Final answer:
The zeros of the polynomial f(x) = x^3 - 2x^2 - 6x + 12 are x = 2, x = 3, and x = -3. The correct option given the choices is a) x = -3, x = 2. The other options do not correctly represent the zeros of the polynomial.
Step-by-step explanation:
To determine which of the provided options represent the zeros of the cubic polynomial f(x) = x^3 - 2x^2 - 6x + 12, we may attempt to factorize the polynomial or use synthetic division. First, we may observe the polynomial and attempt to find rational zeros by using the Rational Root Theorem, which suggests that any rational zero, p/q, must be a factor of the constant term (12) divided by a factor of the leading coefficient (1). Let's try the possible factors of 12:
- x = 1: Not a zero, because f(1) != 0,
- x = -1: Not a zero, because f(-1) != 0,
- x = 2: Is a zero, because f(2) = 0,
- x = -2: Not a zero, because f(-2) != 0,
- x = 3: Is a zero, because f(3) = 0,
- x = -3: Is a zero, because f(-3) = 0.
Now that we have found the zeros x = 2, x = 3, and x = -3, we can conclude that the correct option is a) x = -3, x = 2. There are three zeros and not only two as the multiples choices suggest, which seems to be an error. However, among the given options, option (a) includes two correct zeros.