Question

find the zeros for the given polynomial function and give the multiplicity for each zero. state whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

(f(x) = x^3 - 20x^2 + 100x)

determine the zero(s), if they exist.

the zero(s) is/are 0,10
(type integers or decimals. use a comma to separate answers as needed.)

determine the multiplicities of the zero(s), if they exist. select the correct choice below and, if necessary, fill in the answer box(es) within your choice.

a. there are two zeros. the multiplicity of the smallest zero is 1. the multiplicity of the largest zero is 2.
(simplify your answers.)
b. there is one zero. the multiplicity of the zero is
(simplify your answer.)
c. there are three zeros. the multiplicity of the smallest zero is . the multiplicity of the largest zero is . the multiplicity of the other zero is __.
(simplify your answers.)

determine the behavior of the function at each zero. select the correct choice below and, if necessary, fill in the answer boxes within your choice.

a. the graph crosses the x-axis at x = . the graph touches the x-axis and turns around at x = .
(type integers or decimals. simplify your answers. use a comma to separate answers as needed.)
b. the graph touches the x-axis and turns around at all zeros.
c. the graph crosses the x-axis at all zeros.

Explanation:

Factor the polynomial function

Using the Polynomial Functions knowledge point

$$
LATEXBLOCK0
$$

Find the zeros and multiplicities

Using the Multiplicity of Zeros knowledge point

$$
LATEXBLOCK1
$$

Determine the graph behavior at each zero

Using the Graph Behavior at Zeros knowledge point

$$
LATEXBLOCK2
$$

Answer:

Question 1

The zero(s) is/are 0,10

Question 2

• A. There are two zeros. The multiplicity of the smallest zero is 1. The multiplicity of the largest zero is 2. (Correct answer)
• B. There is one zero. The multiplicity of the zero is [ ].
• C. There are three zeros. The multiplicity of the smallest zero is [ ]. The multiplicity of the largest zero is [ ]. The multiplicity of the other zero is [ ].

Question 3

• A. The graph crosses the x-axis at x = 0. The graph touches the x-axis and turns around at x = 10. (Correct answer)
• B. The graph touches the x-axis and turns around at all zeros.
• C. The graph crosses the x-axis at all zeros.