Do Now

• Let 1. Which of the given polynomials is a factor of

f(x)?

2( ) 3 40 48f x x x

. 2 . 3

. 4 . 6

. 12 . 24

a x b x

c x d x

e x f x

2( ) 3 40 48f x x x

2( ) 3 40 48f x x x

Chapter 9: Polynomial Functions

Lesson 5: The Factor Theorem

Mrs. Parziale

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.

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) = 03) 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.

Example 1:

Let f(x) = x2 + 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?2 5 6

3

x x

x

f(x) = x2 + 5x + 6

2 5 6

3

x x

x

4) Using the Factor – Solution – Intercept Equivalence Theorem, what can we say about this function ?

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

ya. (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. x2 + 5x + 6 divided by (x + 3) has a

remainder of 0.

x2 + 5x + 6 divided by (x + 2) has a remainder

of 0.

2( ) 5 6f x x x

Example 2:

Factor 12x3 – 41x2 +13x + 6 .Graph it first. Are any zeroes obvious? Make a

factor, divide, factor again.

3–3 x

3

6

–3

–6

y

Example 3:

• Find an equation for a polynomial function with zeroes 2

1,3,3

and

Example 4:

• Is (x+1) a factor of ? Is (x+5) a factor?

3 24 5 13 14x x x

Closure

• What is the Factor Theorem?• What does the Factor – Solution – Intercept

Equivalence Theorem say about the function with x-intercepts 2 and 4?

2( ) 6 8f x x x