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, click on "None." Zero(s) of multiplicity one: Zero(s) of multiplicity two:

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Zero(s) of multiplicity one: $$ -4 $$ Zero(s) of multiplicity two: $$ 11, 12 $$ To find the zeros of the polynomial function $$ f(x) = (x-11)^{2}(x+4)(x-12)^{2} $$, we need to identify the values of $$ x $$ that make $$ f(x) = 0 $$ and their corresponding 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 their multiplicities:** - **Zero(s) of multiplicity one:** This is the zero that appears only once. From our analysis, the zero of multiplicity one is: - $$ -4 $$ - **Zero(s) of multiplicity two:** These are the zeros that appear twice. From our analysis, the zeros of multiplicity two are: - $$ 11, 12 $$ Now we can summarize the results: - **Zero(s) of multiplicity one:** $$ -4 $$ - **Zero(s) of multiplicity two:** $$ 11, 12 $$ Thus, the final answers are: - Zero(s) of multiplicity one: $$ -4 $$ - Zero(s) of multiplicity two: $$ 11, 12 $$

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