Let's use the factor theorem, which states that if a polynomial f(x) has a factor x - a, then f(a) = 0.
We can check each of the possible factors by plugging them into the polynomial and seeing if the result is zero:
- Let's try x = 1:
f(1) = (1)^3 - 4(1)^2 + 3(1) + 2 = 0
Since f(1) = 0, we know that x - 1 is a factor of f(x).
- Let's try x = -1:
f(-1) = (-1)^3 - 4(-1)^2 + 3(-1) + 2 = 6
Since f(-1) is not zero, we know that x + 1 is not a factor of f(x).
- Let's try x = 2:
f(2) = (2)^3 - 4(2)^2 + 3(2) + 2 = 0
Since f(2) = 0, we know that x - 2 is a factor of f(x).
- Let's try x = -2:
f(-2) = (-2)^3 - 4(-2)^2 + 3(-2) + 2 = -8 + 16 - 6 + 2 = 4
Since f(-2) is not zero, we know that x + 2 is not a factor of f(x).
Therefore, the factors of the polynomial f(x) are (x - 1) and (x - 2).