\begin{tabular}{l} For the following factored polynomial, find all of the zeros and their multiplicities. \ $$ \qquad f(x)=(x-2)^{5}(x+3)^{9}(x+6)^{8} $$ \ Select the correct answer below: \ x=6 with multiplicity $$ 8 ; x=3 $$ with multiplicity $$ 9 ; x=2 $$ with multiplicity 5 \ $$ x=6 $$ with multiplicity $$ 8 ; x=-3 $$ with multiplicity $$ 9 ; x=-2 $$ with multiplicity 5 \ $$ x=-6 $$ with multiplicity $$ 8 ; x=3 $$ with multiplicity $$ 9 ; x=2 $$ with multiplicity 5 \ $$ x=-6 $$ with multiplicity $$ 8 ; x=-3 $$ with multiplicity $$ 9 ; x=2 $$ with multiplicity 5 \ $$ x=-6 $$ with multiplicity $$ 8 ; x=3 $$ with multiplicity $$ 9 ; x=-2 $$ with multiplicity 5 \ $$ x=-6 $$ with multiplicity $$ 8 ; x=-3 $$ with multiplicity $$ 9 ; x=-2 $$ with multiplicity 5 \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

To determine the zeros of the polynomial $$ f(x) = (x-2)^{5}(x+3)^{9}(x+6)^{8} $$ and their respective multiplicities, we examine each factor:

1. **Factor $$ (x - 2)^5 $$**
   - **Zero:** $$ x = 2 $$
   - **Multiplicity:** 5

2. **Factor $$ (x + 3)^9 $$**
   - **Zero:** $$ x = -3 $$
   - **Multiplicity:** 9

3. **Factor $$ (x + 6)^8 $$**
   - **Zero:** $$ x = -6 $$
   - **Multiplicity:** 8

Putting it all together, the zeros and their multiplicities are:
- $$ x = -6 $$ with multiplicity **8**
- $$ x = -3 $$ with multiplicity **9**
- $$ x = 2 $$ with multiplicity **5**

**Correct Answer:**

$$ x=-6 $$ with multiplicity $$ 8 $$; $$ x=-3 $$ with multiplicity $$ 9 $$; $$ x=2 $$ with multiplicity $$ 5 $$
The zeros and their multiplicities are: - $$ x = -6 $$ with multiplicity 8 - $$ x = -3 $$ with multiplicity 9 - $$ x = 2 $$ with multiplicity 5

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