Final answer:
The function f(x) = x^3 - 3x^2 + 4x - 2 has complex zeros at -1, 1, and 2. These are found by testing rational roots and performing synthetic division, matching option (d).
Step-by-step explanation:
To find all the complex zeros of the function f(x) = x^3 - 3x^2 + 4x - 2, we'll need to factor the polynomial or apply methods such as synthetic division or the Rational Root Theorem. Unfortunately, unlike quadratic equations, there is no straightforward formula for polynomials of higher degrees. However, we can try to identify one root through inspection or by using the Rational Root Theorem, and then apply polynomial division to find the remaining roots.
After testing possible rational roots (which are the factors of the constant term divided by the factors of the leading coefficient), we find that x = 1 is a root, since f(1) = 1 - 3 + 4 - 2 = 0. Using synthetic division or long division with this root, we can factor out (x - 1) from the polynomial, resulting in the remaining quadratic factor.
The given options lead us to try x = -1 and x = 2 as well, which are indeed roots since f(-1) = -1 - 3(-1)^2 + 4(-1) - 2 = 0 and f(2) = 2^3 - 3(2)^2 + 4(2) - 2 = 0. Therefore, the polynomial can be factored as (x - 1)(x + 1)(x - 2). This completely factors the polynomial, indicating the zeros are -1, 1, and 2, which corresponds to option (d).