1. Homework Log Wed 1/6 Lesson 5 – 3 Learning Objective: To apply the Fundamental Theorem of Algebra & Descartes’ Rule of Signs Hw: #505 Pg. 293 #1 – 25 odd

2. 1/6/16 Lesson 5 – 3 Fundamental Theorem of Algebra & Descartes Rule of Signs Day 1 Advanced Math/Trig

3. Learning Objective To apply the Fundamental Theorem of Algebra To apply Descartes Rule of Signs

4. Fundamental Theorem of Algebra

5. State the degree, list the distinct zeros & their multiplicities 1

7. Find Polynomial of least degree that is monic

8. Find Polynomial of least degree 4. Zeroes are 1, – 1, and 2. P(0) = 4 x = 1x = – 1 x = 2 x – 1 = 0x + 1 = 0x – 2 = 0 P(x) = a (x – 1)(x + 1)(x – 2) Plug in x = 0 and make P(x) = to 4 to solve for a 4 = a (0 – 1)(0 + 1)(0 – 2) 4 = 2 a a = 2 Given to find leading coeff P(x) = 2(x – 1)(x + 1)(x – 2)

9. Descartes’ Rule of Signs

11. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots 2 Sign Changes 1 Sign Change Hint: Only odd exponents will change sign when plugging in – x

12. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots + - 2 1 0 0 1 2 Degree 3, so each row should add up to 3  Sum = 3

13. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots 2 Sign Changes 2 Sign Change

14. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots + – 2 2 0 2 0 2 0 2 2 0 0 4

15. 5 Sign Change 4 Sign Changes 5 or 3 or 1 (+) real roots 4 or 2 or 0 (–) real roots

16. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots + – 5 4 0 5 2 2 5 0 4 3 4 2 3 2 4 3 0 6 1 4 4 1 2 6 1 0 8  It could have 5 positive roots, 4 negative roots, & 0 imaginary roots  It could have 3 positive roots, 2 negative roots, & 4 imaginary roots

17. Use Descartes’ Rule of Signs to Discuss the Nature of the Roots 3 Sign Change 1 Sign Change + – 3 1 0 1 1 2

18. Ticket Out the Door

19. Homework #505 Pg. 293 #1 – 25 odd