1. Learning objectives

Learning objectives

Divide a polynomial by a monomial

Divide a polynomial by a binomial

2. Divide a Polynomial by a Monomial

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,	$\frac{y}{5}+\frac{2}{5},$
simplifies to	$\frac{y+2}{5}.$

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.

If $a,b,\,\mathrm{and}\,c$ are numbers where $c\ne{}0$ , then

\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\,\mathrm{and}\,\frac{a+b}{c}=\frac{a}{c}+\frac{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,	$\frac{y+2}{5}$
can be written	$\frac{y}{5}+\frac{2}{5}.$

Table

6.3

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

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

3. Example 6.77

Example 6.77

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

Solution


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

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.

4. Example 6.78

Example 6.78

Find the quotient: $(18{x}^{3}-36{x}^{2})\mathrm{÷}6x.$

Solution


	$(18{x}^{3}-36{x}^{2})\mathrm{÷}6x$
Rewrite as a fraction.	$\frac{18{x}^{3}-36{x}^{2}}{6x}$
Divide each term of the numerator by the denominator.	$\frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}$
Simplify.	$3{x}^{2}-6x$

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

5. Example 6.79

Example 6.79

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

Solution


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

6. Example 6.80

Example 6.80

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

Solution


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

7. Example 6.81

Example 6.81

Find the quotient: $(15{x}^{3}y-35x{y}^{2})\mathrm{÷}(-5xy).$

Solution


	$(15{x}^{3}y-35x{y}^{2})\mathrm{÷}(-5xy)$
Rewrite as a fraction.	$\frac{15{x}^{3}y-35x{y}^{2}}{-5xy}$
Separate the terms.	$\frac{15{x}^{3}y}{-5xy}-\frac{35x{y}^{2}}{-5xy}$
Simplify.	$-3{x}^{2}+7y$

8. Example 6.82

Example 6.82

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

Solution


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

9. Example 6.83

Example 6.83

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

Solution


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

10. Divide a Polynomial by a Binomial

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 check division by multiplying the quotient by the divisor.

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

\begin{array}{c} 35\cdot{}25 \\ 875\,✓ \\ \end{array}

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.

11. Example 6.84

Example 6.84

Find the quotient: $({x}^{2}+9x+20)\mathrm{÷}(x+5).$

Solution

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.

12. Example 6.85

Example 6.85

Find the quotient: $(2{x}^{2}-5x-3)\mathrm{÷}(x-3).$

Solution

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.

13. Example 6.86

Example 6.86

Find the quotient: $({x}^{3}-{x}^{2}+x+4)\mathrm{÷}(x+1).$

Solution

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 ${x}^{4}-{x}^{2}+5x-2$ . It is missing an ${x}^{3}$ term. We will add in $0{x}^{3}$ as a placeholder.

14. Example 6.87

Example 6.87

Find the quotient: $({x}^{4}-{x}^{2}+5x-2)\mathrm{÷}(x+2).$

Solution

Notice that there is no ${x}^{3}$ term in the dividend. We will add $0{x}^{3}$ as a placeholder.

In Example 6.88, we will divide by $2a-3$ . As we divide we will have to consider the constants as well as the variables.

15. Example 6.88

Example 6.88

Find the quotient: $(8{a}^{3}+27)\mathrm{÷}(2a+3).$

Solution

This time we will show the division all in one step. We need to add two placeholders in order to divide.

To check, multiply $(2a+3)(4{a}^{2}-6a+9)$ .

The result should be $8{a}^{3}+27$ .

16. Section 6.6 Exercises

Section 6.6 Exercises

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

Divide a Polynomial by a Binomial

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

Everyday Math

Writing Exercises

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

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

17. Practice 1

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

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

Find the quotient: $(27{b}^{3}-33{b}^{2})\mathrm{÷}3b.$

Find the quotient: $(25{y}^{3}-55{y}^{2})\mathrm{÷}5y.$

18. Practice 2

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

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

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

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

19. Practice 3

Find the quotient: $(32{a}^{2}b-16a{b}^{2})\mathrm{÷}(-8ab).$

Find the quotient: $(-48{a}^{8}{b}^{4}-36{a}^{6}{b}^{5})\mathrm{÷}(-6{a}^{3}{b}^{3}).$

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

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

20. Practice 4

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

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

Find the quotient: $({y}^{2}+10y+21)\mathrm{÷}(y+3).$

Find the quotient: $({m}^{2}+9m+20)\mathrm{÷}(m+4).$

21. Practice 5

Find the quotient: $(2{x}^{2}-3x-20)\mathrm{÷}(x-4).$

Find the quotient: $(3{x}^{2}-16x-12)\mathrm{÷}(x-6).$

Find the quotient: $({x}^{3}+5{x}^{2}+8x+6)\mathrm{÷}(x+2).$

Find the quotient: $(2{x}^{3}+8{x}^{2}+x-8)\mathrm{÷}(x+1).$

22. Practice 6

Find the quotient: $({x}^{3}+3x+14)\mathrm{÷}(x+2).$

Find the quotient: $({x}^{4}-3{x}^{3}-1000)\mathrm{÷}(x+5).$

Find the quotient: $({x}^{3}-64)\mathrm{÷}(x-4).$

Find the quotient: $(125{x}^{3}-8)\mathrm{÷}(5x-2).$

23. Practice 7

$\frac{30b+75}{5}$

$\frac{42{x}^{2}-14x}{7}$

$(55{w}^{2}-10w)\mathrm{÷}5w$

$(8{x}^{3}+6{x}^{2})\mathrm{÷}2x$

24. Practice 8

$\frac{20{b}^{2}-12b}{-4}$

$\frac{51{m}^{4}+72{m}^{3}}{-3}$

$\frac{412{z}^{8}-48{z}^{5}}{4{z}^{3}}$

$\frac{51{y}^{4}+42{y}^{2}}{3{y}^{2}}$

25. Practice 9

$(35{x}^{4}-21x)\mathrm{÷}(-7x)$

$(48{y}^{4}-24{y}^{3})\mathrm{÷}(-8{y}^{2})$

$(45{x}^{3}{y}^{4}+60x{y}^{2})\mathrm{÷}(5xy)$

$\frac{49{c}^{2}{d}^{2}-70{c}^{3}{d}^{3}-35{c}^{2}{d}^{4}}{7c{d}^{2}}$

26. Practice 10

$\frac{72{r}^{5}{s}^{2}+132{r}^{4}{s}^{3}-96{r}^{3}{s}^{5}}{12{r}^{2}{s}^{2}}$

$\frac{12{q}^{2}+3q-1}{3q}$

$\frac{20{y}^{2}+12y-1}{-4y}$

$\frac{63{a}^{3}-108{a}^{2}+99a}{9{a}^{2}}$

27. Practice 11

$({d}^{2}+8d+12)\mathrm{÷}(d+2)$

$({a}^{2}-2a-35)\mathrm{÷}(a+5)$

$({x}^{2}-14x+49)\mathrm{÷}(x-7)$

$(4{x}^{2}-17x-15)\mathrm{÷}(x-5)$

28. Practice 12

$({p}^{2}+11p+16)\mathrm{÷}(p+8)$

$({x}^{2}+2x-30)\mathrm{÷}(x-5)$

$(2{n}^{3}-10n+24)\mathrm{÷}(n+3)$

$(7{q}^{3}-5q-2)\mathrm{÷}(q-1)$

29. Practice 13

$({m}^{3}+1000)\mathrm{÷}(m+10)$

$({x}^{3}-216)\mathrm{÷}(x-6)$

$(125{y}^{3}-64)\mathrm{÷}(5y-4)$

Divide $\frac{10{x}^{2}+x-12}{2x}$ and explain with words how you get each term of the quotient.