Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Elementary Algebra » Adding and Subtracting Rational Expressions

Navigation

Table of Contents

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 collection is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook Collaborative

    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 collection is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange Grove

    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 collection is included inLens: Connexions Featured Content
    By: Connexions

    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.
 

Adding and Subtracting 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 familiar with the basic rule for adding and subtracting rational expressions, be able to add and subtract fractions with the same and with different denominators.

Overview

  • Basic Rule
  • Fractions with the Same Denominator
  • Fractions with Different Denominators

Basic Rule

We are now in a position to study the process of adding and subtracting rational expressions. There is a most basic rule to which we must strictly adhere if we wish to conveniently add or subtract rational expressions.

To add or subtract rational expressions conveniently, they should have the same denominators.

Thus, to add or subtract two or more rational expressions conveniently, we must ensure that they all have the same denominator. The denominator that is most convenient is the LCD.

Fractions With The Same Denominator

The Rule for Adding and Subtracting Rational Expressions

To add (or subtract) two or more rational expressions with the same denominators, add (or subtract) the numerators and place the result over the LCD. Reduce if necessary. Symbolically,

 a c + b c = a+b c a c + b c = a+b c
a c − b c = a−b c a c − b c = a−b c

Note that we combine only the numerators.

Sample Set A

Add or subtract the following rational expressions.

Example 1

1 6 + 3 6 The denominators are the same. Add the numerators. 1 6 + 3 6 = 1+3 6 = 4 6 Reduce. 1 6 + 3 6 = 2 3 1 6 + 3 6 The denominators are the same. Add the numerators. 1 6 + 3 6 = 1+3 6 = 4 6 Reduce. 1 6 + 3 6 = 2 3

Example 2

5 x + 8 x The denominators are the same. Add the numerators. 5 x + 8 x = 5+8 x = 13 x 5 x + 8 x The denominators are the same. Add the numerators. 5 x + 8 x = 5+8 x = 13 x

Example 3

2ab y 2 w − 5b y 2 w The denominators are the same. Subtract the numerators. 2ab y 2 w − 5b y 2 w = 2ab−5b y 2 w 2ab y 2 w − 5b y 2 w The denominators are the same. Subtract the numerators. 2ab y 2 w − 5b y 2 w = 2ab−5b y 2 w

Example 4

3 x 2 +x+2 x−7 + x 2 −4x+1 x−7 The denominators are the same. Add the numerators. 3 x 2 +x+2 x−7 + x 2 −4x+1 x−7 = 3 x 2 +x+2+ x 2 −4x+1 x−7 = 4 x 2 −3x+3 x−7 3 x 2 +x+2 x−7 + x 2 −4x+1 x−7 The denominators are the same. Add the numerators. 3 x 2 +x+2 x−7 + x 2 −4x+1 x−7 = 3 x 2 +x+2+ x 2 −4x+1 x−7 = 4 x 2 −3x+3 x−7

Example 5

5y+3 2y−5 − 2y+4 2y−5 The denominators are the same. Subtract the numerators.  But be careful to subtract the entire numerator. Use parentheses! 5y+3 2y−5 − 2y+4 2y−5 = 5y+3−( 2y+4 ) 2y−5 = 5y+3−2y−4 2y−5 = 3y−1 2y−5 Note: 5y+3 2y−5 − 2y+4 2y−5 ︸ Observe this part . The term− 2y+4 2y−5  could be written as + −( 2y+4 ) 2y−5 = −2y−4 2y−5 5y+3 2y−5 − 2y+4 2y−5 The denominators are the same. Subtract the numerators.  But be careful to subtract the entire numerator. Use parentheses! 5y+3 2y−5 − 2y+4 2y−5 = 5y+3−( 2y+4 ) 2y−5 = 5y+3−2y−4 2y−5 = 3y−1 2y−5 Note: 5y+3 2y−5 − 2y+4 2y−5 ︸ Observe this part . The term− 2y+4 2y−5  could be written as + −( 2y+4 ) 2y−5 = −2y−4 2y−5

A common mistake is to write

− 2y+4 2y−5  as  −2y+4 2y−5 − 2y+4 2y−5  as  −2y+4 2y−5
This is not correct, as the negative sign is not being applied to the entire numerator.

Example 6

3 x 2 +4x+5 ( x+6 )( x−2 ) + 2 x 2 +x+6 x 2 +4x−12 − x 2 −4x−6 x 2 +4x−12 Factor the denominators to determine if they're the same. 3 x 2 +4x+5 ( x+6 )( x−2 ) + 2 x 2 +x+6 ( x+6 )( x−2 ) − x 2 −4x−6 ( x+6 )( x−2 ) The denominators are the same. Combine the numerators being careful to note the negative sign. 3 x 2 +4x+5+2 x 2 +x+6−( x 2 −4x+6 ) ( x+6 )( x−2 ) 3 x 2 +4x+5+2 x 2 +x+6− x 2 +4x+6 ( x+6 )( x−2 ) 4 x 2 +9x+17 ( x+6 )( x−2 ) 3 x 2 +4x+5 ( x+6 )( x−2 ) + 2 x 2 +x+6 x 2 +4x−12 − x 2 −4x−6 x 2 +4x−12 Factor the denominators to determine if they're the same. 3 x 2 +4x+5 ( x+6 )( x−2 ) + 2 x 2 +x+6 ( x+6 )( x−2 ) − x 2 −4x−6 ( x+6 )( x−2 ) The denominators are the same. Combine the numerators being careful to note the negative sign. 3 x 2 +4x+5+2 x 2 +x+6−( x 2 −4x+6 ) ( x+6 )( x−2 ) 3 x 2 +4x+5+2 x 2 +x+6− x 2 +4x+6 ( x+6 )( x−2 ) 4 x 2 +9x+17 ( x+6 )( x−2 )

Practice Set A

Add or Subtract the following rational expressions.

Exercise 1

4 9 + 2 9 4 9 + 2 9

Solution

2 3 2 3

Exercise 2

3 b + 2 b 3 b + 2 b

Solution

5 b 5 b

Exercise 3

5x 2 y 2 − 3x 2 y 2 5x 2 y 2 − 3x 2 y 2

Solution

x y 2 x y 2

Exercise 4

x+y x−y + 2x+3y x−y x+y x−y + 2x+3y x−y

Solution

3x+4y x−y 3x+4y x−y

Exercise 5

4 x 2 −x+4 3x+10 − x 2 +2x+5 3x+10 4 x 2 −x+4 3x+10 − x 2 +2x+5 3x+10

Solution

3 x 2 −3x−1 3x+10 3 x 2 −3x−1 3x+10

Exercise 6

x( x+1 ) x( 2x+3 ) + 3 x 2 −x+7 2 x 2 +3x x( x+1 ) x( 2x+3 ) + 3 x 2 −x+7 2 x 2 +3x

Solution

4 x 2 +7 x( 2x+3 ) 4 x 2 +7 x( 2x+3 )

Exercise 7

4x+3 x 2 −x−6 − 8x−4 ( x+2 )( x−3 ) 4x+3 x 2 −x−6 − 8x−4 ( x+2 )( x−3 )

Solution

−4x+7 ( x+2 )( x−3 ) −4x+7 ( x+2 )( x−3 )

Exercise 8

5 a 2 +a−4 2a( a−6 ) + 2 a 2 +3a+4 2 a 2 −12a + a 2 +2 2 a 2 −12a 5 a 2 +a−4 2a( a−6 ) + 2 a 2 +3a+4 2 a 2 −12a + a 2 +2 2 a 2 −12a

Solution

4 a 2 +2a+1 a( a−6 ) 4 a 2 +2a+1 a( a−6 )

Exercise 9

8 x 2 +x−1 x 2 −6x+8 + 2 x 2 +3x x 2 −6x+8 − 5 x 2 +3x−4 ( x−4 )( x−2 ) 8 x 2 +x−1 x 2 −6x+8 + 2 x 2 +3x x 2 −6x+8 − 5 x 2 +3x−4 ( x−4 )( x−2 )

Solution

5 x 2 +x+3 ( x−4 )( x−2 ) 5 x 2 +x+3 ( x−4 )( x−2 )

Fractions with Different Denominators

Sample Set B

Add or Subtract the following rational expressions.

Example 7

4a 3y + 2a 9 y 2 . The denominators are not the same. Find the LCD. By inspection, the LCD is 9 y 2 . 9 y 2 + 2a 9 y 2 The denominator of the first rational expression has been multiplied by 3y, so the numerator must be multiplied by 3y. 4a·3y=12ay 12ay 9 y 2 + 2a 9 y 2 The denominators are now the same. Add the numerators. 12ay+2a 9 y 2   4a 3y + 2a 9 y 2 . The denominators are not the same. Find the LCD. By inspection, the LCD is 9 y 2 . 9 y 2 + 2a 9 y 2 The denominator of the first rational expression has been multiplied by 3y, so the numerator must be multiplied by 3y. 4a·3y=12ay 12ay 9 y 2 + 2a 9 y 2 The denominators are now the same. Add the numerators. 12ay+2a 9 y 2  

Example 8

3b b+2 + 5b b−3 . The denominators are not the same. The LCD is (b+2)(b−3). (b+2)(b−3) + (b+2)(b−3) The denominator of the first rational expression has been multiplied by b−3, so the numerator must be multiplied by b−3.  3b(b−3) 3b(b−3) (b+2)(b−3) + (b+2)(b−3) The denominator of the second rational expression has been multiplied by b+2, so the numerator must be multiplied by b+2.  5b(b+2) 3b(b−3) (b+2)(b−3) + 5b(b+2) (b+2)(b−3) The denominators are now the same. Add the numerators. 3b(b−3)+5b(b+2) (b−3)(b+2) = 3 b 2 −9b+5 b 2 +10b (b−3)(b+2) = 8 b 2 +b (b−3)(b−2) 3b b+2 + 5b b−3 . The denominators are not the same. The LCD is (b+2)(b−3). (b+2)(b−3) + (b+2)(b−3) The denominator of the first rational expression has been multiplied by b−3, so the numerator must be multiplied by b−3.  3b(b−3) 3b(b−3) (b+2)(b−3) + (b+2)(b−3) The denominator of the second rational expression has been multiplied by b+2, so the numerator must be multiplied by b+2.  5b(b+2) 3b(b−3) (b+2)(b−3) + 5b(b+2) (b+2)(b−3) The denominators are now the same. Add the numerators. 3b(b−3)+5b(b+2) (b−3)(b+2) = 3 b 2 −9b+5 b 2 +10b (b−3)(b+2) = 8 b 2 +b (b−3)(b−2)

Example 9

x+3 x−1 + x−2 4x+4 . The denominators are not the same. Find the LCD. x+3 x−1 + x−2 4( x+1 ) The LCD is ( x+1 )( x−1 ) 4( x+1 )( x−1 ) + 4( x+1 )( x−1 ) The denominator of the first rational expression has been multiplied by 4( x+1 ) so  the numerator must be multiplied by 4( x+1 ). 4( x+3 )( x+1 ) 4( x+3 )( x+1 ) 4( x+1 )( x−1 ) + 4( x+1 )( x−1 ) The denominator of the second rational expression has been multiplied by x−1 so the numerator must be multiplied by x−1.( x−1 )( x−2 ) 4( x+3 )( x+1 ) 4( x+1 )( x−1 ) + ( x−1 )( x−2 ) 4( x+1 )( x−1 ) The denominators are now the same. Add the numerators. 4( x+3 )( x+1 )+( x−1 )( x−2 ) 4( x+1 )( x−1 ) 4( x 2 +4x+3 )+ x 2 −3x+2 4( x+1 )( x−1 ) 4 x 2 +16x+12+ x 2 −3x+2 4( x+1 )( x−1 ) = 5 x 2 +13x+14 4( x+1 )( x−1 ) x+3 x−1 + x−2 4x+4 . The denominators are not the same. Find the LCD. x+3 x−1 + x−2 4( x+1 ) The LCD is ( x+1 )( x−1 ) 4( x+1 )( x−1 ) + 4( x+1 )( x−1 ) The denominator of the first rational expression has been multiplied by 4( x+1 ) so  the numerator must be multiplied by 4( x+1 ). 4( x+3 )( x+1 ) 4( x+3 )( x+1 ) 4( x+1 )( x−1 ) + 4( x+1 )( x−1 ) The denominator of the second rational expression has been multiplied by x−1 so the numerator must be multiplied by x−1.( x−1 )( x−2 ) 4( x+3 )( x+1 ) 4( x+1 )( x−1 ) + ( x−1 )( x−2 ) 4( x+1 )( x−1 ) The denominators are now the same. Add the numerators. 4( x+3 )( x+1 )+( x−1 )( x−2 ) 4( x+1 )( x−1 ) 4( x 2 +4x+3 )+ x 2 −3x+2 4( x+1 )( x−1 ) 4 x 2 +16x+12+ x 2 −3x+2 4( x+1 )( x−1 ) = 5 x 2 +13x+14 4( x+1 )( x−1 )

Example 10

x+5 x 2 −7x+12 + 3x−1 x 2 −2x−3 Determine the LCD. x+5 (x−4)(x−3) + 3x−1 (x−3)(x+1) The LCD is (x−4)(x−3)(x+1). (x−4)(x−3)(x+1) + (x−4)(x−3)(x+1) The first numerator must be multiplied by x+1 and the second by x−4.  (x+5)(x+1) (x−4)(x−3)(x+1) + (3x−1)(x−4) (x−4)(x−3)(x+1) The denominators are now the same. Add the numerators. (x+5)(x+1)+(3x−1)(x−4) (x−4)(x−3)(x+1) x 2 +6x+5+3 x 2 −13x+4 (x−4)(x−3)(x+1) 4 x 2 −7x+9 (x−4)(x−3)(x+1) x+5 x 2 −7x+12 + 3x−1 x 2 −2x−3 Determine the LCD. x+5 (x−4)(x−3) + 3x−1 (x−3)(x+1) The LCD is (x−4)(x−3)(x+1). (x−4)(x−3)(x+1) + (x−4)(x−3)(x+1) The first numerator must be multiplied by x+1 and the second by x−4.  (x+5)(x+1) (x−4)(x−3)(x+1) + (3x−1)(x−4) (x−4)(x−3)(x+1) The denominators are now the same. Add the numerators. (x+5)(x+1)+(3x−1)(x−4) (x−4)(x−3)(x+1) x 2 +6x+5+3 x 2 −13x+4 (x−4)(x−3)(x+1) 4 x 2 −7x+9 (x−4)(x−3)(x+1)

Example 11

a+4 a 2 +5a+6 − a−4 a 2 −5a−24 Determine the LCD. a+4 ( a+3 )( a+2 ) − a−4 ( a+3 )( a−8 ) The LCD is ( a+3 )( a+2 )( a−8 ). ( a+3 )( a+2 )( a−8 ) − ( a+3 )( a+2 )( a−8 ) The first numerator must be multiplied by a−8 and the second by a+2.  ( a+4 )( a−8 ) ( a+3 )( a+2 )( a−8 ) − ( a−4 )( a+2 ) ( a+3 )( a+2 )( a−8 ) The denominators are now the same. Subtract the numerators.  ( a+4 )( a−8 )−( a−4 )( a+2 ) ( a+3 )( a+2 )( a−8 ) a 2 −4a−32−( a 2 −2a−8 ) ( a+3 )( a+2 )( a−8 ) a 2 −4a−32− a 2 +2a+8 ( a+3 )( a+2 )( a−8 ) −2a−24 ( a+3 )( a+2 )( a−8 ) Factor −2 from the numerator.  −2( a+12 ) ( a+3 )( a+2 )( a−8 ) a+4 a 2 +5a+6 − a−4 a 2 −5a−24 Determine the LCD. a+4 ( a+3 )( a+2 ) − a−4 ( a+3 )( a−8 ) The LCD is ( a+3 )( a+2 )( a−8 ). ( a+3 )( a+2 )( a−8 ) − ( a+3 )( a+2 )( a−8 ) The first numerator must be multiplied by a−8 and the second by a+2.  ( a+4 )( a−8 ) ( a+3 )( a+2 )( a−8 ) − ( a−4 )( a+2 ) ( a+3 )( a+2 )( a−8 ) The denominators are now the same. Subtract the numerators.  ( a+4 )( a−8 )−( a−4 )( a+2 ) ( a+3 )( a+2 )( a−8 ) a 2 −4a−32−( a 2 −2a−8 ) ( a+3 )( a+2 )( a−8 ) a 2 −4a−32− a 2 +2a+8 ( a+3 )( a+2 )( a−8 ) −2a−24 ( a+3 )( a+2 )( a−8 ) Factor −2 from the numerator.  −2( a+12 ) ( a+3 )( a+2 )( a−8 )

Example 12

3x 7−x + 5x x−7 . The denominators are nearly the same. They differ only in sign.  Our technique is to factor −1 from one of them.  3x 7−x = 3x −( x−7 ) = −3x x−7 Factor −1 from the first term.  3x 7−x + 5x x−7 = −3x x−7 + 5x x−7 = −3x+5x x−7 = 2x x−7 3x 7−x + 5x x−7 . The denominators are nearly the same. They differ only in sign.  Our technique is to factor −1 from one of them.  3x 7−x = 3x −( x−7 ) = −3x x−7 Factor −1 from the first term.  3x 7−x + 5x x−7 = −3x x−7 + 5x x−7 = −3x+5x x−7 = 2x x−7

Practice Set B

Add or Subtract the following rational expressions.

Exercise 10

3x 4 a 2 + 5x 12 a 3 3x 4 a 2 + 5x 12 a 3

Solution

9ax+5x 12 a 3 9ax+5x 12 a 3

Exercise 11

5b b+1 + 3b b−2 5b b+1 + 3b b−2

Solution

8 b 2 −7b ( b+1 )( b−2 ) 8 b 2 −7b ( b+1 )( b−2 )

Exercise 12

a−7 a+2 + a−2 a+3 a−7 a+2 + a−2 a+3

Solution

2 a 2 −4a−25 ( a+2 )( a+3 ) 2 a 2 −4a−25 ( a+2 )( a+3 )

Exercise 13

4x+1 x+3 − x+5 x−3 4x+1 x+3 − x+5 x−3

Solution

3 x 2 −19x−18 ( x+3 )( x−3 ) 3 x 2 −19x−18 ( x+3 )( x−3 )

Exercise 14

2y−3 y + 3y+1 y+4 2y−3 y + 3y+1 y+4

Solution

5 y 2 +6y−12 y( y+4 ) 5 y 2 +6y−12 y( y+4 )

Exercise 15

a−7 a 2 −3a+2 + a+2 a 2 −6a+8 a−7 a 2 −3a+2 + a+2 a 2 −6a+8

Solution

2 a 2 −10a+26 ( a−2 )( a−1 )( a−4 ) 2 a 2 −10a+26 ( a−2 )( a−1 )( a−4 )

Exercise 16

6 b 2 +6b+9 − 2 b 2 +4b+4 6 b 2 +6b+9 − 2 b 2 +4b+4

Solution

4 b 2 +12b+6 ( b+3 ) 2 ( b+2 ) 2 4 b 2 +12b+6 ( b+3 ) 2 ( b+2 ) 2

Exercise 17

x x+4 − x−2 3x−3 x x+4 − x−2 3x−3

Solution

2 x 2 −5x+8 3( x+4 )( x−1 ) 2 x 2 −5x+8 3( x+4 )( x−1 )

Exercise 18

5x 4−x + 7x x−4 5x 4−x + 7x x−4

Solution

2x x−4 2x x−4

Sample Set C

Combine the following rational expressions.

Example 13

3+ 7 x−1 . Rewrite the expression. 3 1 + 7 x−1 The LCD is x−1. 3( x−1 ) x−1 + 7 x−1 = 3x−3 x−1 + 7 x−1 = 3x−3+7 x−1 = 3x+4 x−1 3+ 7 x−1 . Rewrite the expression. 3 1 + 7 x−1 The LCD is x−1. 3( x−1 ) x−1 + 7 x−1 = 3x−3 x−1 + 7 x−1 = 3x−3+7 x−1 = 3x+4 x−1

Example 14

3y+4− y 2 −y+3 y−6 . Rewrite the expression. 3y+4 1 − y 2 −y+3 y−6 The LCD is y−6. ( 3y+4 )( y−6 ) y−6 − y 2 −y+3 y−6 = ( 3y+4 )( y−6 )−( y 2 −y+3 ) y−6 = 3 y 2 −14y−24− y 2 +y−3 y−6 = 2 y 2 −13y−27 y−6 3y+4− y 2 −y+3 y−6 . Rewrite the expression. 3y+4 1 − y 2 −y+3 y−6 The LCD is y−6. ( 3y+4 )( y−6 ) y−6 − y 2 −y+3 y−6 = ( 3y+4 )( y−6 )−( y 2 −y+3 ) y−6 = 3 y 2 −14y−24− y 2 +y−3 y−6 = 2 y 2 −13y−27 y−6

Practice Set C

Exercise 19

Simplify 8+ 3 x−6 . 8+ 3 x−6 .

Solution

8x−45 x−6 8x−45 x−6

Exercise 20

Simplify 2a−5− a 2 +2a−1 a+3 . 2a−5− a 2 +2a−1 a+3 .

Solution

a 2 −a−14 a+3 a 2 −a−14 a+3

Exercises

For the following problems, add or subtract the rational expressions.

Exercise 21

3 8 + 1 8 3 8 + 1 8

Solution

1 2 1 2

Exercise 22

1 9 + 4 9 1 9 + 4 9

Exercise 23

7 10 − 2 5 7 10 − 2 5

Solution

3 10 3 10

Exercise 24

3 4 − 5 12 3 4 − 5 12

Exercise 25

3 4x + 5 4x 3 4x + 5 4x

Solution

2 x 2 x

Exercise 26

2 7y + 3 7y 2 7y + 3 7y

Exercise 27

6y 5x + 8y 5x 6y 5x + 8y 5x

Solution

14y 5x 14y 5x

Exercise 28

9a 7b + 3a 7b 9a 7b + 3a 7b

Exercise 29

15n 2m − 6n 2m 15n 2m − 6n 2m

Solution

9n 2m 9n 2m

Exercise 30

8p 11q − 3p 11q 8p 11q − 3p 11q

Exercise 31

y+4 y−6 + y+8 y−6 y+4 y−6 + y+8 y−6

Solution

2y+12 y−6 2y+12 y−6

Exercise 32

y−1 y+4 + y+7 y+4 y−1 y+4 + y+7 y+4

Exercise 33

a+6 a−1 + 3a+5 a−1 a+6 a−1 + 3a+5 a−1

Solution

4a+11 a−1 4a+11 a−1

Exercise 34

5a+1 a+7 + 2a−6 a+7 5a+1 a+7 + 2a−6 a+7

Exercise 35

x+1 5x + x+3 5x x+1 5x + x+3 5x

Solution

2x+4 5x 2x+4 5x

Exercise 36

a−6 a+2 + a−2 a+2 a−6 a+2 + a−2 a+2

Exercise 37

b+1 b−3 + b+2 b−3 b+1 b−3 + b+2 b−3

Solution

2b+3 b−3 2b+3 b−3

Exercise 38

a+2 a−5 − a+3 a−5 a+2 a−5 − a+3 a−5

Exercise 39

b+7 b−6 − b−1 b−6 b+7 b−6 − b−1 b−6

Solution

8 b−6 8 b−6

Exercise 40

2b+3 b+1 − b−4 b+1 2b+3 b+1 − b−4 b+1

Exercise 41

3y+4 y+8 − 2y−5 y+8 3y+4 y+8 − 2y−5 y+8

Solution

y+9 y+8 y+9 y+8

Exercise 42

2a−7 a−9 + 3a+5 a−9 2a−7 a−9 + 3a+5 a−9

Exercise 43

8x−1 x+2 − 15x+7 x+2 8x−1 x+2 − 15x+7 x+2

Solution

−7x−8 x+2 −7x−8 x+2

Exercise 44

7 2 x 2 + 1 6 x 3 7 2 x 2 + 1 6 x 3

Exercise 45

2 3x + 4 6 x 2 2 3x + 4 6 x 2

Solution

2( x+1 ) 3 x 2 2( x+1 ) 3 x 2

Exercise 46

5 6 y 3 − 2 18 y 5 5 6 y 3 − 2 18 y 5

Exercise 47

2 5 a 2 − 1 10 a 3 2 5 a 2 − 1 10 a 3

Solution

4a−1 10 a 3 4a−1 10 a 3

Exercise 48

3 x+1 + 5 x−2 3 x+1 + 5 x−2

Exercise 49

4 x−6 + 1 x−1 4 x−6 + 1 x−1

Solution

5( x−2 ) ( x−6 )( x−1 ) 5( x−2 ) ( x−6 )( x−1 )

Exercise 50

2a a+1 − 3a a+4 2a a+1 − 3a a+4

Exercise 51

6y y+4 + 2y y+3 6y y+4 + 2y y+3

Solution

2y( 4y+13 ) ( y+4 )( y+3 ) 2y( 4y+13 ) ( y+4 )( y+3 )

Exercise 52

x−1 x−3 + x+4 x−4 x−1 x−3 + x+4 x−4

Exercise 53

x+2 x−5 + x−1 x+2 x+2 x−5 + x−1 x+2

Solution

2 x 2 −2x+9 ( x−5 )( x+2 ) 2 x 2 −2x+9 ( x−5 )( x+2 )

Exercise 54

a+3 a−3 − a+2 a−2 a+3 a−3 − a+2 a−2

Exercise 55

y+1 y−1 − y+4 y−4 y+1 y−1 − y+4 y−4

Solution

−6y ( y−1 )( y−4 ) −6y ( y−1 )( y−4 )

Exercise 56

x−1 ( x+2 )( x−3 ) + x+4 x−3 x−1 ( x+2 )( x−3 ) + x+4 x−3

Exercise 57

y+2 ( y+1 )( y+6 ) + y−2 y+6 y+2 ( y+1 )( y+6 ) + y−2 y+6

Solution

y 2 ( y+1 )( y+6 ) y 2 ( y+1 )( y+6 )

Exercise 58

2a+1 ( a+3 )( a−3 ) − a+2 a+3 2a+1 ( a+3 )( a−3 ) − a+2 a+3

Exercise 59

3a+5 ( a+4 )( a−1 ) − 2a−1 a−1 3a+5 ( a+4 )( a−1 ) − 2a−1 a−1

Solution

−2 a 2 −4a+9 ( a+4 )( a−1 ) −2 a 2 −4a+9 ( a+4 )( a−1 )

Exercise 60

2x x 2 −3x+2 + 3 x−2 2x x 2 −3x+2 + 3 x−2

Exercise 61

4a a 2 −2a−3 + 3 a+1 4a a 2 −2a−3 + 3 a+1

Solution

7a−9 ( a+1 )( a−3 ) 7a−9 ( a+1 )( a−3 )

Exercise 62

3y y 2 −7y+12 − y 2 y−3 3y y 2 −7y+12 − y 2 y−3

Exercise 63

x−1 x 2 +6x+8 + x+3 x 2 +2x−8 x−1 x 2 +6x+8 + x+3 x 2 +2x−8

Solution

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

Exercise 64

a−4 a 2 +2a−3 + a+2 a 2 +3a−4 a−4 a 2 +2a−3 + a+2 a 2 +3a−4

Exercise 65

b−3 b 2 +9b+20 + b+4 b 2 +b−12 b−3 b 2 +9b+20 + b+4 b 2 +b−12

Solution

2 b 2 +3b+29 ( b−3 )( b+4 )( b+5 ) 2 b 2 +3b+29 ( b−3 )( b+4 )( b+5 )

Exercise 66

y−1 y 2 +4y−12 − y+3 y 2 +6y−16 y−1 y 2 +4y−12 − y+3 y 2 +6y−16

Exercise 67

x+3 x 2 +9x+14 − x−5 x 2 −4 x+3 x 2 +9x+14 − x−5 x 2 −4

Solution

−x+29 ( x−2 )( x+2 )( x+7 ) −x+29 ( x−2 )( x+2 )( x+7 )

Exercise 68

x−1 x 2 −4x+3 + x+3 x 2 −5x+6 + 2x x 2 −3x+2 x−1 x 2 −4x+3 + x+3 x 2 −5x+6 + 2x x 2 −3x+2

Exercise 69

4x x 2 +6x+8 + 3 x 2 +x−6 + x−1 x 2 +x−12 4x x 2 +6x+8 + 3 x 2 +x−6 + x−1 x 2 +x−12

Solution

5 x 4 −3 x 3 −34 x 2 +34x−60 ( x−2 )( x+2 )( x−3 )( x+3 )( x+4 ) 5 x 4 −3 x 3 −34 x 2 +34x−60 ( x−2 )( x+2 )( x−3 )( x+3 )( x+4 )

Exercise 70

y+2 y 2 −1 + y−3 y 2 −3y−4 − y+3 y 2 −5y+4 y+2 y 2 −1 + y−3 y 2 −3y−4 − y+3 y 2 −5y+4

Exercise 71

a−2 a 2 −9a+18 + a−2 a 2 −4a−12 − a−2 a 2 −a−6 a−2 a 2 −9a+18 + a−2 a 2 −4a−12 − a−2 a 2 −a−6

Solution

( a+5 )( a−2 ) ( a+2 )( a−3 )( a−6 ) ( a+5 )( a−2 ) ( a+2 )( a−3 )( a−6 )

Exercise 72

y−2 y 2 +6y + y+4 y 2 +5y−6 y−2 y 2 +6y + y+4 y 2 +5y−6

Exercise 73

a+1 a 3 +3 a 2 − a+6 a 2 −a a+1 a 3 +3 a 2 − a+6 a 2 −a

Solution

− a 3 −8 a 2 −18a−1 a 2 ( a+3 )( a−1 ) − a 3 −8 a 2 −18a−1 a 2 ( a+3 )( a−1 )

Exercise 74

4 3 b 2 −12b − 2 6 b 2 −6b 4 3 b 2 −12b − 2 6 b 2 −6b

Exercise 75

3 2 x 5 −4 x 4 + −2 8 x 3 +24 x 2 3 2 x 5 −4 x 4 + −2 8 x 3 +24 x 2

Solution

− x 3 +2 x 2 +6x+18 4 x 4 ( x−2 )( x+3 ) − x 3 +2 x 2 +6x+18 4 x 4 ( x−2 )( x+3 )

Exercise 76

x+2 12 x 3 + x+1 4 x 2 +8x−12 − x+3 16 x 2 −32x+16 x+2 12 x 3 + x+1 4 x 2 +8x−12 − x+3 16 x 2 −32x+16

Exercise 77

2x x 2 −9 − x+1 4 x 2 −12x − x−4 8 x 3 2x x 2 −9 − x+1 4 x 2 −12x − x−4 8 x 3

Solution

14 x 4 −9 x 3 −2 x 2 +9x−36 8 x 3 ( x+3 )( x−3 ) 14 x 4 −9 x 3 −2 x 2 +9x−36 8 x 3 ( x+3 )( x−3 )

Exercise 78

4+ 3 x+2 4+ 3 x+2

Exercise 79

8+ 2 x+6 8+ 2 x+6

Solution

8x+50 x+6 8x+50 x+6

Exercise 80

1+ 4 x−7 1+ 4 x−7

Exercise 81

3+ 5 x−6 3+ 5 x−6

Solution

3x−13 x−6 3x−13 x−6

Exercise 82

−2+ 4x x+5 −2+ 4x x+5

Exercise 83

−1+ 3a a−1 −1+ 3a a−1

Solution

2a+1 a−1 2a+1 a−1

Exercise 84

6− 4y y+2 6− 4y y+2

Exercise 85

2x+ x 2 −4 x+1 2x+ x 2 −4 x+1

Solution

3 x 2 +2x−4 x+1 3 x 2 +2x−4 x+1

Exercise 86

−3y+ 4 y 2 +2y−5 y+3 −3y+ 4 y 2 +2y−5 y+3

Exercise 87

x+2+ x 2 +4 x−1 x+2+ x 2 +4 x−1

Solution

2 x 2 +x+2 x−1 2 x 2 +x+2 x−1

Exercise 88

b+6+ 2b+5 b−2 b+6+ 2b+5 b−2

Exercise 89

3x−1 x−4 −8 3x−1 x−4 −8

Solution

−5x+31 x−4 −5x+31 x−4

Exercise 90

4y+5 y+1 −9 4y+5 y+1 −9

Exercise 91

2 y 2 +11y−1 y+4 −3y 2 y 2 +11y−1 y+4 −3y

Solution

−( y 2 +y+1 ) y+4 −( y 2 +y+1 ) y+4

Exercise 92

5 y 2 −2y+1 y 2 +y−6 −2 5 y 2 −2y+1 y 2 +y−6 −2

Exercise 93

4 a 3 +2 a 2 +a−1 a 2 +11a+28 +3a 4 a 3 +2 a 2 +a−1 a 2 +11a+28 +3a

Solution

7 a 3 +35 a 2 +85a−1 ( a+7 )( a+4 ) 7 a 3 +35 a 2 +85a−1 ( a+7 )( a+4 )

Exercise 94

2x 1−x + 6x x−1 2x 1−x + 6x x−1

Exercise 95

5m 6−m + 3m m−6 5m 6−m + 3m m−6

Solution

−2m m−6 −2m m−6

Exercise 96

−a+7 8−3a + 2a+1 3a−8 −a+7 8−3a + 2a+1 3a−8

Exercise 97

−2y+4 4−5y − 9 5y−4 −2y+4 4−5y − 9 5y−4

Solution

2y−13 5y−4 2y−13 5y−4

Exercise 98

m−1 1−m − 2 m−1 m−1 1−m − 2 m−1

Exercises For Review

Exercise 99

((Reference)) Simplify ( x 3 y 2 z 5 ) 6 ( x 2 yz ) 2 . ( x 3 y 2 z 5 ) 6 ( x 2 yz ) 2 .

Solution

x 22 y 14 z 32 x 22 y 14 z 32

Exercise 100

((Reference)) Write 6 a −3 b 4 c −2 a −1 b −5 c 3 6 a −3 b 4 c −2 a −1 b −5 c 3 so that only positive exponents appear.

Exercise 101

((Reference)) Construct the graph of y=−2x+4. y=−2x+4.
An xy coordinate plane with gridlines, labeled negative five to five on both axes.

Solution

A graph of a line passing through three points with coordinates zero, four; one, two; and two, zero.

Exercise 102

((Reference)) Find the product: x 2 −3x−4 x 2 +6x+5  ·  x 2 +5x+6 x 2 −2x−8 . x 2 −3x−4 x 2 +6x+5  ·  x 2 +5x+6 x 2 −2x−8 .

Exercise 103

((Reference)) Replace N with the proper quantity: x+3 x−5 = N x 2 −7x+10 . x+3 x−5 = N x 2 −7x+10 .

Solution

( x+3 )( x−2 ) ( x+3 )( x−2 )

Collection Navigation

Content actions

Download:

Collection 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 ...

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:

Collection 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

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