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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:

dividenddivisor=quotient+remainderdivisor\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:

dividend=(divisor×quotient)+remainder\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:

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

• When students cancel the a\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:

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

Then canceling is apparent:

ax+abay=a(x+b)ay=a(x+b)ayx+by\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. ax+abayx+aby\begin{align*}\frac{\cancel{a}x+ab}{\cancel{a}y} \neq \frac{x+ab}{y}\end{align*} (forgetting to remove the canceled factor)

b. ax+a1x\begin{align*}\frac{\cancel{a}}{x+\cancel{a}} \neq \frac{1}{x}\end{align*}

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