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9.4: Polynomial Equations in Factored Form

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 (a + b)(x + y) \Rightarrow ax + bx + ay + by; they will now learn to “put it back together”: ax + bx + ay + by \Rightarrow (a + b)(x + y).
  • 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 (x, y, a, or b) equal to zero.

  • Caution students against dividing by variables. In doing so, they will lose 0 as a solution. See also Example 6.

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

Examples:

a. Solve for x.

(x + 3)(x - 4) = 8

(Are students incorrectly setting each factor equal to 8?)

b. Solve for x.

(x + 3)(x - 4) - 2 = 0

(Are students incorrectly setting each factor equal to 0?)

General Tip: Remind students when factoring the GCF out of itself to leave a 1.

For example, 6ax^2 - 9ax + 3a \neq 3a(2x^2 - 3x); but 6ax^2 - 9ax + 3a = 3a(2x^2 - 3x + 1). 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 6ax^2 - 9ax + 3a = 3a(2x^2 - 3x), students will convince themselves that a 1 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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