Warm Up - For the polynomial listed below:

1. Warm Up-For the polynomial listed below: • Sketch a graph. Label the zero’s on the graph. • State the degree and sign of the leading term. • State the end behavior. • Classify each root.

2. Fundamental Theorem of Algebra

3. Learning Targets • Introduce the FTA • Use FTA to determine how many solutions a polynomial will have • Look at imaginary vs. real solutions • Determine the minimum number of real solutions polynomials will have

4. Determine A Factored form for the following polynomial ***The highest degree a term can be is cubed.

5. Cont. • What is the leading term? • (E.B.  even, positive) • How many roots does this polynomial have? • Eight

6. Determine A Factored form for the following polynomial What is the leading term? How many roots does this polynomial have? ***The highest degree a term can be is cubed.

7. Solution • Leading Term: • (E.B.  odd, negative) • Roots: • Seven

8. Answer the Following • How many roots would you expect the following polynomial to have? • Conjectures: • Rule?

9. Is this always true?

10. Fundamental Theorem of Algebra • The fundamental theorem of algebra states that for each polynomial with complexcoefficients there are as many roots as the degree of the polynomial. • If a polynomial’s leading term has degree n then it will have n roots.

11. FTA • Based on the FTA we must always have the same number of roots as our leading terms degree. • However, this does not mean all of our roots will be real numbers (where the graph crosses the x-axis). • Can you think of any potential functions that do not cross the x-axis the same number of times as their leading terms degree? • When this happens we have what are called imaginary roots (more on these next week)

12. FTA • The combination of real roots and imaginary roots must always equal the same degree of the leading term • Next we will view some graphs and look at the number of real roots and imaginary roots • While we look at these graphs pause and ponder and come up with some skeptical questions or I notice statements…

13. 2nd Degree Polynomial

14. 3rd Degree Polynomial

15. 3rd Degree Polynomial

16. 4th Degree Polynomial

17. 4th Degree Polynomial

18. 5th Degree Polynomial

19. nth Degree Polynomial

20. FTA Rules with Polynomials • Can an even degree polynomial have less than one real root? Why? • Can an even degree polynomial ever have an odd number of real roots? Why? • Can an odd degree polynomial have less than one real root? Why? • Can an odd degree polynomial ever have an even number of real roots? Why?

21. Recap • Fundamental Theorem of Algebra: • Nth degree polynomials have N number of complex roots • A complex root can be real or imaginary • Real root occurs when the graph crosses the x-axis • The degree of a polynomial will determine the lowest amount of real roots that polynomial can have • This depends on odd-even and its end behavior

22. For tonight: • Worksheet • I will be collecting these on Thursday and giving feedback

23. Exit Ticket • Sketch the following graphs: • 5th Degree polynomial with a negative leading coefficient: • 5 real roots • 4 real roots • 3 real roots • 2 real roots • 1 real root • 0 roots • If the graph cannot be sketched put down N/A and state why