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Rational Expressions: 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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

2x+5 x−2 2x+5 x−2

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

Solution

4 x−1 4 x−1

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

Solution

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

Solution

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

Solution

x−1 x−1

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

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

Solution

( y−1 ) ( y−1 )

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

( a−2b ) 2 ( a+b ) ( a−2b ) 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( x−1 ) ( x−3 ) 6( x−1 ) ( x−3 )

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

Solution

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 )( a−6 )( a+1 ) ( 3a+1 )( a−1 )( a−2 ) ( 2a+1 )( a−6 )( a+1 ) ( 3a+1 )( a−1 )( a−2 )

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( x−y ) 1−y x( x−y ) 1−y

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

Solution

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

−( b−8 ) 3( b+2 ) −( b−8 ) 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( x−3 )( x+1 )( 2x−3 ) 2( x 2 −3x−2 )( x−2 ) 3( x−3 )( x+1 )( 2x−3 ) 2( x 2 −3x−2 )( x−2 )

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

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( x−6 )( x−1 ) ( x+2 ) 2 ( x+3 ) 3( x−6 )( x−1 ) ( 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 ( 2y−1 )( 2y+1 ) y 2 ( 2y−1 )( 2y+1 ) .

Exercise 89

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

Solution

2x−4=x+2 2x−4=x+2

Exercise 90

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

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