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.
Terms introduced in this lesson:
factoring, factoring a polynomial
zero product property
greatest common monomial factor
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)⇒ax+bx+ay+by; they will now learn to “put it back together”: ax+bx+ay+by⇒(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.
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.
a. Solve for x.
(Are students incorrectly setting each factor equal to 8?)
b. Solve for x.
(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, 6ax2−9ax+3a≠3a(2x2−3x); but 6ax2−9ax+3a=3a(2x2−3x+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 6ax2−9ax+3a=3a(2x2−3x), 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.