Final answer:
Option B is the correct answer: f(x) = x^3 - 6x^2 + 13x - 10, as it is the polynomial function of the lowest degree with rational coefficients that includes the given zeros 2 + i and 2.
Step-by-step explanation:
To find a polynomial function of lowest degree with rational coefficients that has the numbers 2 + i and 2 as some of its zeros, we first need to recognize that any polynomial with rational coefficients and a complex zero must also have the complex conjugate of that zero as a root. Therefore, the roots of the polynomial must include 2 + i, 2 - i (the complex conjugate of 2 + i), and 2.
The polynomial can be found by taking the product of (x - (2 + i)), (x - (2 - i)), and (x - 2). Expanding this product gives us:
(x - 2 - i)(x - 2 + i)(x - 2) = ((x - 2)^2 - i^2)(x - 2) = ((x^2 - 4x + 4) + 1)(x - 2) = (x^2 - 4x + 5)(x - 2)
Expanding further, we get:
x^3 - 2x^2 - 4x^2 + 8x + 5x - 10 = x^3 - 6x^2 + 13x - 10
Therefore, the correct polynomial is option B: f(x) = x^3 - 6x^2 + 13x - 10.