Question

Given f(x) = 3x^3– 2x + k, and x+2 is a factor of f(x), then what is the value of K

Answer 1

`\displaystyle k = 20`

Explanation:

We are given the function:

`\displaystyle f(x) = 3x^3 - 2x + k`

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:

`\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}`

In conclusion, the value of k is 20.