Find the zeros and their multiplicities

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Local Behavior of Polynomial Functions
Learning Objectives

· Identify intercepts of polynomial functions in factored form

· Understand the relationship between degree, turning points, and x-intercepts

· Understand the intermediate value theorem

· Use factoring to find zeros of polynomial functions

· Identify zeros and their multiplicities from an equation or a graph

Identify intercepts of polynomial functions in factored form

1. Find the x- and y-intercepts of .

2. Find the x- and y-intercepts of .

3. Find the x- and y-intercepts of .

Interpreting Turning Points

A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

A polynomial of degree n has n – 1 turning points.

Understand the relationship between degree, turning points, and x-intercepts

4. Find the maximum number of turning points of the following functions:

a.

b.

c.

Intermediate Value Theorem

Let be a polynomial function. The Intermediate Value Theorem states that if and have opposite signs, then there exists at least one value between and for which .

Understand the intermediate value theorem

5. Show that the function has a real zero between and

6. Show that the function has a real zero between and .

7. Show that the function has a real zero between and .

Use factoring to find zeros of polynomial functions

8. Find the x-intercepts of the following functions:

a.

b.

c.

Identify zeros and their multiplicities from an equation or a graph

9. Find all the zeros and their multiplicities for the function .

10. Find all the zeros and their multiplicities for the function .

11. Use the graph of the function of degree 4 below to find the zeros and their possible multiplicities.

ANSWER KEY

1.

2.

3.

4a. The function will have a maximum of 3 turning points.

4b. The function will have a maximum of 2 turning points.

4c. The function will have a maximum of 4 turning points.

5. . The sign change shows there is a zero between the given values.

6. . The sign change shows there is a zero between the given values.

7. . The sign change shows there is a zero between the given values.

8a.

8b.

8c.

9. have multiplicity of 1. has multiplicity 2.

10. has multiplicity 3, has multiplicity 2, has multiplicity 1.

11. . The graph touches the x-axis at both of the zeros so their multiplicity must be even and 2 since the degree of the function is 4.

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