1. OUTLINE  Homework (and homework questions)  Ask any review questions you want  Review long division, solve by factoring and graphing calculators  BREAK  6.5 Notes (not on quiz)  Homework time / study time

2. Homework:

4. Quiz Friday! About 40-50 pts  Key is online! Class notes are online!  Questions?

5. Review Long division

6. Review Solve by Factoring

7. Graphing calculators  Find all the real zeros of each  How many roots or zeros should each have?

8. Section 6.5 Rational Root Theorem!

9. Objectives  Use the rational, irrational, and imaginary root theorems to locate all the roots of a given polynomial equation

10. Rational Root Theorem  This only works if we have integer coefficients.  Possible rational roots =

11. Find the rational roots of  List the possible rational roots:  Test using substitution or synthetic division

12. Use the rational root theorem to find ALL roots

14. The Irrational Root Theorem  If is a root of a polynomial equation with rational coefficients, then the conjugate is also a root  In other words, irrational roots occur in conjugate pairs.  If is a root, then is a root

15. The Imaginary Root Theorem  If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a – bi also is a root  Imaginary roots also occur in conjugate pairs

16.  If 3, and are roots of a polynomial with integer coefficients, find two additional roots.  What is the degree of this polynomial?

17. Find a third-degree polynomial equation with rational coefficients that has roots 3 and 1 + i

18. Homework  Page 333 (1-3), 7, 8, 11, (16-18), 23, 24 (32-36) even  Due Monday