1. Section 3.3 Real Zeros of Polynomial Functions

2. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational Zeros Theorem – Find the Real Zeros of a Polynomial Function – Solve Polynomial Equations – Use the Theorem for Bounds on Zeros – Use the Intermediate Value Theorem

3. Synthetic and Long Division of Polynomials EXAMPLES

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

5. Factor Theorem Let f be a polynomial function. Then x-c is a factor of f(x) if and only if f(c)=0. So this means that the remainder when the polynomial is divided by x-c is 0. Thus x-c divides into the polynomial evenly. Theorem: A polynomial function of degree n has at most n real zeros.

6. 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 lowest terms, is a rational zero of f, then p must be a factor of and q must be a factor of. We can test each possible solution with synthetic division.

7. Bounds on Zeros f(x) is a polynomial function whose leading coefficient is 1 A bound is the smaller of the following two numbers: OR … Write the bound as plus or minus.

8. Finding Zeros of Polynomials Use the Rational Zeros Theorem and repeated division to find the zeros EXAMPLES