To construct a polynomial function based on the given zeros and their multiplicities, we start with the zeros and multiplicities provided:
- For the zero \( x = -4 \) with multiplicity 2, the corresponding factor of the polynomial is \( (x + 4)^2 \).
- For the zero \( x = 2 \) with multiplicity 3, the corresponding factor of the polynomial is \( (x - 2)^3 \).
Putting these together, the polynomial function can be written as:
\[ f(x) = (x + 4)^2 (x - 2)^3 \]
Now let's compare this result with the provided function options:
- Function #1: \( f(x) = (x + 4)(x - 2) \) (This is incorrect, as it does not account for the correct multiplicities.)
- Function #2: \( f(x) = (x + 4)^2 (x - 2)^3 \) (This is correct.)
- Function #3: \( f(x) = (x - 4)^2 (x + 2)^3 \) (This is incorrect, as the zeros are not correct.)
Thus, Greg should write Function #2.
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