Lynn Marecek; MaryAnne Anthony-Smith

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 6.6

Before you get started, take this readiness quiz.

Add: $\frac{3}{d} + \frac{x}{d} .$
If you missed this problem, review Example 1.77.
Simplify: $\frac{30 x y^{3}}{5 x y} .$
If you missed this problem, review Example 6.72.
Combine like terms: $8 a^{2} + 12 a + 1 + 3 a^{2} - 5 a + 4 .$
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,	$\frac{y}{5} + \frac{2}{5},$
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.

Fraction Addition

If $a, b, and c$ are numbers where $c \neq 0$ , then

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} 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.

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

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.	$y^{2} + 3$

Try It 6.153

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

Try It 6.154

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

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: $(18 x^{3} - 36 x^{2}) \div 6 x .$

Solution

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

Try It 6.155

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

Try It 6.156

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

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

Example 6.79

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

Solution

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

Try It 6.157

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

Try It 6.158

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

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.	$21 y^{3} + 15 y$

Try It 6.159

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

Try It 6.160

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

Example 6.81

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

Solution

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

Try It 6.161

Find the quotient: $(32 a^{2} b - 16 a b^{2}) \div (−8 a b) .$

Try It 6.162

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

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.	$4 x y + 3 y - y^{2}$

Try It 6.163

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

Try It 6.164

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

Example 6.83

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

Solution

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

Try It 6.165

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

Try It 6.166

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

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
We divide the first two digits, 87, by 25.
We multiply 3 times 25 and write the product under the 87.
Now we subtract 75 from 87.
Then we bring down the third digit of the dividend, 5.
Repeat the process, dividing 25 into 125.

We check division by multiplying the quotient by the divisor.

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

\begin{matrix} 35 \cdot 25 \\ 875 ✓ \end{matrix}

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: $(x^{2} + 9 x + 20) \div (x + 5) .$

Solution


Write it as a long division problem.
Be sure the dividend is in standard form.
Divide x² by x. It may help to ask yourself, "What do I need to multiply x by to get x²?"
Put the answer, x, in the quotient over the x term.
Multiply x times x + 5. Line up the like terms under the dividend.
Subtract x² + 5x from x² + 9x.
Then bring down the last term, 20.
Divide 4x by x. It may help to ask yourself, "What do I need to multiply x by to get 4x?"
Put the answer, 4, in the quotient over the constant term.
Multiply 4 times x + 5.
Subtract 4x + 20 from 4x + 20.
Check:
Multiply the quotient by the divisor.
(x + 4)(x + 5)
You should get the dividend.
x² + 9x + 20✓

Try It 6.167

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

Try It 6.168

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

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: $(2 x^{2} - 5 x - 3) \div (x - 3) .$

Solution


Write it as a long division problem.
Be sure the dividend is in standard form.
Divide 2x² by x. Put the answer, 2x, in the quotient over the x term.
Multiply 2x times x − 3. Line up the like terms under the dividend.
Subtract 2x² − 6x from 2x² − 5x. Change the signs and then add. Then bring down the last term.
Divide x by x. Put the answer, 1, in the quotient over the constant term.
Multiply 1 times x − 3.
Subtract x − 3 from x − 3 by changing the signs and adding.
To check, multiply (x − 3)(2x + 1).
The result should be 2x² − 5x − 3.

Try It 6.169

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

Try It 6.170

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

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: $(x^{3} - x^{2} + x + 4) \div (x + 1) .$

Solution


Write it as a long division problem.
Be sure the dividend is in standard form.
Divide x³ by x. Put the answer, x², in the quotient over the x² term. Multiply x² times x + 1. Line up the like terms under the dividend.
Subtract x³ + x² from x³ − x² by changing the signs and adding. Then bring down the next term.
Divide −2x² by x. Put the answer, −2x, in the quotient over the x term. Multiply −2x times x + 1. Line up the like terms under the dividend.
Subtract −2x² − 2x from −2x² + x by changing the signs and adding. Then bring down the last term.
Divide 3x by x. Put the answer, 3, in the quotient over the constant term. Multiply 3 times x + 1. Line up the like terms under the dividend.
Subtract 3x + 3 from 3x + 4 by changing the signs and adding. Write the remainder as a fraction with the divisor as the denominator.	$The sum of 3 x plus 4 and negative 3 x plus negative 3 is 1. Therefore, the polynomial x cubed minus x squared plus x plus 4, divided by the binomial x plus 1, equals x squared minus 2 x plus the fraction 1 over x plus 1.$
To check, multiply $(x + 1) (x^{2} - 2 x + 3 + \frac{1}{x + 1}) .$ The result should be $x^{3} - x^{2} + x + 4$ .

Try It 6.171

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

Try It 6.172

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

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

Example 6.87

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

Solution

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


Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.
Divide x⁴ by x. Put the answer, x³, in the quotient over the x³ term. Multiply x³ times x + 2. Line up the like terms. Subtract and then bring down the next term.
Divide −2x³ by x. Put the answer, −2x², in the quotient over the x² term. Multiply −2x² times x + 1. Line up the like terms. Subtract and bring down the next term.
Divide 3x² by x. Put the answer, 3x, in the quotient over the x term. Multiply 3x times x + 1. Line up the like terms. Subtract and bring down the next term.
Divide −x by x. Put the answer, −1, in the quotient over the constant term. Multiply −1 times x + 1. Line up the like terms. Change the signs, add.
To check, multiply $(x + 2) (x^{3} - 2 x^{2} + 3 x - 1)$ .
The result should be $x^{4} - x^{2} + 5 x - 2$ .

Try It 6.173

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

Try It 6.174

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

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

Example 6.88

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

Solution

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

The figure shows the long division of 8 a cubed plus 27 by 2 a plus 3. In the long division bracket, placeholders 0 a squared and 0 a are added into the polynomial. On the first line under the dividend 8 a cubed plus 12 a squared is subtracted. To the right, an arrow indicates that this value came from multiplying 4 a squared by 2 a plus 3. The subtraction gives negative 12 a squared plus 0 a. From this negative 12 a squared minus 18 a is subtracted. To the right, an arrow indicates that this value came from multiplying 6 a by 2 a plus 3. The subtraction give 18 a plus 27. From this 18 a plus 27 is subtracted. To the right, an arrow indicates that this value came from multiplying 9 by 2 a plus 3. The result is 0.

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

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

Try It 6.175

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

Try It 6.176

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

Media

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.

$\frac{45 y + 36}{9}$

443.

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

444.

$\frac{8 d^{2} - 4 d}{2}$

445.

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

446.

$(16 y^{2} - 20 y) \div 4 y$

447.

$(55 w^{2} - 10 w) \div 5 w$

448.

$(9 n^{4} + 6 n^{3}) \div 3 n$

449.

$(8 x^{3} + 6 x^{2}) \div 2 x$

450.

$\frac{18 y^{2} - 12 y}{−6}$

451.

$\frac{20 b^{2} - 12 b}{−4}$

452.

$\frac{35 a^{4} + 65 a^{2}}{−5}$

453.

$\frac{51 m^{4} + 72 m^{3}}{−3}$

454.

$\frac{310 y^{4} - 200 y^{3}}{5 y^{2}}$

455.

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

456.

$\frac{46 x^{3} + 38 x^{2}}{2 x^{2}}$

457.

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

458.

$(24 p^{2} - 33 p) \div (−3 p)$

459.

$(35 x^{4} - 21 x) \div (−7 x)$

460.

$(63 m^{4} - 42 m^{3}) \div (−7 m^{2})$

461.

$(48 y^{4} - 24 y^{3}) \div (−8 y^{2})$

462.

$(63 a^{2} b^{3} + 72 a b^{4}) \div (9 a b)$

463.

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

464.

$\frac{52 p^{5} q^{4} + 36 p^{4} q^{3} - 64 p^{3} q^{2}}{4 p^{2} q}$

465.

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

466.

$\frac{66 x^{3} y^{2} - 110 x^{2} y^{3} - 44 x^{4} y^{3}}{11 x^{2} y^{2}}$

467.

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

468.

$\frac{4 w^{2} + 2 w - 5}{2 w}$

469.

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

470.

$\frac{10 x^{2} + 5 x - 4}{−5 x}$

471.

$\frac{20 y^{2} + 12 y - 1}{−4 y}$

472.

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

473.

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

Divide a Polynomial by a Binomial

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

474.

$(y^{2} + 7 y + 12) \div (y + 3)$

475.

$(d^{2} + 8 d + 12) \div (d + 2)$

476.

$(x^{2} - 3 x - 10) \div (x + 2)$

477.

$(a^{2} - 2 a - 35) \div (a + 5)$

478.

$(t^{2} - 12 t + 36) \div (t - 6)$

479.

$(x^{2} - 14 x + 49) \div (x - 7)$

480.

$(6 m^{2} - 19 m - 20) \div (m - 4)$

481.

$(4 x^{2} - 17 x - 15) \div (x - 5)$

482.

$(q^{2} + 2 q + 20) \div (q + 6)$

483.

$(p^{2} + 11 p + 16) \div (p + 8)$

484.

$(y^{2} - 3 y - 15) \div (y - 8)$

485.

$(x^{2} + 2 x - 30) \div (x - 5)$

486.

$(3 b^{3} + b^{2} + 2) \div (b + 1)$

487.

$(2 n^{3} - 10 n + 24) \div (n + 3)$

488.

$(2 y^{3} - 6 y - 36) \div (y - 3)$

489.

$(7 q^{3} - 5 q - 2) \div (q - 1)$

490.

$(z^{3} + 1) \div (z + 1)$

491.

$(m^{3} + 1000) \div (m + 10)$

492.

$(a^{3} - 125) \div (a - 5)$

493.

$(x^{3} - 216) \div (x - 6)$

494.

$(64 x^{3} - 27) \div (4 x - 3)$

495.

$(125 y^{3} - 64) \div (5 y - 4)$

Everyday Math

496.

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

ⓐ Find the quotient by dividing the numerator by the denominator.
ⓑ 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 $\frac{n^{2} - n}{2}$ , where $n$ represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

Writing Exercises

498.

James divides $48 y + 6$ by 6 this way: $\frac{48 y + 6}{6} = 48 y$ . What is wrong with his reasoning?

499.

Divide $\frac{10 x^{2} + x - 12}{2 x}$ 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?

6.6 Divide Polynomials

Divide a Polynomial by a Monomial

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Divide a Polynomial by a Binomial

Solution

Solution

Solution

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Solution

Section 6.6 Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check