Suppose that the polynomial function \( f \) is defined as follows. \[ f(x)=(x-11)^{2}(x+4)(x-12)^{2} \] Ust each zero of \( f \) according to its multiplicity in the categories below. If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, dick on "None." Zero(s) of multiplicity one: Zero(s) of multiplicity three:

To find the zeros of the polynomial function \( f(x) = (x-11)^{2}(x+4)(x-12)^{2} \), we need to identify the roots and their respective multiplicities. 1. **Finding the zeros:** - The factor \( (x-11)^{2} \) gives a zero at \( x = 11 \) with multiplicity 2. - The factor \( (x+4) \) gives a zero at \( x = -4 \) with multiplicity 1. - The factor \( (x-12)^{2} \) gives a zero at \( x = 12 \) with multiplicity 2. 2. **Categorizing the zeros by multiplicity:** - **Zero(s) of multiplicity one:** This corresponds to the zero that appears only once. From our analysis, the zero \( x = -4 \) has multiplicity 1. - **Zero(s) of multiplicity three:** There are no zeros with a multiplicity of three in the given polynomial. Now, we can summarize the results: - **Zero(s) of multiplicity one:** \(-4\) - **Zero(s) of multiplicity three:** None Thus, the final answers are: - Zero(s) of multiplicity one: \(-4\) - Zero(s) of multiplicity three: None