Final answer:
After multiplying (x + 2), (x), (x - 2), and (x - 1) which correspond to the zeros -2, 0, 2, and 1 respectively, the polynomial function that fits the given zeros is Option A: f(x) = (x - 1)(x + 2)(x - 2)(x).
Step-by-step explanation:
To find the polynomial function with the given real zeros of -2, 0, 2, and 1, we must first write down the factors associated with each zero. A zero at x = -2 corresponds to the factor (x + 2), a zero at x = 0 corresponds to the factor (x), a zero at x = 2 corresponds to the factor (x - 2), and a zero at x = 1 corresponds to the factor (x - 1). We then multiply these factors to obtain a polynomial that has these zeros.
The correct polynomial based on the given zeros is:
f(x) = (x + 2)(x)(x - 2)(x - 1)
Looking at the options provided:
- Option A: f(x) = (x - 1)(x + 2)(x - 2)(x) is the correct polynomial function.
- Option B: f(x) = (x - 1)(x + 2)(x - 2)(x + 2) contains a repeated factor and is incorrect.
- Option C: f(x) = (x + 2)(x - 1)(x + 2)(x - 2) also has a repeated factor and is incorrect.
- Option D: f(x) = (x + 1)(x - 2)(x + 2)(x - 1) has incorrect factors and is not the polynomial we are looking for.