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Elementary Algebra 2e

6.6 Divide Polynomials

Elementary Algebra 2e6.6 Divide Polynomials
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:

  • Divide a polynomial by a monomial
  • Divide a polynomial by a binomial
Be Prepared 6.12

Before you get started, take this readiness quiz.

Add: 3d+xd.3d+xd.
If you missed this problem, review Example 1.77.

Be Prepared 6.13

Simplify: 30xy35xy.30xy35xy.
If you missed this problem, review Example 6.72.

Be Prepared 6.14

Combine like terms: 8a2+12a+1+3a25a+4.8a2+12a+1+3a25a+4.
If you missed this problem, review Example 1.24.

Divide a Polynomial by a Monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.

The sum, y5+25,y5+25,
simplifies to y+25.y+25.
Table 6.2

Now we will do this in reverse to split a single fraction into separate fractions.

We’ll state the fraction addition property here just as you learned it and in reverse.

Fraction Addition

If a,b,andca,b,andc are numbers where c0c0, then

ac+bc=a+bcanda+bc=ac+bcac+bc=a+bcanda+bc=ac+bc

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

For example, y+25y+25
can be written y5+25.y5+25.
Table 6.3

We use this form of fraction addition to divide polynomials by monomials.

Division of a Polynomial by a Monomial

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

Example 6.77

Find the quotient: 7y2+217.7y2+217.

Try It 6.153

Find the quotient: 8z2+244.8z2+244.

Try It 6.154

Find the quotient: 18z2279.18z2279.

Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

Example 6.78

Find the quotient: (18x336x2)÷6x.(18x336x2)÷6x.

Try It 6.155

Find the quotient: (27b333b2)÷3b.(27b333b2)÷3b.

Try It 6.156

Find the quotient: (25y355y2)÷5y.(25y355y2)÷5y.

When we divide by a negative, we must be extra careful with the signs.

Example 6.79

Find the quotient: 12d216d−4.12d216d−4.

Try It 6.157

Find the quotient: 25y215y−5.25y215y−5.

Try It 6.158

Find the quotient: 42b218b−6.42b218b−6.

Example 6.80

Find the quotient: 105y5+75y35y2.105y5+75y35y2.

Try It 6.159

Find the quotient: 60d7+24d54d3.60d7+24d54d3.

Try It 6.160

Find the quotient: 216p748p56p3.216p748p56p3.

Example 6.81

Find the quotient: (15x3y35xy2)÷(−5xy).(15x3y35xy2)÷(−5xy).

Try It 6.161

Find the quotient: (32a2b16ab2)÷(−8ab).(32a2b16ab2)÷(−8ab).

Try It 6.162

Find the quotient: (−48a8b436a6b5)÷(−6a3b3).(−48a8b436a6b5)÷(−6a3b3).

Example 6.82

Find the quotient: 36x3y2+27x2y29x2y39x2y.36x3y2+27x2y29x2y39x2y.

Try It 6.163

Find the quotient: 40x3y2+24x2y216x2y38x2y.40x3y2+24x2y216x2y38x2y.

Try It 6.164

Find the quotient: 35a4b2+14a4b342a2b47a2b2.35a4b2+14a4b342a2b47a2b2.

Example 6.83

Find the quotient: 10x2+5x205x.10x2+5x205x.

Try It 6.165

Find the quotient: 18c2+6c96c.18c2+6c96c.

Try It 6.166

Find the quotient: 10d25d25d.10d25d25d.

Divide a Polynomial by a Binomial

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

We write the long division The long division of 875 by 25.
We divide the first two digits, 87, by 25. 25 fits into 87 three times. 3 is written above the second digit of 875 in the long division bracket.
We multiply 3 times 25 and write the product under the 87. The product of 3 and 25 is 75, which is written below the first two digits of 875 in the long division bracket.
Now we subtract 75 from 87. 87 minus 75 is 12, which is written under 75.
Then we bring down the third digit of the dividend, 5. The 5 in 875 is brought down next to the 12, making 125.
Repeat the process, dividing 25 into 125. 25 fits into 125 five times. 5 is written to the right of the 3 on top of the long division bracket. 5 times 25 is 125. 125 minus 125 is zero. There is zero remainder, so 25 fits into 125 exactly five times. 875 divided by 25 equals 35.

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

35·2587535·25875

Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

Example 6.84

Find the quotient: (x2+9x+20)÷(x+5).(x2+9x+20)÷(x+5).

Try It 6.167

Find the quotient: (y2+10y+21)÷(y+3).(y2+10y+21)÷(y+3).

Try It 6.168

Find the quotient: (m2+9m+20)÷(m+4).(m2+9m+20)÷(m+4).

When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.

Example 6.85

Find the quotient: (2x25x3)÷(x3).(2x25x3)÷(x3).

Try It 6.169

Find the quotient: (2x23x20)÷(x4).(2x23x20)÷(x4).

Try It 6.170

Find the quotient: (3x216x12)÷(x6).(3x216x12)÷(x6).

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In Example 6.86, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Example 6.86

Find the quotient: (x3x2+x+4)÷(x+1).(x3x2+x+4)÷(x+1).

Try It 6.171

Find the quotient: (x3+5x2+8x+6)÷(x+2).(x3+5x2+8x+6)÷(x+2).

Try It 6.172

Find the quotient: (2x3+8x2+x8)÷(x+1).(2x3+8x2+x8)÷(x+1).

Look back at the dividends in Example 6.84, Example 6.85, and Example 6.86. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in Example 6.87 will be x4x2+5x2x4x2+5x2. It is missing an x3x3 term. We will add in 0x30x3 as a placeholder.

Example 6.87

Find the quotient: (x4x2+5x2)÷(x+2).(x4x2+5x2)÷(x+2).

Try It 6.173

Find the quotient: (x3+3x+14)÷(x+2).(x3+3x+14)÷(x+2).

Try It 6.174

Find the quotient: (x43x31000)÷(x+5).(x43x31000)÷(x+5).

In Example 6.88, we will divide by 2a32a3. As we divide we will have to consider the constants as well as the variables.

Example 6.88

Find the quotient: (8a3+27)÷(2a+3).(8a3+27)÷(2a+3).

Try It 6.175

Find the quotient: (x364)÷(x4).(x364)÷(x4).

Try It 6.176

Find the quotient: (125x38)÷(5x2).(125x38)÷(5x2).

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with dividing polynomials:

Section 6.6 Exercises

Practice Makes Perfect

In the following exercises, divide each polynomial by the monomial.

442.

45y+36945y+369

443.

30b+75530b+755

444.

8d24d28d24d2

445.

42x214x742x214x7

446.

(16y220y)÷4y(16y220y)÷4y

447.

(55w210w)÷5w(55w210w)÷5w

448.

(9n4+6n3)÷3n(9n4+6n3)÷3n

449.

(8x3+6x2)÷2x(8x3+6x2)÷2x

450.

18y212y−618y212y−6

451.

20b212b−420b212b−4

452.

35a4+65a2−535a4+65a2−5

453.

51m4+72m3−351m4+72m3−3

454.

310y4200y35y2310y4200y35y2

455.

412z848z54z3412z848z54z3

456.

46x3+38x22x246x3+38x22x2

457.

51y4+42y23y251y4+42y23y2

458.

(24p233p)÷(−3p)(24p233p)÷(−3p)

459.

(35x421x)÷(−7x)(35x421x)÷(−7x)

460.

(63m442m3)÷(−7m2)(63m442m3)÷(−7m2)

461.

(48y424y3)÷(−8y2)(48y424y3)÷(−8y2)

462.

(63a2b3+72ab4)÷(9ab)(63a2b3+72ab4)÷(9ab)

463.

(45x3y4+60xy2)÷(5xy)(45x3y4+60xy2)÷(5xy)

464.

52p5q4+36p4q364p3q24p2q52p5q4+36p4q364p3q24p2q

465.

49c2d270c3d335c2d47cd249c2d270c3d335c2d47cd2

466.

66x3y2110x2y344x4y311x2y266x3y2110x2y344x4y311x2y2

467.

72r5s2+132r4s396r3s512r2s272r5s2+132r4s396r3s512r2s2

468.

4w2+2w52w4w2+2w52w

469.

12q2+3q13q12q2+3q13q

470.

10x2+5x4−5x10x2+5x4−5x

471.

20y2+12y1−4y20y2+12y1−4y

472.

36p3+18p212p6p236p3+18p212p6p2

473.

63a3108a2+99a9a263a3108a2+99a9a2

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

474.

(y2+7y+12)÷(y+3)(y2+7y+12)÷(y+3)

475.

(d2+8d+12)÷(d+2)(d2+8d+12)÷(d+2)

476.

(x23x10)÷(x+2)(x23x10)÷(x+2)

477.

(a22a35)÷(a+5)(a22a35)÷(a+5)

478.

(t212t+36)÷(t6)(t212t+36)÷(t6)

479.

(x214x+49)÷(x7)(x214x+49)÷(x7)

480.

(6m219m20)÷(m4)(6m219m20)÷(m4)

481.

(4x217x15)÷(x5)(4x217x15)÷(x5)

482.

(q2+2q+20)÷(q+6)(q2+2q+20)÷(q+6)

483.

(p2+11p+16)÷(p+8)(p2+11p+16)÷(p+8)

484.

(y23y15)÷(y8)(y23y15)÷(y8)

485.

(x2+2x30)÷(x5)(x2+2x30)÷(x5)

486.

(3b3+b2+2)÷(b+1)(3b3+b2+2)÷(b+1)

487.

(2n310n+24)÷(n+3)(2n310n+24)÷(n+3)

488.

(2y36y36)÷(y3)(2y36y36)÷(y3)

489.

(7q35q2)÷(q1)(7q35q2)÷(q1)

490.

(z3+1)÷(z+1)(z3+1)÷(z+1)

491.

(m3+1000)÷(m+10)(m3+1000)÷(m+10)

492.

(a3125)÷(a5)(a3125)÷(a5)

493.

(x3216)÷(x6)(x3216)÷(x6)

494.

(64x327)÷(4x3)(64x327)÷(4x3)

495.

(125y364)÷(5y4)(125y364)÷(5y4)

Everyday Math

496.

Average cost Pictures Plus produces digital albums. The company’s average cost (in dollars) to make xx albums is given by the expression 7x+500x7x+500x.

  1. Find the quotient by dividing the numerator by the denominator.
  2. What will the average cost (in dollars) be to produce 20 albums?
497.

Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression n2n2n2n2, where nn represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

Writing Exercises

498.

James divides 48y+648y+6 by 6 this way: 48y+66=48y48y+66=48y. What is wrong with his reasoning?

499.

Divide 10x2+x122x10x2+x122x and explain with words how you get each term of the quotient.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “divide a polynomial by a monomial,” and “divide a polynomial by a binomial.” The rest of the cells are blank.

After reviewing this checklist, what will you do to become confident for all goals?

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