Zeros of Polynomial Functions

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) = 2x3 – 3x2 – 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 a0 and q is a factor of the leading coefficient an Possible zeros = factors of a0 = p factors of an q

5. Example 1 • List all possible zeros of f(x) = -x4 + 3x2 + 4

6. Example 2 • List all possible zeros of f(x) = 15x3 + 14x2 – 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) = x3 + 2x2 – 5x – 6

9. Example 4 • Find all the zeros of f(x) = x3 + 7x2 + 11x – 3

10. Example 5 • Solve: x4 – 6x2 – 8x + 24 = 0

11. Practice • List all possible rational zeros • f(x) = 4x5 + 12x4 – x – 3 • Find all zeros • f(x) = x3 + 8x2 + 11x – 20 • f(x) = x3 + x2 – 5x – 2 • Solve • x4 – 6x3 + 22x2 – 30x + 13 = 0

12. Properties of Polynomial Equations • If a polynomial equation is of degree, n, then counting multiple roots separately, the equation has n roots. • 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 4th 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 • The same as the number of sign changes of f(x) OR • 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 • The same as the number of sign changes of f(-x) OR • 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) = -3x7 + 2x5 – x4 + 7x2 – x – 3 • f(-x) = -4x5 + 2x4 – 3x2 – x + 5 • f(-x) = -7x6 - 5x4 – x + 9

17. Example 7 • Determine the possible numbers of positive and negative real zeros of f(x) = x3 + 2x2 + 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) = x4 - 14x3 + 71x2 – 154x + 120