1. Zeros of Polynomial Functions Section 2.5 Page 312

2. Review Factor Theorem: If x – c is a factor of f(x), then f(c) = 0 Example: x – 3 is a zero of f(x) = 2x 3 – 3x 2 – 11x + 6

3. Information about The Rational Zero Theorem Use to find possible rational zeros Provides a list of possible rational zeros of a polynomial function Not every number will be a zero

4. The Rational Zero Theorem If f(x) has integer coefficients and p/q (where p/q is reduced to lowest terms) is a rational zero of f, the p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n Possible zeros = factors of a 0 = p factors of a n q

5. Example 1 List all possible zeros of f(x) = -x 4 + 3x 2 + 4

6. Example 2 List all possible zeros of f(x) = 15x 3 + 14x 2 – 3x – 2

7. Finding zeros Use the Rational Zero Theorem and trial & error to find a rational zero Once the polynomial is reduced to a quadratic then use factoring or the quadratic formula to find the remaining zeros.

8. Example 3 Find the zeros of f(x) = x 3 + 2x 2 – 5x – 6

9. Example 4 Find all the zeros of f(x) = x 3 + 7x 2 + 11x – 3

10. Example 5 Solve: x 4 – 6x 2 – 8x + 24 = 0

11. Practice List all possible rational zeros 1. f(x) = 4x 5 + 12x 4 – x – 3 Find all zeros 2. f(x) = x 3 + 8x 2 + 11x – 20 3. f(x) = x 3 + x 2 – 5x – 2 Solve 4. x 4 – 6x 3 + 22x 2 – 30x + 13 = 0

12. Properties of Polynomial Equations 1. If a polynomial equation is of degree, n, then counting multiple roots separately, the equation has n roots. 2. If a + bi is a root of a polynomial equation with real coefficients (b ≠ 0), then the complex imaginary number a – bi is also a root. Complex imaginary roots, if they exist, occur in conjugate pairs. If 3i is a root, then –3i is also a root If 2 – 5i is a root, then 2 + 5i is also a root

13. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n ≥ 1, then the equation f(x) = 0 has at least one complex root.

14. Example 6 Find a 4 th degree polynomial function f(x) with real coefficients that has -2, 2, and i as zeros and such that f(3) = -150

15. Descartes’s Rule of Signs 1.The number of positive real zeros of f is either a. The same as the number of sign changes of f(x) OR b. Less than the number of sign changes by an even integer Note: if f(x) has only one sign change, then f has only one positive real zero 2. The number of negative real zeros of f is either a. The same as the number of sign changes of f(-x) OR b. Less than the number of sign changes by an even integer Note: if f(-x) has only one sign change, then f has only one negative real zero

16. Review Table on page 320 Negative Real Zeros f(-x) = -3x 7 + 2x 5 – x 4 + 7x 2 – x – 3 f(-x) = -4x 5 + 2x 4 – 3x 2 – x + 5 f(-x) = -7x 6 - 5x 4 – x + 9

17. Example 7 Determine the possible numbers of positive and negative real zeros of f(x) = x 3 + 2x 2 + 5x + 4

18. Practice Find a third-degree polynomial function with real coefficients that has -3 and i as zeros such that f (1) = 8 Determine the possible numbers of positive and negative real zeros of f(x) = x 4 - 14x 3 + 71x 2 – 154x + 120