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.
\[
f(x)=x^{3}-30 x^{2}+225 x
\]
Determine the zero(s), if they exist.
The zero(s) is/are $\square$
(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 $\square$. The multiplicity of the largest zero is $\square$. The multiplicity of the other zero is $\square$. (Simplify your answers.)
B. There are two zeros. The multiplicity of the smallest zero is $\square$. The multiplicity of the largest zero is $\square$. (Simplify your answers)
C. There is one zero. The multiplicity of the zero is (Simplify your answer.)
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Answer
\(\boxed{\text{The zeros of the function are } x = 0 \text{ and } x = 15\text{. The multiplicity of the smallest zero (0) is 1. The multiplicity of the largest zero (15) is 2.}}\)
Step 1 ：Set the function equal to zero: \(x^{3}-30 x^{2}+225 x = 0\)
Step 2 ：Factor out an \(x\) from each term: \(x(x^{2}-30x+225) = 0\)
Step 3 ：Set each factor equal to zero to find the zeros of the function: \(x = 0\) and \(x^{2}-30x+225 = 0\)
Step 4 ：Factor the quadratic equation further: \(x(x-15)(x-15) = 0\)
Step 5 ：Set each factor equal to zero to find the zeros of the function: \(x = 0\) and \(x-15 = 0\) which gives \(x = 15\)
Step 6 ：Determine the multiplicity of each zero. The multiplicity of a zero is the number of times it appears as a root of the polynomial. In this case, \(x = 0\) appears once and \(x = 15\) appears twice
Step 7 ：\(\boxed{\text{The zeros of the function are } x = 0 \text{ and } x = 15\text{. The multiplicity of the smallest zero (0) is 1. The multiplicity of the largest zero (15) is 2.}}\)
