Answer:
[tex]\displaystyle k = 20[/tex]
Step-by-step explanation:
We are given the function:
[tex]\displaystyle f(x) = 3x^3 - 2x + k[/tex]
Where (x + 2) is a factor of f. And we want to determine the vale of k.
Recall that from the Factor Theorem, if (x - a) is a factor of a polynomial P(x), then P(a) must equal 0.
We can rewrite our factor as (x - (-2)). Hence, a = -2. Our polynomial is f. Since (x + 2) is a factor, then from the Factor Theorem, f(-2) must be 0.
Using this information, we can now determine k:
[tex]\displaystyle\begin{aligned} f(x) & = 3x^3 -2x + k \\ \\ f(-2) = 0 & = 3(-2)^3 -2(-2) + k \\ \\ 0 & = (-24) + (4) + k \\ \\ k & = 20 \end{aligned}[/tex]
In conclusion, the value of k is 20.
