Final answer:
To find a degree 3 polynomial with given roots of 4 and 2 - i, we use the fact that complex roots appear in conjugate pairs. The polynomial is formed by multiplying factors corresponding to these roots. The correct polynomial function is f(x) = (x - 4)(x - (2 - i))(x - (2 + i)).
Step-by-step explanation:
Given that a polynomial function with roots of 4 and 2 - i is sought, we know that for real polynomials, complex roots must come in conjugate pairs. Therefore, the roots of the polynomial would be 4, 2 - i, and its conjugate 2 + i. Using this information, we can construct the polynomial function by taking the product of factors corresponding to these roots.
The correct polynomial function based on the given roots is thus:
f(x) = (x - 4)(x - (2 - i))(x - (2 + i))
When multiplying out these factors, the complex parts will cancel each other out. To show that, let's use the property of complex conjugates where (a + ib)(a - ib) = a² + b², without i. Here:
Applying this to the expression, we get:
f(x) = (x - 4)((x - 2)² + (i)²)
This simplifies to:
f(x) = (x - 4)(x² - 4x + 5)
This product will yield a polynomial of degree 3, which is what we are looking for. Therefore, the correct answer is option A: f(x) = (x - 4)(x - (2 - i))(x - (2 + i))