Connexions

You are here: Home » Content » Elementary Algebra » Multiplying and Dividing Rational Expressions

• Preface
• Acknowledgments

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?

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

Endorsed by (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

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

"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 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

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 to display tags associated with this content.

• Featured Content

This collection is included inLens: Connexions Featured Content
By: Connexions

"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 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.

Inside Collection (Textbook):

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

Multiplying and Dividing 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 multiply and divide rational expressions.

Overview

• Multiplication Of Rational Expressions
• Division Of Rational Expressions

Multiplication Of Rational Expressions

Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following:

Definition 1: Method for Multiplying Rational Numbers
1. Reduce each fraction to lowest terms.
2. Multiply the numerators together.
3. Multiply the denominators together.

Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step.

Definition 2: Method for Multiplying Rational Expressions
1. Factor all numerators and denominators.
2. Reduce to lowest terms first by dividing out all common factors. (It is perfectly legitimate to cancel the numerator of one fraction with the denominator of another.)
3. Multiply numerators together.
4. Multiply denominators. It is often convenient, but not necessary, to leave denominators in factored form.

Sample Set A

Perform the following multiplications.

Example 1

3 4 · 1 2 = 3·1 4·2 = 3 8 3 4 · 1 2 = 3·1 4·2 = 3 8

Example 2

8 9 · 1 6 = 8 4 9 · 1 6 3 = 4·1 9·3 = 4 27 8 9 · 1 6 = 8 4 9 · 1 6 3 = 4·1 9·3 = 4 27

Example 3

3x 5y · 7 12y = 3 1 x 5y · 7 12 4 y = x·7 5y·4y = 7x 20 y 2 3x 5y · 7 12y = 3 1 x 5y · 7 12 4 y = x·7 5y·4y = 7x 20 y 2

Example 4

x+4 x-2 · x+7 x+4 Divide out the common factor  x+4. x+4 x-2 · x+7 x+4 Multiply numerators and denominators together. x+7 x-2 x+4 x-2 · x+7 x+4 Divide out the common factor  x+4. x+4 x-2 · x+7 x+4 Multiply numerators and denominators together. x+7 x-2

Example 5

x 2 +x-6 x 2 -4x+3 · x 2 -2x-3 x 2 +4x-12 . Factor. ( x+3 )( x-2 ) ( x-3 )( x-1 ) · ( x-3 )( x+1 ) ( x+6 )( x-2 ) Divide out the common factors  x-2  and  x-3. ( x+3 ) ( x-2 ) ( x-3 ) ( x-1 ) · ( x-3 ) ( x+1 ) ( x+6 ) ( x-2 ) Multiply. ( x+3 )( x+1 ) ( x-1 )( x+6 ) or x 2 +4x+3 ( x-1 )( x+6 ) or x 2 +4x+3 x 2 +5x-6 Each of these three forms is an acceptable form of the same answer. x 2 +x-6 x 2 -4x+3 · x 2 -2x-3 x 2 +4x-12 . Factor. ( x+3 )( x-2 ) ( x-3 )( x-1 ) · ( x-3 )( x+1 ) ( x+6 )( x-2 ) Divide out the common factors  x-2  and  x-3. ( x+3 ) ( x-2 ) ( x-3 ) ( x-1 ) · ( x-3 ) ( x+1 ) ( x+6 ) ( x-2 ) Multiply. ( x+3 )( x+1 ) ( x-1 )( x+6 ) or x 2 +4x+3 ( x-1 )( x+6 ) or x 2 +4x+3 x 2 +5x-6 Each of these three forms is an acceptable form of the same answer.

Example 6

2x+6 8x-16 · x 2 -4 x 2 -x-12 . Factor. 2( x+3 ) 8( x-2 ) · ( x+2 )( x-2 ) ( x-4 )( x+3 ) Divide out the common factors 2, x+3 and x-2. 2 1 ( x+3 ) 8 4 ( x-2 ) · ( x+2 ) ( x-2 ) ( x+3 ) ( x-4 ) Multiply. x+2 4( x-4 ) or x+2 4x-16 Both these forms are acceptable forms of the same answer. 2x+6 8x-16 · x 2 -4 x 2 -x-12 . Factor. 2( x+3 ) 8( x-2 ) · ( x+2 )( x-2 ) ( x-4 )( x+3 ) Divide out the common factors 2, x+3 and x-2. 2 1 ( x+3 ) 8 4 ( x-2 ) · ( x+2 ) ( x-2 ) ( x+3 ) ( x-4 ) Multiply. x+2 4( x-4 ) or x+2 4x-16 Both these forms are acceptable forms of the same answer.

Example 7

3 x 2 · x+7 x-5 . Rewrite 3 x 2  as  3 x 2 1 . 3 x 2 1 · x+7 x-5 Multiply. 3 x 2 ( x+7 ) x-5 3 x 2 · x+7 x-5 . Rewrite 3 x 2  as  3 x 2 1 . 3 x 2 1 · x+7 x-5 Multiply. 3 x 2 ( x+7 ) x-5

Example 8

( x-3 )· 4x-9 x 2 -6x+9 . ( x-3 ) 1 · 4x-9 ( x-3 ) ( x-3 ) 4x-9 x-3 ( x-3 )· 4x-9 x 2 -6x+9 . ( x-3 ) 1 · 4x-9 ( x-3 ) ( x-3 ) 4x-9 x-3

Example 9

- x 2 -3x-2 x 2 +8x+15 · 4x+20 x 2 +2x . Factor –1 from the first numerator. -( x 2 +3x+2 ) x 2 +8x+15 · 4x+20 x 2 +2x Factor. -( x+1 ) ( x+2 ) ( x+3 ) ( x+5 ) · 4( x+5 ) x ( x+2 ) Multiply. -4( x+1 ) x( x+3 ) = -4x-1 x( x+3 ) or -4x-1 x 2 +3x - x 2 -3x-2 x 2 +8x+15 · 4x+20 x 2 +2x . Factor –1 from the first numerator. -( x 2 +3x+2 ) x 2 +8x+15 · 4x+20 x 2 +2x Factor. -( x+1 ) ( x+2 ) ( x+3 ) ( x+5 ) · 4( x+5 ) x ( x+2 ) Multiply. -4( x+1 ) x( x+3 ) = -4x-1 x( x+3 ) or -4x-1 x 2 +3x

Practice Set A

Perform each multiplication.

Exercise 1

5 3 · 6 7 5 3 · 6 7

10 7 10 7

Exercise 2

a 3 b 2 c 2 · c 5 a 5 a 3 b 2 c 2 · c 5 a 5

Solution

c 3 a 2 b 2 c 3 a 2 b 2

Exercise 3

y-1 y 2 +1 · y+1 y 2 -1 y-1 y 2 +1 · y+1 y 2 -1

Solution

1 y 2 +1 1 y 2 +1

Exercise 4

x 2 -x-12 x 2 +7x+6 · x 2 -4x-5 x 2 -9x+20 x 2 -x-12 x 2 +7x+6 · x 2 -4x-5 x 2 -9x+20

x+3 x+6 x+3 x+6

Exercise 5

x 2 +6x+8 x 2 -6x+8 · x 2 -2x-8 x 2 +2x-8 x 2 +6x+8 x 2 -6x+8 · x 2 -2x-8 x 2 +2x-8

Solution

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

Division Of Rational Expressions

To divide one rational expression by another, we first invert the divisor then multiply the two expressions. Symbolically, if we let P,Q,R, P,Q,R, and S S represent polynomials, we can write

P Q ÷ R S = P Q · S R = P·S Q·R P Q ÷ R S = P Q · S R = P·S Q·R

Sample Set B

Perform the following divisions.

Example 10

6 x 2 5a ÷ 2x 10 a 3 . Invert the divisor and multiply. 6 3 x 2 5 a · 10 2 a 3 2 2 x = 3x·2 a 2 1 =6 a 2 x 6 x 2 5a ÷ 2x 10 a 3 . Invert the divisor and multiply. 6 3 x 2 5 a · 10 2 a 3 2 2 x = 3x·2 a 2 1 =6 a 2 x

Example 11

x 2 +3x-10 2x-2 ÷ x 2 +9x+20 x 2 +3x-4 Invert and multiply. x 2 +3x-10 2x-2 · x 2 +3x-4 x 2 +9x+20 Factor. ( x+5 ) ( x-2 ) 2 ( x-1 ) · ( x+4 ) ( x-1 ) ( x+5 ) ( x+4 ) x-2 2 x 2 +3x-10 2x-2 ÷ x 2 +9x+20 x 2 +3x-4 Invert and multiply. x 2 +3x-10 2x-2 · x 2 +3x-4 x 2 +9x+20 Factor. ( x+5 ) ( x-2 ) 2 ( x-1 ) · ( x+4 ) ( x-1 ) ( x+5 ) ( x+4 ) x-2 2

Example 12

( 4x+7 )÷ 12x+21 x-2 . Write  4x+7  as  4x+7 1 . 4x+7 1 ÷ 12x+21 x-2 Invert and multiply. 4x+7 1 · x-2 12x+21 Factor. 4x+7 1 · x-2 3( 4x+7 ) = x-2 3 ( 4x+7 )÷ 12x+21 x-2 . Write  4x+7  as  4x+7 1 . 4x+7 1 ÷ 12x+21 x-2 Invert and multiply. 4x+7 1 · x-2 12x+21 Factor. 4x+7 1 · x-2 3( 4x+7 ) = x-2 3

Practice Set B

Perform each division.

Exercise 6

8 m 2 n 3 a 5 b 2 ÷ 2m 15 a 7 b 2 8 m 2 n 3 a 5 b 2 ÷ 2m 15 a 7 b 2

Solution

20 a 2 mn 20 a 2 mn

Exercise 7

x 2 -4 x 2 +x-6 ÷ x 2 +x-2 x 2 +4x+3 x 2 -4 x 2 +x-6 ÷ x 2 +x-2 x 2 +4x+3

x+1 x-1 x+1 x-1

Exercise 8

6 a 2 +17a+12 3a+2 ÷( 2a+3 ) 6 a 2 +17a+12 3a+2 ÷( 2a+3 )

Solution

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

Excercises

For the following problems, perform the multiplications and divisions.

Exercise 9

4 a 3 5b · 3b 2a 4 a 3 5b · 3b 2a

6 a 2 5 6 a 2 5

Exercise 10

9 x 4 4 y 3 · 10y x 2 9 x 4 4 y 3 · 10y x 2

Exercise 11

a b · b a a b · b a

1

Exercise 12

2x 5y · 5y 2x 2x 5y · 5y 2x

Exercise 13

12 a 3 7 · 28 15a 12 a 3 7 · 28 15a

Solution

16 a 2 5 16 a 2 5

Exercise 14

39 m 4 16 · 4 13 m 2 39 m 4 16 · 4 13 m 2

Exercise 15

18 x 6 7 · 1 4 x 2 18 x 6 7 · 1 4 x 2

Solution

9 x 4 14 9 x 4 14

Exercise 16

34 a 6 21 · 42 17 a 5 34 a 6 21 · 42 17 a 5

Exercise 17

16 x 6 y 3 15 x 2 · 25x 4y 16 x 6 y 3 15 x 2 · 25x 4y

Solution

20 x 5 y 2 3 20 x 5 y 2 3

Exercise 18

27 a 7 b 4 39b · 13 a 4 b 2 16 a 5 27 a 7 b 4 39b · 13 a 4 b 2 16 a 5

Exercise 19

10 x 2 y 3 7 y 5 · 49y 15 x 6 10 x 2 y 3 7 y 5 · 49y 15 x 6

Solution

14 3 x 4 y 14 3 x 4 y

Exercise 20

22 m 3 n 4 11 m 6 n · 33mn 4m n 3 22 m 3 n 4 11 m 6 n · 33mn 4m n 3

Exercise 21

-10 p 2 q 7 a 3 b 2 · 21 a 5 b 3 2p -10 p 2 q 7 a 3 b 2 · 21 a 5 b 3 2p

Solution

15 a 2 bpq 15 a 2 bpq

Exercise 22

-25 m 4 n 3 14 r 3 s 3 · 21r s 4 10mn -25 m 4 n 3 14 r 3 s 3 · 21r s 4 10mn

Exercise 23

9 a ÷ 3 a 2 9 a ÷ 3 a 2

3a 3a

Exercise 24

10 b 2 ÷ 4 b 3 10 b 2 ÷ 4 b 3

Exercise 25

21 a 4 5 b 2 ÷ 14a 15 b 3 21 a 4 5 b 2 ÷ 14a 15 b 3

Solution

9 a 3 b 2 9 a 3 b 2

Exercise 26

42 x 5 16 y 4 ÷ 21 x 4 8 y 3 42 x 5 16 y 4 ÷ 21 x 4 8 y 3

Exercise 27

39 x 2 y 2 55 p 2 ÷ 13 x 3 y 15 p 6 39 x 2 y 2 55 p 2 ÷ 13 x 3 y 15 p 6

Solution

9 p 4 y 11x 9 p 4 y 11x

Exercise 28

14m n 3 25 n 6 ÷ 32m 20 m 2 n 3 14m n 3 25 n 6 ÷ 32m 20 m 2 n 3

Exercise 29

12 a 2 b 3 -5x y 4 ÷ 6 a 2 15 x 2 12 a 2 b 3 -5x y 4 ÷ 6 a 2 15 x 2

Solution

6 b 3 x y 4 6 b 3 x y 4

Exercise 30

24 p 3 q 9m n 3 ÷ 10pq -21 n 2 24 p 3 q 9m n 3 ÷ 10pq -21 n 2

Exercise 31

x+8 x+1 · x+2 x+8 x+8 x+1 · x+2 x+8

x+2 x+1 x+2 x+1

Exercise 32

x+10 x-4 · x-4 x-1 x+10 x-4 · x-4 x-1

Exercise 33

2x+5 x+8 · x+8 x-2 2x+5 x+8 · x+8 x-2

2x+5 x2 2x+5 x2

Exercise 34

y+2 2y-1 · 2y-1 y-2 y+2 2y-1 · 2y-1 y-2

Exercise 35

x-5 x-1 ÷ x-5 4 x-5 x-1 ÷ x-5 4

4 x1 4 x1

Exercise 36

x x-4 ÷ 2x 5x+1 x x-4 ÷ 2x 5x+1

Exercise 37

a+2b a-1 ÷ 4a+8b 3a-3 a+2b a-1 ÷ 4a+8b 3a-3

3 4 3 4

Exercise 38

6m+2 m-1 ÷ 4m-4 m-1 6m+2 m-1 ÷ 4m-4 m-1

Exercise 39

x 3 · 4ab x x 3 · 4ab x

4ab x 2 4ab x 2

Exercise 40

y 4 · 3 x 2 y 2 y 4 · 3 x 2 y 2

Exercise 41

2 a 5 ÷ 6 a 2 4b 2 a 5 ÷ 6 a 2 4b

Solution

4 a 3 b 3 4 a 3 b 3

Exercise 42

16 x 2 y 3 ÷ 10xy 3 16 x 2 y 3 ÷ 10xy 3

Exercise 43

21 m 4 n 2 ÷ 3m n 2 7n 21 m 4 n 2 ÷ 3m n 2 7n

Solution

49 m 3 n 49 m 3 n

Exercise 44

( x+8 )· x+2 x+8 ( x+8 )· x+2 x+8

Exercise 45

( x-2 )· x-1 x-2 ( x-2 )· x-1 x-2

x1 x1

Exercise 46

( a-6 ) 3 · ( a+2 ) 2 a-6 ( a-6 ) 3 · ( a+2 ) 2 a-6

Exercise 47

( b+1 ) 4 · ( b-7 ) 3 b+1 ( b+1 ) 4 · ( b-7 ) 3 b+1

Solution

( b+1 ) 3 ( b7 ) 3 ( b+1 ) 3 ( b7 ) 3

Exercise 48

( b 2 +2 ) 3 · b-3 ( b 2 +2 ) 2 ( b 2 +2 ) 3 · b-3 ( b 2 +2 ) 2

Exercise 49

( x 3 -7 ) 4 · x 2 -1 ( x 3 -7 ) 2 ( x 3 -7 ) 4 · x 2 -1 ( x 3 -7 ) 2

Solution

( x 3 7 ) 2 ( x+1 )( x1 ) ( x 3 7 ) 2 ( x+1 )( x1 )

Exercise 50

( x-5 )÷ x-5 x-2 ( x-5 )÷ x-5 x-2

Exercise 51

( y-2 )÷ y-2 y-1 ( y-2 )÷ y-2 y-1

( y1 ) ( y1 )

Exercise 52

( y+6 ) 3 ÷ ( y+6 ) 2 y-6 ( y+6 ) 3 ÷ ( y+6 ) 2 y-6

Exercise 53

( a-2b ) 4 ÷ ( a-2b ) 2 a+b ( a-2b ) 4 ÷ ( a-2b ) 2 a+b

Solution

( a2b ) 2 ( a+b ) ( a2b ) 2 ( a+b )

Exercise 54

x 2 +3x+2 x 2 -4x+3 · x 2 -2x-3 2x+2 x 2 +3x+2 x 2 -4x+3 · x 2 -2x-3 2x+2

Exercise 55

6x-42 x 2 -2x-3 · x 2 -1 x-7 6x-42 x 2 -2x-3 · x 2 -1 x-7

Solution

6( x1 ) ( x3 ) 6( x1 ) ( x3 )

Exercise 56

3a+3b a 2 -4a-5 ÷ 9a+9b a 2 -3a-10 3a+3b a 2 -4a-5 ÷ 9a+9b a 2 -3a-10

Exercise 57

a 2 -4a-12 a 2 -9 ÷ a 2 -5a-6 a 2 +6a+9 a 2 -4a-12 a 2 -9 ÷ a 2 -5a-6 a 2 +6a+9

Solution

( a+2 )( a+3 ) ( a3 )( a+1 ) ( a+2 )( a+3 ) ( a3 )( a+1 )

Exercise 58

b 2 -5b+6 b 2 -b-2 · b 2 -2b-3 b 2 -9b+20 b 2 -5b+6 b 2 -b-2 · b 2 -2b-3 b 2 -9b+20

Exercise 59

m 2 -4m+3 m 2 +5m-6 · m 2 +4m-12 m 2 -5m+6 m 2 -4m+3 m 2 +5m-6 · m 2 +4m-12 m 2 -5m+6

1

Exercise 60

r 2 +7r+10 r 2 -2r-8 ÷ r 2 +6r+5 r 2 -3r-4 r 2 +7r+10 r 2 -2r-8 ÷ r 2 +6r+5 r 2 -3r-4

Exercise 61

2 a 2 +7a+3 3 a 2 -5a-2 · a 2 -5a+6 a 2 +2a-3 2 a 2 +7a+3 3 a 2 -5a-2 · a 2 -5a+6 a 2 +2a-3

Solution

( 2a+1 )( a6 )( a+1 ) ( 3a+1 )( a1 )( a2 ) ( 2a+1 )( a6 )( a+1 ) ( 3a+1 )( a1 )( a2 )

Exercise 62

6 x 2 +x-2 2 x 2 +7x-4 · x 2 +2x-12 3 x 2 -4x-4 6 x 2 +x-2 2 x 2 +7x-4 · x 2 +2x-12 3 x 2 -4x-4

Exercise 63

x 3 y- x 2 y 2 x 2 y- y 2 · x 2 -y x-xy x 3 y- x 2 y 2 x 2 y- y 2 · x 2 -y x-xy

Solution

x( xy ) 1y x( xy ) 1y

Exercise 64

4 a 3 b-4 a 2 b 2 15a-10 · 3a-2 4ab-2 b 2 4 a 3 b-4 a 2 b 2 15a-10 · 3a-2 4ab-2 b 2

Exercise 65

x+3 x-4 · x-4 x+1 · x-2 x+3 x+3 x-4 · x-4 x+1 · x-2 x+3

x2 x+1 x2 x+1

Exercise 66

x-7 x+8 · x+1 x-7 · x+8 x-2 x-7 x+8 · x+1 x-7 · x+8 x-2

Exercise 67

2a-b a+b · a+3b a-5b · a-5b 2a-b 2a-b a+b · a+3b a-5b · a-5b 2a-b

Solution

a+3b a+b a+3b a+b

Exercise 68

3a ( a+1 ) 2 a-5 · 6 ( a-5 ) 2 5a+5 · 15a+30 4a-20 3a ( a+1 ) 2 a-5 · 6 ( a-5 ) 2 5a+5 · 15a+30 4a-20

Exercise 69

-3 a 2 4b · -8 b 3 15a -3 a 2 4b · -8 b 3 15a

Solution

2a b 2 5 2a b 2 5

Exercise 70

-6 x 3 5 y 2 · 20y -2x -6 x 3 5 y 2 · 20y -2x

Exercise 71

-8 x 2 y 3 -5x ÷ 4 -15xy -8 x 2 y 3 -5x ÷ 4 -15xy

Solution

6 x 2 y 4 6 x 2 y 4

Exercise 72

-4 a 3 3b ÷ 2a 6 b 2 -4 a 3 3b ÷ 2a 6 b 2

Exercise 73

-3a-3 2a+2 · a 2 -3a+2 a 2 -5a-6 -3a-3 2a+2 · a 2 -3a+2 a 2 -5a-6

Solution

3( a2 )( a1 ) 2( a6 )( a+1 ) 3( a2 )( a1 ) 2( a6 )( a+1 )

Exercise 74

x 2 -x-2 x 2 -3x-4 · - x 2 +2x+3 -4x-8 x 2 -x-2 x 2 -3x-4 · - x 2 +2x+3 -4x-8

Exercise 75

-5x-10 x 2 -4x+3 · x 2 +4x+1 x 2 +x-2 -5x-10 x 2 -4x+3 · x 2 +4x+1 x 2 +x-2

Solution

5( x 2 +4x+1 ) ( x3 ) ( x1 ) 2 5( x 2 +4x+1 ) ( x3 ) ( x1 ) 2

Exercise 76

- a 2 -2a+15 -6a-12 ÷ a 2 -2a-8 -2a-10 - a 2 -2a+15 -6a-12 ÷ a 2 -2a-8 -2a-10

Exercise 77

- b 2 -5b+14 3b-6 ÷ - b 2 -9b-14 -b+8 - b 2 -5b+14 3b-6 ÷ - b 2 -9b-14 -b+8

Solution

( b8 ) 3( b+2 ) ( b8 ) 3( b+2 )

Exercise 78

3a+6 4a-24 · 6-a 3a+15 3a+6 4a-24 · 6-a 3a+15

Exercise 79

4x+12 x-7 · 7-x 2x+2 4x+12 x-7 · 7-x 2x+2

Solution

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

Exercise 80

-2b-2 b 2 +b-6 · -b+2 b+5 -2b-2 b 2 +b-6 · -b+2 b+5

Exercise 81

3 x 2 -6x-9 2 x 2 -6x-4 ÷ 3 x 2 -5x-2 6 x 2 -7x-3 3 x 2 -6x-9 2 x 2 -6x-4 ÷ 3 x 2 -5x-2 6 x 2 -7x-3

Solution

3( x3 )( x+1 )( 2x3 ) 2( x 2 3x2 )( x2 ) 3( x3 )( x+1 )( 2x3 ) 2( x 2 3x2 )( x2 )

Exercise 82

-2 b 2 -2b+4 8 b 2 -28b-16 ÷ b 2 -2b+1 2 b 2 -5b-3 -2 b 2 -2b+4 8 b 2 -28b-16 ÷ b 2 -2b+1 2 b 2 -5b-3

Exercise 83

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

Solution

( x+4 )( x1 ) ( x+3 )( x 2 4x3 ) ( x+4 )( x1 ) ( x+3 )( x 2 4x3 )

Exercise 84

x 2 -3x+2 x 2 -4x+3 ÷( x-3 ) x 2 -3x+2 x 2 -4x+3 ÷( x-3 )

Exercise 85

3 x 2 -21x+18 x 2 +5x+6 ÷( x+2 ) 3 x 2 -21x+18 x 2 +5x+6 ÷( x+2 )

Solution

3( x6 )( x1 ) ( x+2 ) 2 ( x+3 ) 3( x6 )( x1 ) ( x+2 ) 2 ( x+3 )

Exercises For Review

Exercise 86

((Reference)) If a<0 a<0 , then | a |= | a |=


.

Exercise 87

((Reference)) Classify the polynomial 4xy+2y 4xy+2y as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

Solution

binomial; 2; 4, 2

Exercise 88

((Reference)) Find the product: y 2 ( 2y1 )( 2y+1 ) y 2 ( 2y1 )( 2y+1 ) .

Exercise 89

((Reference)) Translate the sentence “four less than twice some number is two more than the number” into an equation.

2x4=x+2 2x4=x+2

Exercise 90

((Reference)) Reduce the fraction x 2 -4x+4 x 2 -4 x 2 -4x+4 x 2 -4 .

Content actions

PDF | EPUB (?)

What is an EPUB file?

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

PDF | EPUB (?)

What is an EPUB file?

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

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?

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?

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