To determine the correct polynomial function based on the given zeros and multiplicities, we need to identify the indications of zero values (roots) and their corresponding multiplicities.
Typically, if a polynomial has a zero \( r \) with a multiplicity of \( m \), the term represented would be \( (x - r)^m \).
From the provided options, let’s summarize the zeros and multiplicities based on how they are structured:
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Option A: \( f(x) = (x + 24)^7 (x + 9)^{12} \)
- Zeros: -24 (multiplicity 7) and -9 (multiplicity 12)
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Option B: \( f(x) = (x - 24)^7 (x - 9)^{12} \)
- Zeros: +24 (multiplicity 7) and +9 (multiplicity 12)
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Option C: \( f(x) = (x - 24)^{12} (x - 9)^7 \)
- Zeros: +24 (multiplicity 12) and +9 (multiplicity 7)
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Option D: \( f(x) = (x + 24)^{12} (x + 9)^7 \)
- Zeros: -24 (multiplicity 12) and -9 (multiplicity 7)
Since the problem does not explicitly state the required zeros and their corresponding multiplicities, you should identify which of the above options would best match the specifications given (if any) once they are clarified.
However, if the intention is to provide a polynomial based on common practices, it generally employs negative roots (i.e., \( (x - r) \)) unless specified otherwise.
Assuming the task is to express these zeros negatively, Option B \( f(x) = (x - 24)^7 (x - 9)^{12} \) would typically be the most appropriate choice as it clearly states the classic formation involving positive zeros converted into polynomial factors.
Therefore, based on common conventions and unless instructed otherwise, the answer would be:
B: \( f(x) = (x - 24)^7 (x - 9)^{12} \)
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