Question

If f(–2) = 0, what are all the factors of the function f (x) = x cubed minus 2 x squared minus 68 x minus 120? Use the Remainder Theorem.
(x + 2)(x + 60)
(x – 2)(x – 60)
(x – 10)(x + 2)(x + 6)
(x + 10)(x – 2)(x – 6)

Answer 1

Since f(-2)=0, (x + 2) is a factor. Divide f(x) by (x + 2) (synthetic division with -2):

Coefficients: 1 -2 -68 -120
Bring down 1 → multiply -2 → -2 → add to -2 → -4 → multiply -2 → 8 → add to -68 → -60 → multiply -2 → 120 → add to -120 → 0.

Quotient: x^2 - 4x - 60 = (x - 10)(x + 6).

So f(x) = (x + 2)(x - 10)(x + 6), which matches:
(x – 10)(x + 2)(x + 6).