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

\\(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 the multiplicity of the largest zero is
(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.)

Explanation:

Factor the polynomial function

Using the Polynomial Functions knowledge point, we set \(f(x) = 0\) to find the zeros:

$$ x^3 - 20x^2 + 100x = 0 $$

First, factor out the greatest common factor, \(x\):

$$ x(x^2 - 20x + 100) = 0 $$

Find the zeros and their multiplicities

Using the Perfect Square Trinomials and Quadratic Equations knowledge points, we factor the quadratic expression inside the parentheses:

$$ x^2 - 20x + 100 = (x - 10)^2 $$

Thus, the fully factored form of the function is:

$$ f(x) = x(x - 10)^2 $$

Setting each factor to zero gives the zeros of the function:

• From the factor \(x\), we get the zero \(x = 0\). Since the exponent of this factor is \(1\), its multiplicity is \(1\).
• From the factor \((x - 10)^2\), we get the zero \(x = 10\). Since the exponent of this factor is \(2\), its multiplicity is \(2\).

Using the Multiplicity of Zeros knowledge point, we identify that there are two zeros:

• The smallest zero is \(0\) with a multiplicity of \(1\).
• The largest zero is \(10\) with a multiplicity of \(2\).

Determine the graph behavior at each zero

Using the Graph Behavior at Zeros knowledge point:

• For the zero \(x = 0\) (multiplicity \(1\), which is odd), the graph crosses the x-axis.
• For the zero \(x = 10\) (multiplicity \(2\), which is even), the graph touches the x-axis and turns around.

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

The graph crosses the x-axis at the smaller zero, \(x = 0\).
The graph touches the x-axis and turns around at the larger zero, \(x = 10\).