1. 3.7 The Real Zeros of a Polynomial Function

2. Theorem: Division Algorithm for Polynomials
If f(x) and g(x) denote polynomial functions and if g(x) is not the zero polynomial, then there are unique polynomial functions q(x) and r(x) such that
where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x).

3. Remainder Theorem
Let f be a polynomial function. If f(x) is divided by x - c, then the remainder is f(c).

4. Find the remainder if
is divided by x + 3.
x + 3 = x - (-3) 30

5. 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.

6. (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).

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

8. Theorem 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.

9. List the potential rational zeros ofq:

10. 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.

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

12. Thus f(x) factors as:

13. Theorem Bounds on Zeros

15. Intermediate Value Theorem
Let f denote a continuous function. If a<b
And if f(a) and f(b) are of opposite sign, then the graph of f has at least one zero between a and b.

16. Use the Intermediate Value Theorem to show that the graph of function
has an x-intercept in the interval [-3, -2].
f(-3) = < 0
f(-2) = 1.8 > 0