x y 4 ( x−1 )

Exercise 27

22 a 4 b 6 c 7 ( a+2 )( a-7 ) 4c( a+2 )( a-5 ) 22 a 4 b 6 c 7 ( a+2 )( a-7 ) 4c( a+2 )( a-5 )

Exercise 28

( x+10 ) 3 x+10 ( x+10 ) 3 x+10

Solution

( x+10 ) 2 ( x+10 ) 2

Exercise 29

( y-6 ) 7 y-6 ( y-6 ) 7 y-6

Exercise 30

( x-8 ) 2 ( x+6 ) 4 ( x-8 )( x+6 ) ( x-8 ) 2 ( x+6 ) 4 ( x-8 )( x+6 )

Solution

( x−8 ) ( x+6 ) 3 ( x−8 ) ( x+6 ) 3

Exercise 31

( a+1 ) 5 ( a-1 ) 7 ( a+1 ) 3 ( a-1 ) 4 ( a+1 ) 5 ( a-1 ) 7 ( a+1 ) 3 ( a-1 ) 4

Exercise 32

( y-2 ) 6 ( y-1 ) 4 ( y-2 ) 3 ( y-1 ) 2 ( y-2 ) 6 ( y-1 ) 4 ( y-2 ) 3 ( y-1 ) 2

Solution

( y−2 ) 3 ( y−1 ) 2 ( y−2 ) 3 ( y−1 ) 2

Exercise 33

( x+10 ) 5 ( x-6 ) 3 ( x-6 ) ( x+10 ) 2 ( x+10 ) 5 ( x-6 ) 3 ( x-6 ) ( x+10 ) 2

Exercise 34

( a+6 ) 2 ( a-7 ) 6 ( a+6 ) 5 ( a-7 ) 2 ( a+6 ) 2 ( a-7 ) 6 ( a+6 ) 5 ( a-7 ) 2

Solution

( a−7 ) 4 ( a+6 ) 3 ( a−7 ) 4 ( a+6 ) 3

Exercise 35

( m+7 ) 4 ( m-8 ) 5 ( m+7 ) 7 ( m-8 ) 2 ( m+7 ) 4 ( m-8 ) 5 ( m+7 ) 7 ( m-8 ) 2

Exercise 36

( a+2 ) ( a-1 ) 3 ( a+1 )( a-1 ) ( a+2 ) ( a-1 ) 3 ( a+1 )( a-1 )

Solution

( a+2 ) ( a−1 ) 2 ( a+1 ) ( a+2 ) ( a−1 ) 2 ( a+1 )

Exercise 37

( b+6 ) ( b-2 ) 4 ( b-1 )( b-2 ) ( b+6 ) ( b-2 ) 4 ( b-1 )( b-2 )

Exercise 38

8 ( x+2 ) 3 ( x-5 ) 6 2( x+2 ) ( x-5 ) 2 8 ( x+2 ) 3 ( x-5 ) 6 2( x+2 ) ( x-5 ) 2

Solution

4 ( x+2 ) 2 ( x−5 ) 4 4 ( x+2 ) 2 ( x−5 ) 4

Exercise 39

14 ( x-4 ) 3 ( x-10 ) 6 -7 ( x-4 ) 2 ( x-10 ) 2 14 ( x-4 ) 3 ( x-10 ) 6 -7 ( x-4 ) 2 ( x-10 ) 2

Exercise 40

x 2 +x-12 x 2 -4x+3 x 2 +x-12 x 2 -4x+3

Solution

( x+4 ) ( x−1 ) ( x+4 ) ( x−1 )

Exercise 41

x 2 +3x-10 x 2 +2x-15 x 2 +3x-10 x 2 +2x-15

Exercise 42

x 2 -10x+21 x 2 -6x-7 x 2 -10x+21 x 2 -6x-7

Solution

( x−3 ) ( x+1 ) ( x−3 ) ( x+1 )

Exercise 43

x 2 +10x+24 x 2 +6x x 2 +10x+24 x 2 +6x

Exercise 44

x 2 +9x+14 x 2 +7x x 2 +9x+14 x 2 +7x

Solution

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

Exercise 45

6 b 2 -b 6 b 2 +11b-2 6 b 2 -b 6 b 2 +11b-2

Exercise 46

3 b 2 +10b+3 3 b 2 +7b+2 3 b 2 +10b+3 3 b 2 +7b+2

Solution

b+3 b+2 b+3 b+2

Exercise 47

4 b 2 -1 2 b 2 +5b-3 4 b 2 -1 2 b 2 +5b-3

Exercise 48

16 a 2 -9 4 a 2 -a-3 16 a 2 -9 4 a 2 -a-3

Solution

( 4a−3 ) ( a−1 ) ( 4a−3 ) ( a−1 )

Exercise 49

20 x 2 +28xy+9 y 2 4 x 2 +4xy+ y 2 20 x 2 +28xy+9 y 2 4 x 2 +4xy+ y 2

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms.

Exercise 50

x+3 x+4 x+3 x+4

Solution

( x+3 ) ( x+4 ) ( x+3 ) ( x+4 )

Exercise 51

a+7 a-1 a+7 a-1

Exercise 52

3a+6 3 3a+6 3

Solution

a+2 a+2

Exercise 53

4x+12 4 4x+12 4

Exercise 54

5a-5 -5 5a-5 -5

Solution

−( a−1 ) or −a+1 −( a−1 ) or −a+1

Exercise 55

6b-6 -3 6b-6 -3

Exercise 56

8x-16 -4 8x-16 -4

Solution

−2( x−2 ) −2( x−2 )

Exercise 57

4x-7 -7 4x-7 -7

Exercise 58

-3x+10 10 -3x+10 10

Solution

−3x+10 10 −3x+10 10

Exercise 59

x-2 2-x x-2 2-x

Exercise 60

a-3 3-a a-3 3-a

Solution

−1 −1

Exercise 61

x 3 -x x x 3 -x x

Exercise 62

y 4 -y y y 4 -y y

Solution

y 3 −1 y 3 −1

Exercise 63

a 5 - a 2 a a 5 - a 2 a

Exercise 64

a 6 - a 4 a 3 a 6 - a 4 a 3

Solution

a( a+1 )( a−1 ) a( a+1 )( a−1 )

Exercise 65

4 b 2 +3b b 4 b 2 +3b b

Exercise 66

2 a 3 +5a a 2 a 3 +5a a

Solution

2 a 2 +5 2 a 2 +5

Exercise 67

a a 3 +a a a 3 +a

Exercise 68

x 4 x 5 -3x x 4 x 5 -3x

Solution

x 3 x 4 −3 x 3 x 4 −3

Exercise 69

-a - a 2 -a -a - a 2 -a

Excercises For Review

Exercise 70

((Reference)) Write ( 4 4 a 8 b 10 4 2 a 6 b 2 ) −1 ( 4 4 a 8 b 10 4 2 a 6 b 2 ) −1 so that only positive exponents appear.

Solution

1 16 a 2 b 8 1 16 a 2 b 8

Exercise 71

((Reference)) Factor y 4 −16 y 4 −16 .

Exercise 72

((Reference)) Factor 10 x 2 −17x+3 10 x 2 −17x+3 .

Solution

( 5x−1 )( 2x−3 ) ( 5x−1 )( 2x−3 )

Exercise 73

((Reference)) Supply the missing word. An equation expressed in the form ax+by=c ax+by=c is said to be expressed in
form.

Exercise 74

((Reference)) Find the domain of the rational expression 2 x 2 −3x−18 2 x 2 −3x−18 .

Solution

x≠−3, 6 x≠−3, 6

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks