1. Objectives: 1. Use the factor theorem. 2. Factor a polynomial completely.

2. Finding Real Zeros Recall that the zeros of a function are where the graph crosses the x-axis. They are also solutions to the equation f(x) = 0. When working with linear equations, such as f(x) = 2x+3, finding the zeros are as simples as setting it equal to zero and solving for x. To find the zeros of a quadratic equation, one must use either factoring (if possible) or the quadratic equation. To find the zeros of higher degree polynomials, other methods may be necessary. 0 = 2x + 3  -3 = 2x  x = -3/2

3. Rational Zero Test The Rational Zero Test, is a definitive way to identify every rational root of a polynomial, but when used alone it can be very cumbersome and cannot identify irrational or complex roots. To speed up the process, we will use it only when necessary, and after pulling out all the integer roots by graphing.

4. Example #1 Completely Factoring a Polynomial Write the following polynomial function in completely factored form. Graph the polynomial & identify the zeros. x = −2, x = −1, x = 3 Since every zero is an integer, we can use synthetic division to factor the polynomial and can do it successively.

5. Example #1 Completely Factoring a Polynomial Write the following polynomial function in completely factored form. When we put the x back into the quotient we get: This means that the factor x + 2 occurs twice since we already divided by it on the very first step. So, for the completely factored form of the polynomial we would write: Where each factor comes from one of the zeros except (x + 2) which actually occurs twice.

6. Example #2 Completely Factoring a Polynomial Write the following polynomial function in completely factored form. x = −2, x = 1 This time there is a GCF left after the synthetic division, which is then factored and left out front of the polynomial. The factor (x – 1) occurred twice.

7. Example #3 Completely Factoring a Polynomial A. Identify all the possible rational roots by using the rational zero test. Factors of 7: Factors of 2: Identify all the factors of the constant term, 7, and all the factors of the leading coefficient, 2, and divide them.

8. Example #3 Completely Factoring a Polynomial B. Find all the zeros of the following polynomial. From the rational zero test all possible roots are: Using a graph this list can be narrowed further. From the graph it appears that − 1 is definitely a real zero and ½ as well. The remaining two zeros are most likely irrational.

9. Example #3 Completely Factoring a Polynomial B. Find all the zeros of the following polynomial. Synthetic division confirms the two zeros: So far the factored form looks like this: The remaining trinomial is not factorable and requires the quadratic formula to obtain the remaining zeros.

10. Example #3 Completely Factoring a Polynomial B. Find all the zeros of the following polynomial. In completely factored form: