Final answer:
The polynomial with a degree of 3 and zeros -2, 3+i is f(x) = (x + 2)(x - 3 - i)(x - 3 + i). This is correct because complex zeros come in conjugate pairs, and the polynomial is constructed accordingly.
Step-by-step explanation:
To create a polynomial, f(x), with a degree of 3 and given zeros -2 and 3+i, we need to consider that the zeros of a polynomial with real coefficients that are complex must come in conjugate pairs. This means that if 3+i is a zero, then its conjugate 3-i is also a zero. Using this information, we can construct the polynomial.
The correct polynomial with the given zeros -2, 3+i, and 3-i is expressed as f(x) = (x + 2)(x - 3 - i)(x - 3 + i). To confirm, we can multiply out the factors:
(x + 2)(x - 3 - i)(x - 3 + i) = (x + 2)[(x - 3) - i][(x - 3) + i] = (x + 2)((x - 3)^2 - (i)^2) = (x + 2)(x^2 - 6x + 9 + 1) = (x + 2)(x^2 - 6x + 10)
Expanding and simplifying these expressions, we find:
f(x) = x^3 - 4x^2 + 2x + 20
This confirms that the correct answer is (a), which is the polynomial having a degree of 3 and zeros -2, 3+i, and 3-i.