Final answer:
The correct polynomial function of the lowest degree with rational coefficients that has 2 + i and 2 as some of its zeros is f(x) = x^3 - 6x^2 + 13x - 10, as it accounts for the complex conjugate root 2 - i, which must also be a root by the conjugate pairs theorem.
Step-by-step explanation:
To find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros, namely 2 + i and 2, we need to use the fact that complex roots of polynomials with rational coefficients always come in conjugate pairs. This means if 2 + i is a root, then its conjugate 2 - i must also be a root. Since we also have a real root which is 2, our polynomial must at least be a third degree polynomial.
The polynomial with real coefficients that has 2 + i and 2 - i as roots can be constructed by multiplying the factors (x - (2 + i)) and (x - (2 - i)). Doing that yields:
(x - 2 - i)(x - 2 + i) = (x - 2)2 - i2 = x2 - 4x + 4 + 1 = x2 - 4x + 5
Now we include the real root 2, giving us another factor (x - 2), so our polynomial is:
(x2 - 4x + 5)(x - 2)
Expanding this out gives us:
x3 - 2x2 - 4x2 + 8x + 5x - 10 = x3 - 6x2 + 13x - 10
Comparing this with the provided options, the correct polynomial is:
B) f(x) = x3 - 6x2 + 13x - 10