2. 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write a polynomial function. Standard: 2.8.11.N. Solve equations both symbolically and graphically.

3. The Rational Root Theorem can be used to identify possible roots of polynomial equations with integer coefficients. Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If p/q (in lowest terms) is a root of P(x) = 0, then p is a factor of the constant term of P and q is a factor of the leading coefficient of P.

7. * 8x 3 + 10x 2 - 11x + 2 = 0

9. * Q(x) = x 3 - 6x 2 + 7x + 2

10. * Q(x) = x 3 + 4x 2 – 6x – 12

11. Ex 3. Find all of the zeros of: Same as above, but you will get an imaginary #. * P(x) = 3x 3 – 10x 2 + 10x – 4 * P(x) = 3x 3 – 10x 2 + 10x – 4

12. * P(x) = x 3 - 9x 2 + 49x – 145

13. * P(x) = -4x 3 + 2x 2 – x + 3

14. Complex Conjugate Root Theorem If P is a polynomial function with real-number coefficients and a + bi (where b ≠ 0) is a root of P(x) = 0, then a – bi is also a root of P(x) = 0.

16. Fundamental Theorem of Algebra: Every polynomial function of degree n ≥ 1 has at least one complex zero Corollary: Every polynomial function of degree n ≥ 1 has exactly n complex zeros, counting multiplicities.

18. Writing Activities

19. PSSA Warm-Up Question Algebra II Chp 7 Standard 2.8.11 S Analyze linear, polynomial, and rational functions. How can you identify and describe functions and their graphs? What are the functions zero(s)? 1). Linear Function y = ½x + 2 2). Quadratic Function y = x 2 – 2x – 3 3). Cubic Function y = x 3 – 4x

20. Review of Zeros of Polynomial Functions

22. Homework Integrated Algebra II- Section 7.5 Level A Academic Algebra II- Section 7.5 Level B