CHAPTER 8 Polynomials

8.6 Divide Polynomials

Learning Objectives

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

  • Divide a polynomial by a monomial

Divide a Polynomial by a Monomial

In the last chapter, 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, \dfrac{y}{5}+\dfrac{2}{5},
simplifies to \dfrac{y+2}{5}.

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, and c are numbers where c\ne 0, then

\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} and \dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}

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, \dfrac{y+2}{5}
can be written \dfrac{y}{5}+\dfrac{2}{5}.

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 1

Find the quotient: \dfrac{7{y}^{2}+21}{7}.

Solution
\dfrac{7{y}^{2}+21}{7}
Divide each term of the numerator by the denominator. \dfrac{7{y}^{2}}{7}+\dfrac{21}{7}
Simplify each fraction. {y}^{2}+3

TRY IT 1.1

Find the quotient: \dfrac{8{z}^{2}+24}{4}.

Show answer

2{z}^{2}+6

TRY IT 1.2

Find the quotient: \dfrac{18{z}^{2}-27}{9}.

Show answer

2{z}^{2}-3

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 2

Find the quotient: \left(18{x}^{3}-36{x}^{2}\right)\div 6x.

Solution
\left(18{x}^{3}-36{x}^{2}\right)\div 6x
Rewrite as a fraction. \dfrac{18{x}^{3}-36{x}^{2}}{6x}
Divide each term of the numerator by the denominator. \dfrac{18{x}^{3}}{6x}-\dfrac{36{x}^{2}}{6x}
Simplify. 3{x}^{2}-6x

TRY IT 2.1

Find the quotient: \left(27{b}^{3}-33{b}^{2}\right)\div 3b.

Show answer

9{b}^{2}-11b

TRY IT 2.2

Find the quotient: \left(25{y}^{3}-55{y}^{2}\right)\div 5y.

Show answer

5{y}^{2}-11y

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

EXAMPLE 3

Find the quotient: \dfrac{12{d}^{2}-16d}{-4}.

Solution
\dfrac{12{d}^{2}-16d}{-4}
Divide each term of the numerator by the denominator. \dfrac{12{d}^{2}}{-4}-\dfrac{16d}{-4}
Simplify. Remember, subtracting a negative is like adding a positive! -3{d}^{2}+4d

TRY IT 3.1

Find the quotient: \dfrac{25{y}^{2}-15y}{-5}.

Show answer

-5{y}^{2}+3y

TRY IT 3.2

Find the quotient: \dfrac{42{b}^{2}-18b}{-6}.

Show answer

-7{b}^{2}+3b

EXAMPLE 4

Find the quotient: \dfrac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}.

Solution
\dfrac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}
Separate the terms. \dfrac{105{y}^{5}}{5{y}^{2}}+\dfrac{75{y}^{3}}{5{y}^{2}}
Simplify. 21{y}^{3}+15y

TRY IT 4.1

Find the quotient: \dfrac{60{d}^{7}+24{d}^{5}}{4{d}^{3}}.

Show answer

15{d}^{4}+6{d}^{2}

TRY IT 4.2

Find the quotient: \dfrac{216{p}^{7}-48{p}^{5}}{6{p}^{3}}.

Show answer

36{p}^{4}-8{p}^{2}

EXAMPLE 5

Find the quotient: \left(15{x}^{3}y-35x{y}^{2}\right)\div \left(-5xy\right).

Solution
\left(15{x}^{3}y-35x{y}^{2}\right)\div \left(-5xy\right)
Rewrite as a fraction. \dfrac{15{x}^{3}y-35x{y}^{2}}{-5xy}
Separate the terms. \dfrac{15{x}^{3}y}{-5xy}-\dfrac{35x{y}^{2}}{-5xy}
Simplify. -3{x}^{2}+7y

TRY IT 5.1

Find the quotient: \left(32{a}^{2}b-16a{b}^{2}\right)\div\left(-8ab\right).

Show answer

-4a+2b

TRY IT 5.2

Find the quotient: \left(-48{a}^{8}{b}^{4}-36{a}^{6}{b}^{5}\right)\div \left(-6{a}^{3}{b}^{3}\right).

Show answer

8{a}^{5}b+6{a}^{3}{b}^{2}

EXAMPLE 6

Find the quotient: \dfrac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}.

Solution
\dfrac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}
Separate the terms. \dfrac{36{x}^{3}{y}^{2}}{9{x}^{2}y}+\dfrac{27{x}^{2}{y}^{2}}{9{x}^{2}y}-\dfrac{9{x}^{2}{y}^{3}}{9{x}^{2}y}
Simplify. 4xy+3y-{y}^{2}

 

TRY IT 6.1

Find the quotient: \dfrac{40{x}^{3}{y}^{2}+24{x}^{2}{y}^{2}-16{x}^{2}{y}^{3}}{8{x}^{2}y}.

Show answer

5xy+3y-2{y}^{2}

TRY IT 6.2

Find the quotient: \dfrac{35{a}^{4}{b}^{2}+14{a}^{4}{b}^{3}-42{a}^{2}{b}^{4}}{7{a}^{2}{b}^{2}}.

Show answer

5{a}^{2}+2{a}^{2}b-6{b}^{2}

EXAMPLE 7

Find the quotient: \dfrac{10{x}^{2}+5x-20}{5x}.

Solution
\dfrac{10{x}^{2}+5x-20}{5x}
Separate the terms. \dfrac{10{x}^{2}}{5x}+\dfrac{5x}{5x}-\dfrac{20}{5x}
Simplify. 2x+1+\dfrac{4}{x}

TRY IT 7.1

Find the quotient: \dfrac{18{c}^{2}+6c-9}{6c}.

Show answer

3c+1-\dfrac{3}{2c}

TRY IT 7.2

Find the quotient: \dfrac{10{d}^{2}-5d-2}{5d}.

Show answer

2d-1-\dfrac{2}{5d}

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

Key Concepts

  • Fraction Addition
    • If a,b, and c are numbers where c\ne 0, then
      \dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} and \dfrac{a+b}{c}=\frac{a}{c}+\dfrac{b}{c}
  • Division of a Polynomial by a Monomial
    • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Practice Makes Perfect

Dividing Polynomial by Monomial

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

1. \frac{30b+75}{5} 2.\frac{45y+36}{9}
3. \frac{42{x}^{2}-14x}{7} 4. \frac{8{d}^{2}-4d}{2}
5. \left(55{w}^{2}-10w\right)\div5w 6. \left(16{y}^{2}-20y\right)\div4y
7. \left(8{x}^{3}+6{x}^{2}\right)\div2x 8. \left(9{n}^{4}+6{n}^{3}\right)\div3n
9. \frac{20{b}^{2}-12b}{-4} 10. \frac{18{y}^{2}-12y}{-6}
11. \frac{51{m}^{4}+72{m}^{3}}{-3} 12. \frac{35{a}^{4}+65{a}^{2}}{-5}
13. \frac{412{z}^{8}-48{z}^{5}}{4{z}^{3}} 14. \frac{310{y}^{4}-200{y}^{3}}{5{y}^{2}}
15. \frac{51{y}^{4}+42{y}^{2}}{3{y}^{2}} 16. \frac{46{x}^{3}+38{x}^{2}}{2{x}^{2}}
17. \left(35{x}^{4}-21x\right)\div \left(-7x\right) 18. \left(24{p}^{2}-33p\right)\div \left(-3p\right)
19. \left(48{y}^{4}-24{y}^{3}\right)\div \left(-8{y}^{2}\right) 20. \left(63{m}^{4}-42{m}^{3}\right)\div \left(-7{m}^{2}\right)
21. \left(45{x}^{3}{y}^{4}+60x{y}^{2}\right)\div \left(5xy\right) 22. \left(63{a}^{2}{b}^{3}+72a{b}^{4}\right)\div \left(9ab\right)
23. \frac{49{c}^{2}{d}^{2}-70{c}^{3}{d}^{3}-35{c}^{2}{d}^{4}}{7c{d}^{2}} 24. \frac{52{p}^{5}{q}^{4}+36{p}^{4}{q}^{3}-64{p}^{3}{q}^{2}}{4{p}^{2}q}
25. \frac{72{r}^{5}{s}^{2}+132{r}^{4}{s}^{3}-96{r}^{3}{s}^{5}}{12{r}^{2}{s}^{2}} 26. \frac{66{x}^{3}{y}^{2}-110{x}^{2}{y}^{3}-44{x}^{4}{y}^{3}}{11{x}^{2}{y}^{2}}
27. \frac{12{q}^{2}+3q-1}{3q} 28. \frac{4{w}^{2}+2w-5}{2w}
29. \frac{20{y}^{2}+12y-1}{-4y} 30. \frac{10{x}^{2}+5x-4}{-5x}
31. \frac{63{a}^{3}-108{a}^{2}+99a}{9{a}^{2}} 32. \frac{36{p}^{3}+18{p}^{2}-12p}{6{p}^{2}}

Everyday Math

33. Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression \frac{{n}^{2}-n}{2}, where n represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

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

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

Writing Exercises

35. Divide \frac{10{x}^{2}+x-12}{2x} and explain with words how you get each term of the quotient. 36. James divides 48y+6 by 6 this way: \frac{48y+\overline{)6}}{\overline{)6}}=48y. What is wrong with his reasoning?

Answers:

1. 6b+15 3. 6{x}^{2}-2x 5. 11w-2
7. 4{x}^{2}+3x 9. -5{b}^{2}+3b 11. -17{m}^{4}-24{m}^{3}
13. 103{z}^{5}-12{z}^{2} 15. 17{y}^{2}+14 17. -5{x}^{3}+3
19. -6{y}^{2}+3y 21. 9{x}^{2}{y}^{3}+12y 23. 7c-10{c}^{2}d-5c{d}^{2}
25. 6{r}^{3}+11{r}^{2}s-8r{s}^{3} 27. 4q+1-\frac{1}{3q} 29. -5y-3+\frac{1}{4y}
31. 7a-12+\frac{11}{a} 33. 45 35. Answers will vary.

Attributions

This chapter has been adapted from “Divide Polynomials” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information. 

License

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Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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