12.3: Division of Polynomials
Suppose that you know that the area of a rectangular mural in square feet is represented by the polynomial
Watch This
Multimedia Link: For more help with using long division to simplify rational expressions, visit this http://www.purplemath.com/modules/polydiv2.htm  website or watch this: CK12 Basic Algebra: 6 7 Polynomial long division with Mr. Nystrom
 YouTube video.
Guidance
We will begin with a property that is the converse of the Adding Fractions Property presented in previous Concepts.
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.
Example A
Simplify
Solution:
Using the property above, separate the polynomial into its individual fractions.
Example B
Simplify
Solution:
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.
Example C
Simplify
Solution:
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
Guided Practice
Divide
Solution:
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
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Polynomial Division (12:09)
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 
3−6x+x3−9x3 
x2+3x+6x+1 
x2−9x+6x−1 
x2+5x+4x+4 
x2−10x+25x−5 
x2−20x+12x−3  \begin{align*}\frac{3x^2x+5}{x2}\end{align*}
 \begin{align*}\frac{9x^2+2x8}{x+4}\end{align*}
 \begin{align*}\frac{3x^24}{3x+1}\end{align*}
 \begin{align*}\frac{5x^2+2x9}{2x1}\end{align*}
 \begin{align*}\frac{x^26x12}{5x+4}\end{align*}
 \begin{align*}\frac{x^42x}{8x+24}\end{align*}
 \begin{align*}\frac{x^3+1}{4x1}\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 200pound gas is 16.75 psi, find the pressure if the amount of gas is 60 pounds.
 Is \begin{align*}5x^3+x^2x^{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 twoperson teams can be made from a group of nine individuals?
 Solve for \begin{align*}m: 4= \frac{\sqrt{m3}}{2}\end{align*}.
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.
Concept Nodes:
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.