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12.3: Division of Polynomials

Difficulty Level: At Grade Created by: CK-12
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We will begin with a property that is the converse of the Adding Fractions Property presented in Chapter 2.

For all real numbers \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}, and \begin{align*}c \neq 0\end{align*}, \begin{align*}\frac{a+b}{c}\end{align*} = \begin{align*}\frac{a}{c}+\frac{b}{c}\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.

Example: Simplify \begin{align*}\frac{8x^2-4x+16}{2}\end{align*}

Solution: 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*}

Example 1: Simplify \begin{align*}\frac{-3m^2-18m+6}{9m}\end{align*}.

Solution: 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.

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

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 \begin{align*}\frac{x^2}{x}=x\end{align*}. Place the answer on the line above the \begin{align*}x\end{align*} term.

Next, multiply the \begin{align*}x\end{align*} term in the answer by each of the \begin{align*}x+3\end{align*} terms in the divisor and place the result under the divided, matching like terms.

Now subtract \begin{align*}x^2+3x\end{align*} from \begin{align*}x^2+4x+5\end{align*}. It is useful to change the signs of the terms of \begin{align*}x^2+3x\end{align*} to \begin{align*}-x^2-3x\end{align*} and add like terms vertically.

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*} by the first term of the divisor \begin{align*}\left(\frac{x}{x}\right)=1\end{align*}. Place this answer on the line above the constant term of the dividend.

Multiply 1 by the divisor \begin{align*}x+3\end{align*} and write the answer below \begin{align*}x+5\end{align*}, matching like terms.

Subtract \begin{align*}x+3\end{align*} from \begin{align*}x+5\end{align*} by changing the signs of \begin{align*}x+3\end{align*} to \begin{align*}-x-3\end{align*} and adding like terms.

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

The answer is \begin{align*}x+1\end{align*} with a remainder of 2.

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 CK-12 Basic Algebra: 6 7 Polynomial long division with Mr. Nystrom

- YouTube video.

Practice Set

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. CK-12 Basic Algebra: Polynomial Division (12:09)

Divide the following polynomials.

  1. \begin{align*}\frac{2x+4}{2}\end{align*}
  2. \begin{align*}\frac{x-4}{x}\end{align*}
  3. \begin{align*}\frac{5x-35}{5x}\end{align*}
  4. \begin{align*}\frac{x^2+2x-5}{x}\end{align*}
  5. \begin{align*}\frac{4x^2+12x-36}{-4x}\end{align*}
  6. \begin{align*}\frac{2x^2+10x+7}{2x^2}\end{align*}
  7. \begin{align*}\frac{x^3-x}{-2x^2}\end{align*}
  8. \begin{align*}\frac{5x^4-9}{3x}\end{align*}
  9. \begin{align*}\frac{x^3-12x^2+3x-4}{12x^2}\end{align*}
  10. \begin{align*}\frac{3-6x+x^3}{-9x^3}\end{align*}
  11. \begin{align*}\frac{x^2+3x+6}{x+1}\end{align*}
  12. \begin{align*}\frac{x^2-9x+6}{x-1}\end{align*}
  13. \begin{align*}\frac{x^2+5x+4}{x+4}\end{align*}
  14. \begin{align*}\frac{x^2-10x+25}{x-5}\end{align*}
  15. \begin{align*}\frac{x^2-20x+12}{x-3}\end{align*}
  16. \begin{align*}\frac{3x^2-x+5}{x-2}\end{align*}
  17. \begin{align*}\frac{9x^2+2x-8}{x+4}\end{align*}
  18. \begin{align*}\frac{3x^2-4}{3x+1}\end{align*}
  19. \begin{align*}\frac{5x^2+2x-9}{2x-1}\end{align*}
  20. \begin{align*}\frac{x^2-6x-12}{5x+4}\end{align*}
  21. \begin{align*}\frac{x^4-2x}{8x+24}\end{align*}
  22. \begin{align*}\frac{x^3+1}{4x-1}\end{align*}

Mixed Review

  1. 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.
  2. Is \begin{align*}5x^3+x^2-x^{-1}+8\end{align*} an example of a polynomial? Explain your answer.
  3. Find the slope of the line perpendicular to \begin{align*}y=-\frac{3}{4} x+5\end{align*}.
  4. How many two-person teams can be made from a group of nine individuals?
  5. What is a problem with face-to-face interviews? What do you think is a potential solution to this problem?
  6. Solve for \begin{align*}m: -4= \frac{\sqrt{m-3}}{-2}\end{align*}.

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