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 :\(\boxed{\text{The zeros of the function are } x = 0 \text{ and } x = 15\text{. The multiplicity of the smallest zero (0) is 1. The multiplicity of the largest zero (15) is 2.}}\)