1. The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine possible positive and negative real zeros

2. Why? We can easily find the zeros of a quadratic by factoring or using the Quadratic Formula. In this section we will learn more about the real zeros of a polynomial and in 4.5 we will learn about the complex zeros of a polynomial. In 4.2 it was easy to find the zeros because the polynomials given were easily factorable. In the next two sections, we will find the zeros when factoring is not as apparent….

3. Use the Factor Theorem Show that x – 3 is a factor of x 3 + 4x 2 – 15x – 18. If so, factor the polynomial completely.

4. The Number of Real Zeros A polynomial function cannot have more real zeros than its degree. Not all zeros of a polynomial are real, either! (we will see this in 4.5) Example: The following functions illustrate that a polynomial function of degree n can have at most n real zeros. P(x)DegreeReal ZerosComments f(x) = x 2 – 9 f(x) = x 2 + 4 f(x) = x 3 – 1 f(x) = x 3 – x 2 – 6x

5. Rational Zero Theorem

6. B. List all of the possible rational zeros of f(x) = x 3 + 3x + 24.

7. Identify Possible Zeros List all of the possible rational zeros of f(x) = 3x 4 – x 3 + 4.

8. Descartes’ Rule of Signs

9. State the possible number of positive real zeros, negative real zeros of p(x) = –x 6 + 4x 3 – 2x 2 – x – 1. Use Descartes’ Rule of Signs to determine the number and type of real zeros. Count the number of changes in sign for the coefficients of p(x). Find Numbers of Positive and Negative Zeros p(x) =–x 6 +4x 3 –2x 2 –x–1 yes – to + yes + to – no – to – no – to – 2 or 0 positive real zeros

10. Since there are two sign changes, there are 2 or 0 positive real zeros. Find p(–x) and count the number of sign changes for its coefficients. Find Numbers of Positive and Negative Zeros p(–x) =–(–x) 6 +4(–x) 3 –2(–x) 2 –(–x)–1

11. Find all of the zeros of f(x) = x 3 – x 2 + 2x + 4.