# 9.4: Polynomial Equations in Factored Form

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Use the zero-product property.
- Find greatest common monomial factor.
- Solve simple polynomial equations by factoring.

## Vocabulary

Terms introduced in this lesson:

- factoring, factoring a polynomial
- expanded form
- factored form
- zero product property
- factoring completely
- common factor
- greatest common monomial factor
- polynomial equation

## Teaching Strategies and Tips

Use the introduction to motivate factoring.

- The reverse of distribution is called factoring.
- Whereas before students were learning the direction \begin{align*}(a + b)(x + y) \Rightarrow ax + bx + ay + by\end{align*}; they will now learn to “put it back together”: \begin{align*}ax + bx + ay + by \Rightarrow (a + b)(x + y)\end{align*}.
- Students realize that polynomials can be expressed in expanded or factored form

Teachers may decide to have their students pull common factors out one at a time, instead of factoring the GCF in one step.

## Error Troubleshooting

In *Review Questions* 9 and 12-16, remind students to set the monomial factor \begin{align*}(x, y, a,\end{align*} or \begin{align*}b)\end{align*} equal to zero.

- Caution students against dividing by variables. In doing so, they will lose \begin{align*}0\end{align*} as a solution. See also Example 6.

General Tip: Check that students are using the zero-product property correctly.

Examples:

a. *Solve for* \begin{align*}x\end{align*}.

\begin{align*}(x + 3)(x - 4) = 8\end{align*}

(Are students incorrectly setting each factor equal to \begin{align*}8\end{align*}?)

b. *Solve for* \begin{align*}x\end{align*}.

\begin{align*}(x + 3)(x - 4) - 2 = 0\end{align*}

(Are students incorrectly setting each factor equal to \begin{align*}0\end{align*}?)

General Tip: Remind students when factoring the GCF out of itself to leave a \begin{align*}1\end{align*}.

For example, \begin{align*}6ax^2 - 9ax + 3a \neq 3a(2x^2 - 3x)\end{align*}; but \begin{align*}6ax^2 - 9ax + 3a = 3a(2x^2 - 3x + 1)\end{align*}. See Example 5b and *Review Questions* 3 and 15.

General Tip: Have students check their work by expanding the factored polynomial.

- By checking a problem worked out as \begin{align*}6ax^2 - 9ax + 3a = 3a(2x^2 - 3x)\end{align*}, students will convince themselves that a \begin{align*}1\end{align*} is missing.

General Tip: Suggest that students look carefully over the remaining terms after having factored out the GCF so as to not leave any other common factors.

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