1. Polynomial Multiplicity

2. Review Zeros of Polynomial Functions

3. If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. These values of x are the roots, or solutions, of the polynomial equation f(x)=0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function.

4. Find all zeros of f(x)= x3+4x2- 3x - 12

5. Example
Find all zeros of x3+2x2- 4x-8=0

6. Find all zeros of x3+2x2- 4x-8=0
Answer
Find all zeros of x3+2x2- 4x-8=0
Zeros: x = 2, -2

7. Multiplicity of x-Intercepts

10. Multiplicity Video
Watch both!!! Create a table for multiplicity to graph!
Find the zeros, multiplicity, end behavior, and y-intercept to sketch a graph.

11. Graphing Calculator- Finding the Zeros x3+2x2- 4x-8=0
One zero of the function
One of the zeros
Other zero
The other zero
The x-intercepts are the zeros of the function. To find the zeros, press 2nd Trace then #2. The zero -2 has multiplicity of 2.

12. Example
Find the zeros of f(x)=(x- 3)2(x-1)3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
Continued on the next slide.

13. Zeros: x=3 multiplicity 2 – touches and turns around
Answer
Find the zeros of f(x)=(x- 3)2(x-1)3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
Zeros: x=3 multiplicity 2 – touches and turns around
x=1 multiplicity 3 – crosses
Continued on the next slide.

14. Example
Now sketch this function on your INB.
f(x)=(x- 3)2(x-1)3
Remember end behavior!!!

15. Answer
Now sketch this function on your INB.
f(x)=(x- 3)2(x-1)3
Remember end behavior!!!

16. A Strategy for Graphing Polynomial Functions

18. Example
Graph f(x)=x4- 4x2 using what you have learned in this section.

19. Answer
Graph f(x)=x4- 4x2 using what you have learned in this section.

20. Example
Graph f(x)=x3- 9x2 using what you have learned in this section.

21. Answer
Graph f(x)=x3- 9x2 using what you have learned in this section.

22. State whether the graph crosses the x-axis, or touches the x-axis and turns around at the zeros of 1, and - 3.
f(x)=(x-1)2(x+3)3
(a)
(b)
(c)
(d)

23. State whether the graph crosses the x-axis, or touches the x-axis and turns around at the zeros of 1, and - 3.
f(x)=(x-1)2(x+3)3
(a)
(b)
(c)
(d)