do now let 1. which of the given polynomials is a factor of f(x)?
TRANSCRIPT
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