Zeros of Polynomial Functions

1. Zeros of Polynomial Functions Section 2.5

2. Objectives • Use the Factor Theorem to show that x-c is a factor a polynomial. • Find all real zeros of a polynomial given one or more zeros. • Find all the rational zeros of a polynomial using the Rational Zero Test. • Find all real zeros of a polynomial using the Rational Zero Test. • Find all zeros of a polynomial. • Write the equation of a polynomial given some of its zeros.

3. Vocabulary • rational zero • real zero • multiplicity

4. Factor Theorem • Let f (x) be a polynomial • If f(c) = 0, then x – c is a factor of f (x). • If x – c,is a factor of f(x), then f(c) = 0.

5. If c = 3 is a zero of the polynomial find all other zeros of P(x).

6. Use synthetic division to show that x = 6 is a solutions of the equation

7. If has integercoefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term, a0, and q is a factor of the leading coefficient, an. Rational Root (Zero) Theorem (Test)

8. Find all the rational zeros of the polynomial

9. Find all the real zeros of the polynomial

10. Linear Factorization Theorem If , where n ≥ 1 and an ≠ 0, then Where c1, c2, . . ., cn are complex numbers (possibly real and not necessarily distinct).

11. Factor into linear and irreducible quadratic factors with real coefficients.

12. Find all the zeros of the polynomial

13. Find all the zeros of the polynomial

14. Find the equation of a polynomial of degree 4 with integer coefficients and leading coefficient 1 that had zeros x = -2-3i, and at x = 1 with x = 1 a zero of multiplicity 2.

15. Descartes’s Rule of Signs • Let , • Be a polynomial with real coefficients. • 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 of f(x) by a positive even integer. If f(x) has only one variation in sign, then f has exactly one positive real zero.

16. Descartes’s Rule of Signs • Let , • Be a polynomial with real coefficients. • 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 of f(—x) by a positive even integer. If f(—x) has only one variation in sign, then f has exactly one negative real zero.

17. 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.