1. Warm-Up 2/26 1. J

3. Rigor: You will learn how to find the real and complex zeros of polynomial functions. Relevance: You will be able to use graphs and equations of polynomial functions to solve real world problems.

4. 2-4 Zeros of Polynomial Functions

6. Example 1a: List possible rational zeros and determine which, if any are zeros. Step 1 Identify possible rational zeros. Step 2 Test possible rational zeros to determine if they are rational zeros. There are no rational zeros.

7. Example 1b: List possible rational zeros and determine which, if any, are zeros. Step 1 Identify possible rational zeros. Step 2 Test possible rational zeros to determine if they are rational zeros. 0 3 3 1 4– 12 ↓ – 9 – 1– 3 1 – 9 9 0 – 1 – 3 9 0 1 3– 9 ↓ – 30 10

8. Example 2: List possible rational zeros and determine which, if any, are zeros. Step 1 Identify possible rational zeros. Step 2 Test possible rational zeros to determine if they are rational zeros. – 22 – 8 – 13 3 – 7 8 ↓ – 626 43 0 – 2 4 – 1 3 – 13 ↓ 12 – 4 0 3 4

10. y = (x + 2)(x – 4)[x – (3 – i)][x – (3 + i)] y = (x + 2)(x – 4)[(x – 3) + i][(x – 3) – i] y = (x² – 4x + 2x – 8)[(x – 3)² – i(x – 3) + i(x – 3) – i²] y = (x² – 2x – 8)[(x – 3)² + 1] y = (x² – 2x – 8)(x² – 6x + 9 + 1) y = (x² – 2x – 8)(x² – 6x + 10) y = x 4 – 6x 3 + 10x² – 2x 3 + 12x² – 20x – 8x² + 48x – 80 y = x 4 – 8x 3 + 14x² + 28x – 80 (x – c)3 + i

11. Example 7: Write function as (a) the product of linear and irreducible quadratic factors and (b) the product of linear factors. Then (c) list all zeros. (a) the product of linear and irreducible quadratic factors

12. Example 7: Write function as (a) the product of linear and irreducible quadratic factors and (b) the product of linear factors. Then (c) list all zeros. (b) the product of linear factors (c) List all zeros

13. Example 8: Use given zeros to find all complex zeros. Then write the linear factorization of the function. 20 24 + 3 i – 4 – 3 i 1 – 6 – 22 ↓ – 13 2 – 3 i – 17 + 6 i 3 + 6 i 1 2 + 3 i 13 0 2 – 3 i 3 + 6 i – 2 – 3 i – 2 1 – 4 – 3 i 2 + 3 i ↓ – 4 – 6 i – 1 1 0 2 + 3 i

14. Example 8: Use given zeros to find all complex zeros. Then write the linear factorization of the function.

15. math! 2-4 Assignment: TX p127, 4-16 EOE & 32-52 EOE