1. Do Now Let 1. Which of the given polynomials is a factor of f(x)?

4. Chapter 9: Polynomial Functions Lesson 5: The Factor Theorem Mrs. Parziale

5. Factor Theorem: For a polynomial f(x), the number c is a solution to f(x) = 0 if and only if (x-c) is a factor of x.

6. Factor – Solution – Intercept Equivalence Theorem: For any polynomial f(x), the following are logically equivalent: 1) (x-c) is a factor of f(x) 2) f(c) = 0 3) c is an x-intercept of the graph of f(x) 4) c is a zero of f(x) 5) The remainder when f(x) is divided by (x-c) is 0.

7. Example 1: Let f(x) = x 2 + 5x + 6. Show why the theorem above holds here: 1)Factor f(x). What are the two values of c in this problem? 2)Graph. Where are the zeroes? 3)Divide using long division. What is the remainder?

8. f(x) = x 2 + 5x + 6

10. 4) Using the Factor – Solution – Intercept Equivalence Theorem, what can we say about this function ? a. (x + 3) and (x + 2) are factors b. f(-3) = 0, and f(-2) = 0 c. -3 and -2 are x-intercepts d. -3 is a zero of f(x), -2 is a zero of the graph e. x 2 + 5x + 6 divided by (x + 3) has a remainder of 0. x 2 + 5x + 6 divided by (x + 2) has a remainder of 0.

11. Example 2: Factor 12x 3 – 41x 2 +13x + 6. Graph it first. Are any zeroes obvious? Make a factor, divide, factor again.

12. Example 3: Find an equation for a polynomial function with zeroes

14. Example 4: Is (x+1) a factor of ? Is (x+5) a factor?

17. Closure What is the Factor Theorem? What does the Factor – Solution – Intercept Equivalence Theorem say about the function with x-intercepts 2 and 4?