# 5.6 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 binomial

**Try It**

Before you get started, take this readiness quiz:

1) Add: [latex]\frac{3}{d}+\frac{x}{d}[/latex]

2) Simplify: [latex]\frac{30x{y}^{3}}{5xy}[/latex]

3) Combine like terms: [latex]8{a}^{2}+12a+1+3{a}^{2}-5a+4[/latex]

**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.

[latex]\displaystyle\frac{y}{5}+\frac{2}{5}[/latex]

The sum simplifies to

[latex]\displaystyle\frac{y+2}{5}[/latex]

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 [latex]a[/latex]*, *[latex]b[/latex] and [latex]c[/latex] are numbers where [latex]c\neq{0}[/latex], then

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, [latex]\frac{y+2}{5}[/latex] can be written [latex]\frac{y}{5}+\frac{2}{5}[/latex]

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 5.6.1**

Find the quotient: [latex]\frac{7{y}^{2}+21}{7}[/latex]

**Solution**

**Step 1:****Divide each term of the numerator by the denominator.**

[latex]\frac{7{y}^{2}}{7}+\frac{21}{7}[/latex]

**Step 2:****Simplify each fraction.**

[latex]{y}^{2}+3[/latex]

**Try It**

4) Find the quotient: [latex]\frac{8{z}^{2}+24}{4}[/latex]

**Solution**

[latex]2{z}^{2}+6[/latex]

5) Find the quotient: [latex]\frac{18{z}^{2}-27}{9}[/latex]

**Solution**

[latex]2{z}^{2}-3[/latex]

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 5.6.2**

Find the quotient: [latex](18{x}^{3}-36{x}^{2})\div{6x}[/latex]

**Solution**

*Step 1:* Rewrite as a fraction.

[latex]\frac{18{x}^{3}-36{x}^{2}}{6x}[/latex]

*Step 2:* Divide each term of the numerator by the denominator.

[latex]\begin{eqnarray*}&=&\frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}\\[1ex]\text{Simplify.}\;\;&=&3{x}^{2}-6x\end{eqnarray*}[/latex]

**Try It**

6) Find the quotient: [latex](27{b}^{3}-33{b}^{2})\div{3b}[/latex]

**Solution**

[latex]9{b}^{2}-11b[/latex]

7) Find the quotient: [latex](25{y}^{3}-55{y}^{2})\div{5y}[/latex]

**Solution**

[latex]5{y}^{2}-11y[/latex]

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

**Example 5.6.3**

Find the quotient: [latex]\frac{12{d}^{2}-16d}{-4}[/latex]

**Solution**

*Step 1: *Divide each term of the numerator by the denominator.

[latex]\frac{12{d}^{2}}{-4}-\frac{16d}{-4}[/latex]

*Step 2: *Simplify. Remember, subtracting a negative is like adding a positive!

[latex]-3{d}^{2}+4d[/latex]

**Try It**

8) Find the quotient: [latex]\frac{25{y}^{2}-15y}{-5}[/latex]

**Solution**

[latex]-5{y}^{2}+3y[/latex]

9) Find the quotient: [latex]\frac{42{b}^{2}-18b}{-6}[/latex]

**Solution**

[latex]-7{b}^{2}+3b[/latex]

**Example 5.6.4**

Find the quotient: [latex]\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}[/latex]

**Solution**

**Step 1:****Separate the terms.**

[latex]\begin{eqnarray*}&=&\frac{105{y}^{5}}{5{y}^{2}}+\frac{75{y}^{3}}{5{y}^{2}}\\[1ex]\text{Simplify.}\;\;&=&21{y}^{3}+15y\end{eqnarray*}[/latex]

**Try It**

10) Find the quotient: [latex]\frac{60{d}^{7}+24{d}^{5}}{4{d}^{3}}[/latex]

**Solution**

[latex]15{d}^{4}+6{d}^{2}[/latex]

11) Find the quotient: [latex]\frac{216{p}^{7}-48{p}^{5}}{6{p}^{3}}[/latex]

**Solution**

[latex]36{p}^{4}-8{p}^{2}[/latex]

**Example 5.6.5**

Find the quotient: [latex](15{x}^{3}y-35x{y}^{2})\div(-5xy)[/latex]

**Solution**

*Step 1:* Rewrite as a fraction.

[latex]\frac{15{x}^{3}y-35x{y}^{2}}{-5xy}[/latex]

*Step 2:* Separate the terms.

[latex]\begin{eqnarray*}&=&\frac{15{x}^{3}y}{-5xy}-\frac{35x{y}^{2}}{-5xy}\\[1ex]\text{Simplify.}\;\;&=&-3{x}^{2}+7y\end{eqnarray*}[/latex]

**Try It**

12) Find the quotient: [latex](32{a}^{2}b-16a{b}^{2})\div(-8ab)[/latex]

**Solution**

[latex]-4a+2b[/latex]

13) Find the quotient: [latex](-48{a}^{8}{b}^{4}-36{a}^{6}{b}^{5})\div(-6{a}^{3}{b}^{3})[/latex]

**Solution**

[latex]8{a}^{5}b+6{a}^{3}{b}^{2}[/latex]

**Example 5.6.6**

**Solution**

*Step 1:* Separate the terms.

[latex]\begin{eqnarray*}&=&\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}\\[1ex]\text{Simplify.}\;\;&=&4xy+3y-{y}^{2}\end{eqnarray*}[/latex]

**Try It**

**Solution**

15) Find the quotient: [latex]\frac{35{a}^{4}{b}^{2}+14{a}^{4}{b}^{3}-42{a}^{2}{b}^{4}}{7{a}^{2}{b}^{2}}[/latex]

**Solution**

**Example 5.6.7**

Find the quotient: [latex]\frac{10{x}^{2}+5x-20}{5x}[/latex]

**Solution**

*Step 1:* Separate the terms.

[latex]\begin{eqnarray*}&=&\frac{10{x}^{2}}{5x}+\frac{5x}{5x}-\frac{20}{5x}\\[1ex]\text{Simplify.}\;\;&=&2x+1+\frac{4}{x}\end{eqnarray*}[/latex]

**Try It**

16) Find the quotient: [latex]\frac{18{c}^{2}+6c-9}{6c}[/latex]

**Solution**

[latex]3c+1-\frac{3}{2c}[/latex]

17) Find the quotient: [latex]\frac{10{d}^{2}-5d-2}{5d}[/latex]

**Solution**

[latex]2d-1-\frac{2}{5d}[/latex]

**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, [latex]875[/latex], by a 2-digit number, [latex]25[/latex].

We write the long division | |

We divide the first two digits, [latex]87[/latex], by [latex]25[/latex]. | |

We multiply [latex]3[/latex] times [latex]25[/latex] and write the product under the [latex]87[/latex]. | |

Now we subtract [latex]75[/latex] from [latex]87[/latex]. | |

Then we bring down the third digit of the dividend, [latex]5[/latex]. | |

Repeat the process, dividing [latex]25[/latex] into [latex]125[/latex]. |

We check division by multiplying the quotient by the divisor.

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

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 5.6.8**

Find the quotient: [latex]({x}^{2}+9x+20)\div(x+5)[/latex]

**Solution**

*Step 1:* Write it as a long division problem.

*Step 2:* Be sure the dividend is in standard form.

*Step 3:* Divide [latex]x^2[/latex] by [latex]x[/latex].

It may help to ask yourself, “What do I need to multiply [latex]x[/latex] by to get [latex]x^2[/latex]?”

*Step 4:* Put the answer, [latex]x[/latex], in the quotient over the [latex]x[/latex] term.

*Step 5:* Multiply [latex]x[/latex] times [latex]x + 5[/latex].

Line up the like terms under the dividend.

*Step 6:* Subtract [latex]x^2+5x[/latex] from [latex]x^2+9x[/latex].

*Step 7: *Add, changing the signs might make it easier to do this.

*Step 8:* Then bring down the last term, [latex]20[/latex].

*Step 9:* Divide [latex]4x[/latex] by [latex]x[/latex].

It may help to ask yourself, “What do I need to multiply [latex]x[/latex] by to get [latex]4x[/latex]?”

*Step 10:* Put the answer, [latex]4[/latex], in the quotient over the constant term.

*Step 11:* Multiply [latex]4[/latex] times [latex]x+5[/latex].

*Step 12:* Subtract [latex]4x+20[/latex] from [latex]4x+20[/latex].

*Step 13:* Check:

Multiply the quotient by the divisor.

You should get the dividend.

[latex](x+4)(x+5)\\x^2+9x+20\checkmark[/latex]

**Try It**

18) Find the quotient: [latex]({y}^{2}+10y+21)\div(y+3)[/latex]

**Solution**

[latex]y+7[/latex]

19) Find the quotient: [latex]({m}^{2}+9m+20)\div(m+4)[/latex]

**Solution**

[latex]m+5[/latex]

When the divisor has a 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 5.6.9**

Find the quotient: ([latex]2{x}^{2}-5x-3)\div(x-3)[/latex]

**Solution**

*Step 1:* Write it as a long division problem.

*Step 2:* Be sure the dividend is in standard form.

*Step 3:* Divide [latex]2x^2[/latex] by [latex]x[/latex].

*Step 4:* Put the answer, [latex]2x[/latex], in the quotient over the [latex]x[/latex] term.

*Step 5:* Multiply [latex]2x[/latex] times [latex]x-3[/latex].

Line up the like terms under the dividend.

*Step 6:* Subtract [latex]2x^2-6x[/latex] from [latex]2x^2-5x[/latex].

*Step 7:* Change the signs and then add.

*Step 8:* Then bring down the last term.

*Step 9:* Divide [latex]x[/latex] by [latex]x[/latex].

*Step 10:* Put the answer, [latex]1[/latex], in the quotient over the constant term.

*Step 11:* Multiply [latex]1[/latex] times [latex]x-3[/latex].

*Step 12:* Subtract [latex]x-3[/latex] from [latex]x-3[/latex] by changing the signs and adding.

*Step 13:* To check, multiply [latex](x-3)(2x+1)[/latex].

The result should be [latex]2x^2-5x-3[/latex].

**Try It**

20) Find the quotient: [latex](2{x}^{2}-3x-20)\div(x-4)[/latex]

**Solution**

[latex]2x+5[/latex]

21) Find the quotient: [latex](3{x}^{2}-16x-12)\div(x-6)[/latex]

**Solution**

[latex]3x+2[/latex]

When we divided [latex]875[/latex] by [latex]25[/latex], we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In example 5.6.10 we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

**Example 5.6.10**

Find the quotient: [latex]({x}^{3}-{x}^{2}+x+4)\div(x+1)[/latex]

**Solution**

*Step 1:* Write it as a long division problem.

*Step 2:* Be sure the dividend is in standard form.

*Step 3:* Divide [latex]x^3[/latex] by [latex]x[/latex].

*Step 4:* Put the answer, [latex]x^2[/latex], in the quotient over the [latex]x^2[/latex] term.

*Step 5:* Multiply [latex]x^2[/latex] times [latex]x+1[/latex].

Line up the like terms under the dividend.

*Step 6:* Subtract [latex]x^3+x^2[/latex] from [latex]x^3-x^2[/latex] by changing the signs and adding.

*Step 7:* Then bring down the next term.

*Step 9:* Divide [latex]-2x^2[/latex] by [latex]x[/latex].

*Step 10:* Put the answer, [latex]-2x[/latex], in the quotient over the [latex]x[/latex] term.

*Step 11: *Multiply [latex]-2x[/latex] times [latex]x+1[/latex].

Line up the like terms under the dividend.

*Step 12:* Subtract [latex]-2x^2-2x[/latex] from [latex]-2x^2+x[/latex] by changing the signs and adding.

*Step 13:* Then bring down the last term.

*Step 14:* Divide [latex]3x[/latex] by [latex]x[/latex].

*Step 15:* Put the answer, [latex]3[/latex], in the quotient over the constant term.

*Step 16: *Multiply [latex]3[/latex] times [latex]x+1[/latex]. Line up the like terms under the dividend.

*Step 17:* Subtract [latex]3x+3[/latex] from [latex]3x+4[/latex] by changing the signs and adding.

*Step 18:* Write the remainder as a fraction with the divisor as the denominator.

**Step 19:** **To check, multiply [latex](x+1)({x}^{2}-2x+3+\frac{1}{x+1})[/latex].**

The result should be [latex]{x}^{3}-{x}^{2}+x+4[/latex]

**Try It**

22) Find the quotient: [latex]({x}^{3}+5{x}^{2}+8x+6)\div(x+2)[/latex]

**Solution**

[latex]\left({x}^{2}+3x+2+\frac{2}{x+2}\right)[/latex]

23) Find the quotient: [latex](2{x}^{3}+8{x}^{2}+x-8)\div(x+1)[/latex]

**Solution**

[latex]\left(2{x}^{2}+6x-5-\frac{3}{x+1}\right)[/latex]

Look back at the dividends in Examples 5.6.8, 5.6.9, and 5.6.10 The terms were written in descending order of degrees, and there were no missing degrees. The dividend in Example 5.6.11 will be [latex]{x}^{4}-{x}^{2}+5x-2[/latex]. It is missing an [latex]{x}^{3}[/latex] term. We will add in [latex]0{x}^{3}[/latex] as a placeholder.

**Example 5.6.11**

Find the quotient: [latex]({x}^{4}-{x}^{2}+5x-2)\div(x+2)[/latex]

**Solution**

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

*Step 1: *Write it as a long division problem.

Be sure the dividend is in standard form with placeholders for missing terms.

*Step 2:* Divide [latex]x^4[/latex] by [latex]x[/latex].

*Step 3:* Put the answer, [latex]x^3[/latex], in the quotient over the [latex]x^3[/latex] term.

*Step 4:* Multiply [latex]x^3[/latex] times [latex]x+2[/latex].

Line up the like terms.

*Step 5:* Subtract and then bring down the next term.

*Step 6:* Divide [latex]-2x^3[/latex] by [latex]x[/latex].

*Step 7:* Put the answer, [latex]-2x^2[/latex], in the quotient over the [latex]x^2[/latex] term.

*Step 8:* Multiply [latex]-2x^2[/latex] times [latex]x+1[/latex].

Line up the like terms.

*Step 9:* Subtract and bring down the next term.

*Step 10:* Divide [latex]3x^2[/latex] by [latex]x[/latex].

*Step 11: *Put the answer, [latex]3x[/latex], in the quotient over the [latex]x[/latex] term.

*Step 12: *Multiply [latex]3x[/latex] times [latex]x+1[/latex].

Line up the like terms.

*Step 13:* Subtract and bring down the next term.

*Step 14:* Divide [latex]-x[/latex] by [latex]x[/latex].

*Step 15:* Put the answer, [latex]-1[/latex], in the quotient over the constant term.

*Step 16:* Multiply [latex]-1[/latex] times [latex]x+1[/latex].

Line up the like terms.

*Step 17: *Change the signs, add.

*Step 18:* To check, multiply [latex](x+2)({x}^{3}-2{x}^{2}+3x-1)[/latex].

The result should be [latex]({x}^{4}-{x}^{2}+5x-2)[/latex]

**Try It**

24) Find the quotient: [latex]({x}^{3}+3x+14)\div(x+2)[/latex]

**Solution**

[latex]{x}^{2}-2x+7[/latex]

25) Find the quotient: [latex]({x}^{4}-3{x}^{3}-1000)\div(x+5)[/latex]

**Solution**

[latex]{x}^{3}-8{x}^{2}+40x-200[/latex]

In Example 5.6.12, we will divide by [latex]2a-3[/latex]. As we divide we will have to consider the constants as well as the variables.

**Example 5.6.12**

Find the quotient: [latex](8{a}^{3}+27)\div(2a+3)[/latex]

**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 [latex](2a+3)(4{a}^{2}-6a+9)[/latex]

The result should be [latex]8{a}^{3}+27[/latex]

**Try It**

26) Find the quotient: [latex]({x}^{3}-64)\div(x-4)[/latex]

**Solution**

[latex]{x}^{2}+4x+16[/latex]

27) Find the quotient: [latex](125{x}^{3}-8)\div(5x-2)[/latex]

**Solution**

[latex]25{x}^{2}+10x+4[/latex]

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

**Key Concepts**

**Fraction Addition**- If [latex]a[/latex]
*,*[latex]b[/latex] and [latex]c[/latex] are numbers where [latex]c\neq{0}[/latex], then

[latex]\displaystyle\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}[/latex] and [latex]\displaystyle\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}[/latex]

- If [latex]a[/latex]
**Division of a Polynomial by a Monomial**- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

**Self Check**

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?