1. 2.5 The Real Zeros of a Polynomial Function

2. Factor Theorem 1. If f(c)=0, then x - c is a factor of f(x).
2. If x - c is a factor of f(x), then f(c)=0.

3. (a) x + 3 (b) x + 4 Use the Factor Theorem to determine whether the
function
has the factor
(a) x + 3
(b) x + 4
x +3 is not a factor of f(x).
(b) f (-4) = 0
x + 4 is a factor of f(x).

4. Theorem Number of Zeros
A polynomial function cannot have more zeros than its degree.

5. Rational Zeros Theorem
Let f be a polynomial function of degree 1 or higher of the form
where each coefficient is an integer. If p/q in the lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

6. List the potential rational zeros ofq:

7. Find the real zeros of
Factor f over the reals.
There are at most five zeros.
Write factors of -12 and 1 to obtain the potential rational zeros.

8. Thus, -3 is a zero of f and x + 3 is a factor of f.

9. Thus f(x) factors as: