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:
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.
General Tip: Students often incorrectly cancel a factor not common to all the terms.
- When students cancel the 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. (forgetting to remove the canceled factor)