1. 1 What you will learn today…  How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function  How to use your graphing calculator to find real zeros

2. Objective: 6.7 Using the Fundamental Theorem of Algebra 2 A Calculator Investigation Use your graphing calculator to graph the following: 1. x 2 – 2 2. x 3 – 1 3. x 4 + 7x 3 – x 2 – 67x – 60 What do you notice about the number of solutions and the number of “roots” or zeros?

3. Objective: 6.7 Using the Fundamental Theorem of Algebra 3 OK…What About  Graph f(x) = x 4 + 6x 3 + 14x 2 + 54x + 45.  How many solutions or zeros does it look like it has?

4. Objective: 6.7 Using the Fundamental Theorem of Algebra 4 OK…What About  Graph f(x) = x 4 + 6x 3 + 14x 2 + 54x + 45.  How many solutions or zeros does it look likes it has?  Surprise! It has four roots…two of them aren’t real.  Let’s see!

5. Objective: 6.7 Using the Fundamental Theorem of Algebra 5 How to Find Complex Roots f(x) = x 4 + 6x 3 + 14x 2 + 54x + 45

6. Objective: 6.7 Using the Fundamental Theorem of Algebra 6 You Try!  Find all of the zeros of: f(x) = x 3 – 5x 2 + 4x - 20

7. Objective: 6.7 Using the Fundamental Theorem of Algebra 7 You May Have to Resort To…  The quadratic formula…egad.  Find all of the zeros of: f(x) = x 3 + x 2 – x + 15

8. Objective: 6.7 Using the Fundamental Theorem of Algebra 8 Working Backwards…Oh Boy  We can also turn this process around and use the zeros of a function to write the equation of the function.  The roots or zeros are: 2, 1, 4

9. Objective: 6.7 Using the Fundamental Theorem of Algebra 9 One More Time  Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. (2, -2, -6i)

10. Objective: 6.7 Using the Fundamental Theorem of Algebra 10 Taking it up a notch!  Write a polynomial function “f” of least degree that has real coefficients, a leading coefficient of 1, and 2 and 1+i as zeros.

11. Objective: 6.7 Using the Fundamental Theorem of Algebra 11 You Try!  Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1 and 1, -2+i, and -2-i as zeros.

12. Objective: 6.7 Using the Fundamental Theorem of Algebra 12 Homework  page 369, 16-20 even, 21, 23, 29, 31, 35, 41, 45, 47, 49 and 56