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12.3: Division of Polynomials

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

At the end of this lesson, students will be able to:

  • Divide a polynomial by a monomial.
  • Divide a polynomial by a binomial.
  • Rewrite and graph rational functions.


Terms introduced in this lesson:

rational expression
common denominator

Teaching Strategies and Tips

Students learned in chapter Factoring Polynomials how to add, subtract, and multiply polynomials. This lesson completes that discussion with dividing polynomials.

  • Emphasize that the quotient of two polynomials forms a rational expression which is studied in its own right (rational functions).

Use Example 1 to demonstrate dividing a polynomial by a monomial.

  • Remind students that each term in the numerator must be divided by the monomial in the denominator. See Example 2.

Use Example 3 to motivate long division of polynomials.

  • To write the answer, remind students that:

\begin{align*}\frac{\text{dividend}}{\text{divisor}} = \text{quotient} + \frac{\text{remainder}}{\text{divisor}}\end{align*}dividenddivisor=quotient+remainderdivisor

  • To check an answer, have students use the equivalent form:

\begin{align*}\text{dividend} = (\text{divisor} \times \text{quotient}) + \text{remainder}\end{align*}dividend=(divisor×quotient)+remainder

Have students rewrite for themselves the four cases for graphing rational functions preceding Example 5.

Error Troubleshooting

General Tip: Students often incorrectly cancel a factor not common to all the terms.

  • Example:

\begin{align*}\frac{\cancel{a}x+b}{\cancel{a}y} \neq \frac{x+b}{y}\end{align*}ax+bayx+by

  • When students cancel the \begin{align*}a\end{align*}a above, they violate order of operations. Remind students that the fraction sign is a grouping symbol (parentheses) and therefore the numerator and denominator must be simplified before dividing.
  • Otherwise, if the numerator and denominator are completely factored, then the order of operations says to multiply or divide; therefore, canceling is justified.
  • Have students write out the step preceding the canceling:


\begin{align*}\frac{ax+ab}{ay} = \frac{a(x+b)}{ay}\end{align*}ax+abay=a(x+b)ay

Then canceling is apparent:

\begin{align*}\frac{ax+ab}{ay} = \frac{a(x+b)}{ay} = \frac{\cancel{a}(x+b)}{\cancel{a}y} - \frac{x+b}{y}\end{align*}

  • Other common cancelling errors are:

a. \begin{align*}\frac{\cancel{a}x+ab}{\cancel{a}y} \neq \frac{x+ab}{y}\end{align*} (forgetting to remove the canceled factor)

b. \begin{align*}\frac{\cancel{a}}{x+\cancel{a}} \neq \frac{1}{x}\end{align*}

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