Final answer:
To factor f(x) = x^3 + 5x^2 - 128x - 528, we use the Rational Root Theorem to find a root of -4. Synthetic division is then used to factor the quadratic equation, giving us the factored form (x + 4)(x + 12)(x - 11). The x-values for f(x) = 0 are -4, -12, and 11.
Step-by-step explanation:
To factor the function f(x) = x^3 + 5x^2 - 128x - 528, we can use the Rational Root Theorem to find possible rational roots. By trying out different values, we find that x = -4 is a root.
Using synthetic division with x = -4, we get (x + 4)(x^2 + x - 132). Now, we can factor the quadratic equation x^2 + x - 132, which gives us (x + 12)(x - 11).
Therefore, the factored form of f(x) is (x + 4)(x + 12)(x - 11).
To find the x-values for which f(x) = 0, we set the factored expression equal to 0 and solve for x. This gives us three solutions: x = -4, x = -12, and x = 11.