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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:


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


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:


  • When students cancel the 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:



Then canceling is apparent:


  • Other common cancelling errors are:

a. ax+abayx+aby (forgetting to remove the canceled factor)

b. ax+a1x

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