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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.

## 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*} 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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