\begin{tabular}{l} For the following factored polynomial, find all of the zeros and their multiplicities. \\ \( \qquad f(x)=(x+2)^{8}(x+7)^{7} \) \\ Select the correct answer below: \\ \( x=7 \) with multiplicity 7 , and \( x=-2 \) with multiplicity 8 \\ \( x=-7 \) with multiplicity 8 , and \( x=-2 \) with multiplicity 7 \\ \( x=-7 \) with multiplicity 7 , and \( x=2 \) with multiplicity 8 \\ \( x=7 \) with multiplicity -7 , and \( x=8 \) with multiplicity -2 \\ \( x=-7 \) with multiplicity 7 , and \( x=-2 \) with multiplicity 8 \\ \( x=7 \) with multiplicity 7 , and \( x=2 \) with multiplicity 8 \\ 8 \\ \( x=1 \) \\ \hline\end{tabular}

To find the zeros of the polynomial \( f(x) = (x+2)^{8}(x+7)^{7} \) and their multiplicities, we can follow these steps: 1. **Identify the factors of the polynomial**: The polynomial is factored into two parts: \( (x+2)^{8} \) and \( (x+7)^{7} \). 2. **Set each factor to zero**: - For \( (x+2)^{8} = 0 \): \[ x + 2 = 0 \implies x = -2 \] - For \( (x+7)^{7} = 0 \): \[ x + 7 = 0 \implies x = -7 \] 3. **Determine the multiplicities**: - The factor \( (x+2)^{8} \) has a zero at \( x = -2 \) with a multiplicity of 8. - The factor \( (x+7)^{7} \) has a zero at \( x = -7 \) with a multiplicity of 7. 4. **Summarize the results**: - The zeros of the polynomial are: - \( x = -2 \) with multiplicity 8 - \( x = -7 \) with multiplicity 7 Now, let's match this with the provided options: - \( x=7 \) with multiplicity 7, and \( x=-2 \) with multiplicity 8 - \( x=-7 \) with multiplicity 8, and \( x=-2 \) with multiplicity 7 - \( x=-7 \) with multiplicity 7, and \( x=2 \) with multiplicity 8 - \( x=7 \) with multiplicity -7, and \( x=8 \) with multiplicity -2 - \( x=-7 \) with multiplicity 7, and \( x=-2 \) with multiplicity 8 - \( x=7 \) with multiplicity 7, and \( x=2 \) with multiplicity 8 The correct answer is: - \( x=-7 \) with multiplicity 7, and \( x=-2 \) with multiplicity 8.