8.6 Divide Polynomials

Izabela Mazur

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

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:

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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. $\frac{36{p}^{3}+18{p}^{2}-12p}{6{p}^{2}}$

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.

8.6 Divide Polynomials

Dividing Polynomial by Monomial

Everyday Math

Writing Exercises

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Divide a Polynomial by a Monomial

Key Concepts

Practice Makes Perfect

Dividing Polynomial by Monomial

Everyday Math

Writing Exercises

Answers:

Attributions

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