# 12.3: Division of Polynomials

**Basic**Created by: CK-12

**Practice**Division of Polynomials

Suppose that you know that the area of a rectangular mural in square feet is represented by the polynomial \begin{align*}x^2+2x-24\end{align*}

### Dividing Polynomials

We will begin with a property that is the converse of the Adding Fractions Property presented in previous sections.

The **converse of the Adding Fractions Property** states that for all real numbers \begin{align*}a, b\end{align*}

This property allows you to separate the numerator into its individual fractions. This property is used when dividing a polynomial by a monomial.

#### Let's take a look at a couple of problems that use the Adding Fractions Property:

- Simplify \begin{align*}\frac{8x^2-4x+16}{2}\end{align*}
8x2−4x+162 .

Using the property above, separate the polynomial into its individual fractions.

\begin{align*}&& \frac{8x^2}{2}-\frac{4x}{2}+\frac{16}{2}\\
\text{Reduce.} && 4x^2-2x+8\end{align*}

- Simplify \begin{align*}\frac{-3m^2-18m+6}{9m}\end{align*}
−3m2−18m+69m .

Separate the trinomial into its individual fractions and reduce.

\begin{align*}& -\frac{3m^2}{9m}-\frac{18m}{9m}+\frac{6}{9m}\\
& -\frac{m}{3}-2+\frac{2}{3m}\end{align*}

Polynomials can also be divided by binomials. However, instead of separating into its individual fractions, we use a process called long division.

#### Let's look at the process of polynomial long division by solving the following problem:

Simplify \begin{align*}\frac{x^2+4x+5}{x+3}\end{align*}

When we perform division, the expression in the numerator is called the **dividend** and the expression in the denominator is called the **divisor.**

To start the division we rewrite the problem in the following form.

Start by dividing the first term in the dividend by the first term in the divisor \begin{align*}\frac{x^2}{x}=x\end{align*}

Next, multiply the \begin{align*}x\end{align*}

Now subtract \begin{align*}x^2+3x\end{align*}

Now, bring down 5, the next term in the dividend.

Repeat the process. First divide the first term of \begin{align*}x+5\end{align*}

Multiply 1 by the divisor \begin{align*}x+3\end{align*}

\begin{align*}w\end{align*}

Subtract \begin{align*}x+3\end{align*}

Since there are no more terms from the dividend to bring down, we are done.

The answer is \begin{align*}x+1\end{align*}

### Examples

#### Example 1

Earlier, you were asked to calculate the width of a rectangular mural. You know that the area of the mural in square feet is represented by \begin{align*}x^2+2x-24\end{align*}

We know that the area of a rectangle is equal to it's width times the length. Let \begin{align*}w=\end{align*}

#### \begin{align*}& Area=wl\\
& Plug\ and\ chug \rightarrow x^2-2x-24=(w)(x+6)\\
& Solve\ for\ w\ \rightarrow w=\frac{x^2-2x-24}{x+6}\\
& Factor\ the \ numerator\rightarrow w= \frac{(x-8)(x+6)}{x+6}\\
& Take\ out\ common\ factor \rightarrow w=x-8 \end{align*}Area=wlPlug and chug→x2−2x−24=(w)(x+6)Solve for w →w=x2−2x−24x+6Factor the numerator→w=(x−8)(x+6)x+6Take out common factor→w=x−8

#### Example 2

Divide \begin{align*}9x^2-16\end{align*}

You are being asked to simplify:

\begin{align*} \frac{9x^2-16}{3x+4}.\end{align*}

You could use long division to find the answer. You can also use patterns of polynomials to simplify and cancel.

Recall that \begin{align*}a^2-b^2=(a+b)(a-b).\end{align*}

\begin{align*} \frac{9x^2-16}{3x+4}&=\frac{(3x)^2-4^2}{3x+4}\\
&=\frac{(3x-4)(3x+4)}{3x+4}\\
&=3x-4 \end{align*}

### Review

Divide the following polynomials.

- \begin{align*}\frac{2x+4}{2}\end{align*}
2x+42 - \begin{align*}\frac{x-4}{x}\end{align*}
x−4x - \begin{align*}\frac{5x-35}{5x}\end{align*}
5x−355x - \begin{align*}\frac{x^2+2x-5}{x}\end{align*}
x2+2x−5x - \begin{align*}\frac{4x^2+12x-36}{-4x}\end{align*}
4x2+12x−36−4x - \begin{align*}\frac{2x^2+10x+7}{2x^2}\end{align*}
- \begin{align*}\frac{x^3-x}{-2x^2}\end{align*}
- \begin{align*}\frac{5x^4-9}{3x}\end{align*}
- \begin{align*}\frac{x^3-12x^2+3x-4}{12x^2}\end{align*}
- \begin{align*}\frac{3-6x+x^3}{-9x^3}\end{align*}
- \begin{align*}\frac{x^2+3x+6}{x+1}\end{align*}
- \begin{align*}\frac{x^2-9x+6}{x-1}\end{align*}
- \begin{align*}\frac{x^2+5x+4}{x+4}\end{align*}
- \begin{align*}\frac{x^2-10x+25}{x-5}\end{align*}
- \begin{align*}\frac{x^2-20x+12}{x-3}\end{align*}
- \begin{align*}\frac{3x^2-x+5}{x-2}\end{align*}
- \begin{align*}\frac{9x^2+2x-8}{x+4}\end{align*}
- \begin{align*}\frac{3x^2-4}{3x+1}\end{align*}
- \begin{align*}\frac{5x^2+2x-9}{2x-1}\end{align*}
- \begin{align*}\frac{x^2-6x-12}{5x+4}\end{align*}
- \begin{align*}\frac{x^4-2x}{8x+24}\end{align*}
- \begin{align*}\frac{x^3+1}{4x-1}\end{align*}

**Mixed Review**

- Boyle’s Law states that the pressure of a compressed gas varies inversely as its pressure. If the pressure of a 200-pound gas is 16.75 psi, find the pressure if the amount of gas is 60 pounds.
- Is \begin{align*}5x^3+x^2-x^{-1}+8\end{align*} an example of a polynomial? Explain your answer.
- Find the slope of the line perpendicular to \begin{align*}y=-\frac{3}{4} x+5\end{align*}.
- How many two-person teams can be made from a group of nine individuals?
- Solve for \begin{align*}m: -4= \frac{\sqrt{m-3}}{-2}\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3.

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
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Term | Definition |
---|---|

Adding Fraction Property |
For all real numbers , and , and , = . |

Denominator |
The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator . |

Dividend |
In a division problem, the dividend is the number or expression that is being divided. |

divisor |
In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend. |

Polynomial long division |
Polynomial long division is the standard method of long division, applied to the division of polynomials. |

Rational Expression |
A rational expression is a fraction with polynomials in the numerator and the denominator. |

Rational Root Theorem |
The rational root theorem states that for a polynomial, , where are integers, the rational roots can be determined from the factors of and . More specifically, if is a factor of and is a factor of , then all the rational factors will have the form . |

Remainder Theorem |
The remainder theorem states that if , then is the remainder when dividing by . |

Synthetic Division |
Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used. |

### Image Attributions

Here you'll learn how to perform division problems involving polynomials.

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