\begin{tabular}{l} For the following factored polynomial, find all of the zeroes and their multiplicities. \ $$ \qquad f(x)=(x-7)^{3}(x-6)^{2}(x-3)^{6} $$ \ Select the correct answer below: \ x=3 with multiplicity $$ 6 ; x=-6 $$ with multiplicity $$ 2 ; x=-7 $$ with multiplicity 3 \ $$ x=3 $$ with multiplicity $$ 6 ; x=6 $$ with multiplicity $$ 2 ; x=7 $$ with multiplicity 3 \ $$ x=3 $$ with multiplicity $$ 6 ; x=6 $$ with multiplicity $$ 2 ; x=-7 $$ with multiplicity 3 \ $$ x=3 $$ with multiplicity $$ 6 ; x=-6 $$ with multiplicity $$ 2 ; x=7 $$ with multiplicity 3 \ $$ x=-3 $$ with multiplicity $$ 6 ; x=-6 $$ with multiplicity $$ 2 ; x=-7 $$ with multiplicity 3 \ \hline$$ x=-3 $$ with multiplicity $$ 6 ; x=6 $$ with multiplicity $$ 2 ; x=-7 $$ with multiplicity 3 \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

To determine the zeros of the polynomial $$ f(x) = (x-7)^3 (x-6)^2 (x-3)^6 $$ and their respective multiplicities, let's analyze each factor:

1. **$$ (x-7)^3 $$**
   - **Zero:** $$ x = 7 $$
   - **Multiplicity:** 3

2. **$$ (x-6)^2 $$**
   - **Zero:** $$ x = 6 $$
   - **Multiplicity:** 2

3. **$$ (x-3)^6 $$**
   - **Zero:** $$ x = 3 $$
   - **Multiplicity:** 6

**Summary of Zeros and Their Multiplicities:**
- $$ x = 3 $$ with multiplicity **6**
- $$ x = 6 $$ with multiplicity **2**
- $$ x = 7 $$ with multiplicity **3**

**Correct Answer:**
$$
\boxed{x=3 \text{ with multiplicity } 6;\ x=6 \text{ with multiplicity } 2;\ x=7 \text{ with multiplicity } 3}
$$
The zeros and their multiplicities are: - $$ x = 3 $$ with multiplicity 6 - $$ x = 6 $$ with multiplicity 2 - $$ x = 7 $$ with multiplicity 3

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