# 12.3: Division of Polynomials

**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.

## Vocabulary

Terms introduced in this lesson:

- rational expression
- numerator
- denominator
- common denominator
- dividend
- divisor
- quotient
- remainder

## 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*}

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

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

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*}

- 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:

Example:

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

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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