To construct a polynomial from given zeros and their multiplicities, we follow the rule that each zero contributes a factor to the polynomial, raised to the power of its multiplicity.
Given the zeros and their multiplicities:
- Zero at \( x = -4 \) with multiplicity 1
- Zero at \( x = 2 \) with multiplicity 2
- Zero at \( x = 2 \) (again) with multiplicity 3
Considering the multiplicities:
- The factor for \( x = -4 \) is \( (x + 4) \).
- The factor for \( x = 2 \) with multiplicity 2 is \( (x - 2)^2 \).
- The factor for \( x = 2 \) with additional multiplicity 3 (which is already covered in the earlier multiplicity) implies that we need to raise it to the combined multiplicity of 5 in total.
So, the complete polynomial can be expressed as: \[ f(x) = (x + 4)(x - 2)^{5} \]
This polynomial incorporates the zero at \( x = -4 \) once, and the zero at \( x = 2 \) a total of five times.
Therefore, the polynomial Greg should write is: \[ f(x) = (x + 4)(x - 2)^{5} \]