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Rational Expressions: Dividing Polynomials

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 divide a polynomial by a monomial, understand the process and be able to divide a polynomial by a polynomial.

Overview

  • Dividing a Polynomial by a Monomial
  • The Process of Division
  • Review of Subtraction of Polynomials
  • Dividing a Polynomial by a Polynomial

Dividing A Polynomial By A Monomial

The following examples illustrate how to divide a polynomial by a monomial. The division process is quite simple and is based on addition of rational expressions.

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

Turning this equation around we get

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

Now we simply divide c c into aa, and c c into b b. This should suggest a rule.

Dividing a Polynomial By a Monomial

To divide a polynomial by a monomial, divide every term of the polynomial by the monomial.

Sample Set A

Example 1

3 x 2 +x−11 x . Divide every term of 3 x 2  + x−11 by x. 3 x 2 x + x x − 11 x =3x+1− 11 x 3 x 2 +x−11 x . Divide every term of 3 x 2  + x−11 by x. 3 x 2 x + x x − 11 x =3x+1− 11 x

Example 2

8 a 3 +4 a 2 −16a+9 2 a 2 . Divide every term of 8 a 3  + 4 a 2 −16a +9 by 2 a 2 . 8 a 3 2 a 2 + 4 a 2 2 a 2 − 16a 2 a 2 + 9 2 a 2 =4a+2− 8 a + 9 2 a 2 8 a 3 +4 a 2 −16a+9 2 a 2 . Divide every term of 8 a 3  + 4 a 2 −16a +9 by 2 a 2 . 8 a 3 2 a 2 + 4 a 2 2 a 2 − 16a 2 a 2 + 9 2 a 2 =4a+2− 8 a + 9 2 a 2

Example 3

4 b 6 −9 b 4 −2b+5 −4 b 2 . Divide every term of 4 b 6 −9 b 4 −2b+5 by−4 b 2 . 4 b 6 −4 b 2 − 9 b 4 −4 b 2 − 2b −4 b 2 + 5 −4 b 2 =− b 4 + 9 4 b 2 + 1 2b − 5 4 b 2 4 b 6 −9 b 4 −2b+5 −4 b 2 . Divide every term of 4 b 6 −9 b 4 −2b+5 by−4 b 2 . 4 b 6 −4 b 2 − 9 b 4 −4 b 2 − 2b −4 b 2 + 5 −4 b 2 =− b 4 + 9 4 b 2 + 1 2b − 5 4 b 2

Practice Set A

Perform the following divisions.

Exercise 1

2 x 2 +x−1 x 2 x 2 +x−1 x

Solution

2x+1− 1 x 2x+1− 1 x

Exercise 2

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

Solution

3x+4+ 10 x − 4 x 2 3x+4+ 10 x − 4 x 2

Exercise 3

a 2 b+3a b 2 +2b ab a 2 b+3a b 2 +2b ab

Solution

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

Exercise 4

14 x 2 y 2 −7xy 7xy 14 x 2 y 2 −7xy 7xy

Solution

2xy−1 2xy−1

Exercise 5

10 m 3 n 2 +15 m 2 n 3 −20mn −5m 10 m 3 n 2 +15 m 2 n 3 −20mn −5m

Solution

−2 m 2 n 2 −3m n 3 +4n −2 m 2 n 2 −3m n 3 +4n

The Process Of Division

In Section (Reference) we studied the method of reducing rational expressions. For example, we observed how to reduce an expression such as

x 2 −2x−8 x 2 −3x−4 x 2 −2x−8 x 2 −3x−4

Our method was to factor both the numerator and denominator, then divide out common factors.

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

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

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

When the numerator and denominator have no factors in common, the division may still occur, but the process is a little more involved than merely factoring. The method of dividing one polynomial by another is much the same as that of dividing one number by another. First, we’ll review the steps in dividing numbers.

  1. 35 8 . 35 8 .  We are to divide 35 by 8.
  2. Long division showing eight dividing thirty five. This division is not performed completely.   We try 4, since 32 divided by 8 is 4.
  3. Long division showing eight dividing thirty five, with four at quotient's place. This division is not performed completely. Multiply 4 and 8.
  4. Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five. This division is not performed completely Subtract 32 from 35.
  5. Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five and three is written as the subtraction of thirty five and thirty two. Since the remainder 3 is less than the divisor 8, we are done with the 32 division.
  6. 4 3 8 . 4 3 8 .   The quotient is expressed as a mixed number.

The process was to divide, multiply, and subtract.

Review Of Subtraction Of Polynomials

A very important step in the process of dividing one polynomial by another is subtraction of polynomials. Let’s review the process of subtraction by observing a few examples.

1. Subtract x−2 x−2 from x−5; x−5; that is, find ( x−5 )−( x−2 ). ( x−5 )−( x−2 ).

 Since   x−2   x−2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

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

The result is −3. −3.

2. Subtract x 3 +3 x 2 x 3 +3 x 2 from x 3 +4 x 2 +x−1. x 3 +4 x 2 +x−1.

 Since x 3 +3 x 2 x 3 +3 x 2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

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

The result is x 2 +x−1. x 2 +x−1.

3. Subtract x 2 +3x x 2 +3x from x 2 +1. x 2 +1.

 We can write x 2 +1 x 2 +1 as x 2 +0x+1. x 2 +0x+1.

x 2 +1 −( x 2 +3x ) = x 2 +0x+1 −( x 2 +3x ) = x 2 +0x+1 − x 2 −3x −3x+1 x 2 +1 −( x 2 +3x ) = x 2 +0x+1 −( x 2 +3x ) = x 2 +0x+1 − x 2 −3x −3x+1

Dividing A Polynomial By A Polynomial

Now we’ll observe some examples of dividing one polynomial by another. The process is the same as the process used with whole numbers: divide, multiply, subtract, divide, multiply, subtract,....

The division, multiplication, and subtraction take place one term at a time. The process is concluded when the polynomial remainder is of lesser degree than the polynomial divisor.

Sample Set B

Perform the division.

Example 4

x−5 x−2 . We are to divide x−5 by x−2. x−5 x−2 . We are to divide x−5 by x−2.

Long division showing x minus two dividing x minus five with the comment 'Divide x into x' on the right side. This division is not performed completely. See the longdesc for a full description.

1− 3 x−2 Thus, x−5 x−2 =1− 3 x−2 1− 3 x−2 Thus, x−5 x−2 =1− 3 x−2

Example 5

x 3 +4 x 2 +x−1 x+3 . We are to divide  x 3 +4 x 2 +x−1 by x+3. x 3 +4 x 2 +x−1 x+3 . We are to divide  x 3 +4 x 2 +x−1 by x+3.

Long division showing x plus three dividing x cube plus four x square plus x minus one with the comment 'Divide x into x cube' on the right side. This division is not performed completely. See the longdesc for a full description

 x 2 +x−2+ 5 x+3 Thus, x 3 +4 x 2 +x−1 x+3 = x 2 +x−2+ 5 x+3 x 2 +x−2+ 5 x+3 Thus, x 3 +4 x 2 +x−1 x+3 = x 2 +x−2+ 5 x+3

Practice Set B

Perform the following divisions.

Exercise 6

x+6 x−1 x+6 x−1

Solution

1+ 7 x−1 1+ 7 x−1

Exercise 7

x 2 +2x+5 x+3 x 2 +2x+5 x+3

Solution

x−1+ 8 x+3 x−1+ 8 x+3

Exercise 8

x 3 + x 2 −x−2 x+8 x 3 + x 2 −x−2 x+8

Solution

x 2 −7x+55− 442 x+8 x 2 −7x+55− 442 x+8

Exercise 9

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

Solution

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

Sample Set C

Example 6

Divide 2 x 3 −4x+1 by x+6. 2 x 3 −4x+1 x+6 Notice that the  x 2  term in the numerator is missing.  We can avoid any confusion by writing 2 x 3 +0 x 2 −4x+1 x+6 Divide, multiply, and subtract. Divide 2 x 3 −4x+1 by x+6. 2 x 3 −4x+1 x+6 Notice that the  x 2  term in the numerator is missing.  We can avoid any confusion by writing 2 x 3 +0 x 2 −4x+1 x+6 Divide, multiply, and subtract.

Steps of long division showing the quantity x plus six dividing the quantity two x cubed plus zero x squared minus four x minus plus one. See the longdesc for a full description

2 x 3 −4x+1 x+6 =2 x 3 −12x+68− 407 x+6 2 x 3 −4x+1 x+6 =2 x 3 −12x+68− 407 x+6

Practice Set C

Perform the following divisions.

Exercise 10

x 2 −3 x+2 x 2 −3 x+2

Solution

x−2+ 1 x+2 x−2+ 1 x+2

Exercise 11

4 x 2 −1 x−3 4 x 2 −1 x−3

Solution

4x+12+ 35 x−3 4x+12+ 35 x−3

Exercise 12

x 3 +2x+2 x−2 x 3 +2x+2 x−2

Solution

x 2 +2x+6+ 14 x−2 x 2 +2x+6+ 14 x−2

Exercise 13

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

Solution

3 x 2 −2x+3− 10 2x+3 3 x 2 −2x+3− 10 2x+3

Exercises

For the following problems, perform the divisions.

Exercise 14

6a+12 2 6a+12 2

Solution

3a+6 3a+6

Exercise 15

12b−6 3 12b−6 3

Exercise 16

8y−4 −4 8y−4 −4

Solution

−2y+1 −2y+1

Exercise 17

21a−9 −3 21a−9 −3

Exercise 18

3 x 2 −6x −3 3 x 2 −6x −3

Solution

−x( x−2 ) −x( x−2 )

Exercise 19

4 y 2 −2y 2y 4 y 2 −2y 2y

Exercise 20

9 a 2 +3a 3a 9 a 2 +3a 3a

Solution

3a+1 3a+1

Exercise 21

20 x 2 +10x 5x 20 x 2 +10x 5x

Exercise 22

6 x 3 +2 x 2 +8x 2x 6 x 3 +2 x 2 +8x 2x

Solution

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

Exercise 23

26 y 3 +13 y 2 +39y 13y 26 y 3 +13 y 2 +39y 13y

Exercise 24

a 2 b 2 +4 a 2 b+6a b 2 −10ab ab a 2 b 2 +4 a 2 b+6a b 2 −10ab ab

Solution

ab+4a+6b−10 ab+4a+6b−10

Exercise 25

7 x 3 y+8 x 2 y 3 +3x y 4 −4xy xy 7 x 3 y+8 x 2 y 3 +3x y 4 −4xy xy

Exercise 26

5 x 3 y 3 −15 x 2 y 2 +20xy −5xy 5 x 3 y 3 −15 x 2 y 2 +20xy −5xy

Solution

− x 2 y 2 +3xy−4 − x 2 y 2 +3xy−4

Exercise 27

4 a 2 b 3 −8a b 4 +12a b 2 −2a b 2 4 a 2 b 3 −8a b 4 +12a b 2 −2a b 2

Exercise 28

6 a 2 y 2 +12 a 2 y+18 a 2 24 a 2 6 a 2 y 2 +12 a 2 y+18 a 2 24 a 2

Solution

1 4 y 2 + 1 2 y+ 3 4 1 4 y 2 + 1 2 y+ 3 4

Exercise 29

3 c 3 y 3 +99 c 3 y 4 −12 c 3 y 5 3 c 3 y 3 3 c 3 y 3 +99 c 3 y 4 −12 c 3 y 5 3 c 3 y 3

Exercise 30

16a x 2 −20a x 3 +24a x 4 6 a 4 16a x 2 −20a x 3 +24a x 4 6 a 4

Solution

8 x 2 −10 x 3 +12 x 4 3 a 3 or 12 x 4 −10 x 3 +8 x 2 3 a 3 8 x 2 −10 x 3 +12 x 4 3 a 3 or 12 x 4 −10 x 3 +8 x 2 3 a 3

Exercise 31

21a y 3 −18a y 2 −15ay 6a y 2 21a y 3 −18a y 2 −15ay 6a y 2

Exercise 32

−14 b 2 c 2 +21 b 3 c 3 −28 c 3 −7 a 2 c 3 −14 b 2 c 2 +21 b 3 c 3 −28 c 3 −7 a 2 c 3

Solution

2 b 2 −3 b 3 c+4c a 2 c 2 b 2 −3 b 3 c+4c a 2 c

Exercise 33

−30 a 2 b 4 −35 a 2 b 3 −25 a 2 −5 b 3 −30 a 2 b 4 −35 a 2 b 3 −25 a 2 −5 b 3

Exercise 34

x+6 x−2 x+6 x−2

Solution

1+ 8 x−2 1+ 8 x−2

Exercise 35

y+7 y+1 y+7 y+1

Exercise 36

x 2 −x+4 x+2 x 2 −x+4 x+2

Solution

x−3+ 10 x+2 x−3+ 10 x+2

Exercise 37

x 2 +2x−1 x+1 x 2 +2x−1 x+1

Exercise 38

x 2 −x+3 x+1 x 2 −x+3 x+1

Solution

x−2+ 5 x+1 x−2+ 5 x+1

Exercise 39

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

Exercise 40

x 2 −2 x+1 x 2 −2 x+1

Solution

x−1− 1 x+1 x−1− 1 x+1

Exercise 41

a 2 −6 a+2 a 2 −6 a+2

Exercise 42

y 2 +4 y+2 y 2 +4 y+2

Solution

y−2+ 8 y+2 y−2+ 8 y+2

Exercise 43

x 2 +36 x+6 x 2 +36 x+6

Exercise 44

x 3 −1 x+1 x 3 −1 x+1

Solution

x 2 −x+1− 2 x+1 x 2 −x+1− 2 x+1

Exercise 45

a 3 −8 a+2 a 3 −8 a+2

Exercise 46

x 3 −1 x−1 x 3 −1 x−1

Solution

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

Exercise 47

a 3 −8 a−2 a 3 −8 a−2

Exercise 48

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

Solution

x 2 +5x+11+ 20 x−2 x 2 +5x+11+ 20 x−2

Exercise 49

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

Exercise 50

a 3 +a+6 a−1 a 3 +a+6 a−1

Solution

a 2 +a+2+ 8 a−1 a 2 +a+2+ 8 a−1

Exercise 51

x 3 +2x+1 x−3 x 3 +2x+1 x−3

Exercise 52

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

Solution

y 2 +y−2+ 8 y+2 y 2 +y−2+ 8 y+2

Exercise 53

y 3 +5 y 2 −3 y−1 y 3 +5 y 2 −3 y−1

Exercise 54

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

Solution

x 2 x 2

Exercise 55

a 2 +2a a+2 a 2 +2a a+2

Exercise 56

x 2 −x−6 x 2 −2x−3 x 2 −x−6 x 2 −2x−3

Solution

1+ 1 x+1 1+ 1 x+1

Exercise 57

a 2 +5a+4 a 2 −a−2 a 2 +5a+4 a 2 −a−2

Exercise 58

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

Solution

2+ 11 y−4 2+ 11 y−4

Exercise 59

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

Exercise 60

2 x 2 −x+4 2x−1 2 x 2 −x+4 2x−1

Solution

x+ 4 2x−1 x+ 4 2x−1

Exercise 61

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

Exercise 62

6 x 2 +8x−1 3x+4 6 x 2 +8x−1 3x+4

Solution

2x− 1 3x+4 2x− 1 3x+4

Exercise 63

20 y 2 +15y−4 4y+3 20 y 2 +15y−4 4y+3

Exercise 64

4 x 3 +4 x 2 −3x−2 2x−1 4 x 3 +4 x 2 −3x−2 2x−1

Solution

2 x 2 +3x− 2 2x−1 2 x 2 +3x− 2 2x−1

Exercise 65

9 a 3 −18 a 2 +8a−1 3a−2 9 a 3 −18 a 2 +8a−1 3a−2

Exercise 66

4 x 4 −4 x 3 +2 x 2 −2x−1 x−1 4 x 4 −4 x 3 +2 x 2 −2x−1 x−1

Solution

4 x 3 +2x− 1 x−1 4 x 3 +2x− 1 x−1

Exercise 67

3 y 4 +9 y 3 −2 y 2 −6y+4 y+3 3 y 4 +9 y 3 −2 y 2 −6y+4 y+3

Exercise 68

3 y 2 +3y+5 y 2 +y+1 3 y 2 +3y+5 y 2 +y+1

Solution

3+ 2 y 2 +y+1 3+ 2 y 2 +y+1

Exercise 69

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

Exercise 70

8 z 6 −4 z 5 −8 z 4 +8 z 3 +3 z 2 −14z 2z−3 8 z 6 −4 z 5 −8 z 4 +8 z 3 +3 z 2 −14z 2z−3

Solution

4 z 5 +4 z 4 +2 z 3 +7 z 2 +12z+11+ 33 2z−3 4 z 5 +4 z 4 +2 z 3 +7 z 2 +12z+11+ 33 2z−3

Exercise 71

9 a 7 +15 a 6 +4 a 5 −3 a 4 − a 3 +12 a 2 +a−5 3a+1 9 a 7 +15 a 6 +4 a 5 −3 a 4 − a 3 +12 a 2 +a−5 3a+1

Exercise 72

(2 x 5 +5 x 4 −1) ÷ (2x+5) (2 x 5 +5 x 4 −1) ÷ (2x+5)

Solution

x 4 − 1 2x+5 x 4 − 1 2x+5

Exercise 73

(6 a 4 −2 a 3 −3 a 2 +a+4) ÷ (3a−1) (6 a 4 −2 a 3 −3 a 2 +a+4) ÷ (3a−1)

Exercises For Review

Exercise 74

((Reference)) Find the product. x 2 +2x−8 x 2 −9 · 2x+6 4x−8 . x 2 +2x−8 x 2 −9 · 2x+6 4x−8 .

Solution

x+4 2( x−3 ) x+4 2( x−3 )

Exercise 75

((Reference)) Find the sum. x−7 x+5 + x+4 x−2 . x−7 x+5 + x+4 x−2 .

Exercise 76

((Reference)) Solve the equation 1 x+3 + 1 x−3 = 1 x 2 −9 . 1 x+3 + 1 x−3 = 1 x 2 −9 .

Solution

x= 1 2 x= 1 2

Exercise 77

((Reference)) When the same number is subtracted from both the numerator and denominator of 3 10 3 10 , the result is 1 8 1 8 . What is the number that is subtracted?

Exercise 78

((Reference)) Simplify 1 x+5 4 x 2 −25 . 1 x+5 4 x 2 −25 .

Solution

x−5 4 x−5 4

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