Answer:
k = -18.
But See below.
Step-by-step explanation:
f(x) = x^4 - 3x^3 - 6x^2 + kx - 16
If x^2 + 3x + 2 is a factor then so is (x + 1) and (x + 2) as
x^2 + 3x + 2 = (x + 1)(x + 2)
So by the Factor Theorem f(-1) = 0:
f(-1) = (-1)^4 - 3(-1)^3 - 6(-1)^2 - 1k - 16 = 0
1 + 3 - 6 - k - 16 = 0
-18 - k = 0
k = -18.
Checking:
f(-2) = (-2)^4 - 3(-2)^3 - 6(-2)^2 - 18(-2) - 16
= 16 + 24 - 24 + 36 - 16 = 36 (NOT 0).
