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}-4x^{2}+4x

determine the zero(s), if they exist.
the zero(s) is/are 0,2.
(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 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.)
b. there is one zero. the multiplicity of the zero is
(simplify your answer.)
c. there are two zeros. the multiplicity of the smallest zero is 1. the multiplicity of the largest zero is 2.
(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 crosses the x - axis at all zeros.
c. the graph touches the x - axis and turns around at all zeros.

Explanation:

Step1: Factor the polynomial

First, factor out the greatest - common factor from \(f(x)=x^{3}-4x^{2}+4x\). We can factor out \(x\) to get \(f(x)=x(x^{2}-4x + 4)\). Then, factor the quadratic \(x^{2}-4x + 4=(x - 2)^{2}\). So, \(f(x)=x(x - 2)^{2}\).

Step2: Find the zeros

Set \(f(x)=0\). Then \(x(x - 2)^{2}=0\). Using the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\). So \(x=0\) or \((x - 2)^{2}=0\). Solving \((x - 2)^{2}=0\) gives \(x = 2\). The zeros are \(x = 0\) and \(x = 2\).

Step3: Determine the multiplicities

For the factor \(x\), the exponent is 1, so the multiplicity of \(x = 0\) is 1. For the factor \((x - 2)^{2}\), the exponent is 2, so the multiplicity of \(x = 2\) is 2.

Step4: Determine the behavior at the zeros

If the multiplicity of a zero is odd, the graph of the function crosses the \(x\) - axis at that zero. If the multiplicity of a zero is even, the graph of the function touches the \(x\) - axis and turns around at that zero. Since the multiplicity of \(x = 0\) is 1 (odd), the graph crosses the \(x\) - axis at \(x = 0\). Since the multiplicity of \(x = 2\) is 2 (even), the graph touches the \(x\) - axis and turns around at \(x = 2\).

Answer:

The zeros are \(0\) and \(2\). The multiplicity of the smallest zero (\(x = 0\)) is \(1\), and the multiplicity of the largest zero (\(x = 2\)) is \(2\). The graph crosses the \(x\) - axis at \(x = 0\) and touches the \(x\) - axis and turns around at \(x = 2\). So the answers for the multiple - choice parts are:
For the multiplicity part: C. There are two zeros. The multiplicity of the smallest zero is 1. The multiplicity of the largest zero is 2.
For the behavior part: A. The graph crosses the \(x\) - axis at \(x = 0\). The graph touches the \(x\) - axis and turns around at \(x = 2\).