# 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

### 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

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
8x2−4x+162 .

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

- Simplify
−3m2−18m+69m .

Separate the trinomial into its individual fractions and reduce.

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

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

Next, multiply the

Now subtract

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

Repeat the process. First divide the first term of

Multiply 1 by the divisor

Subtract

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

The answer is

### 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

We know that the area of a rectangle is equal to it's width times the length. Let

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

You are being asked to simplify:

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

Recall that

### Review

Divide the following polynomials.

2x+42 x−4x 5x−355x x2+2x−5x 4x2+12x−36−4x 2x2+10x+72x2 x3−x−2x2 5x4−93x x3−12x2+3x−412x2 - \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.

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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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