Answer:
[tex]\displaystyle k = \frac{34}{3}[/tex]
Step-by-step explanation:
We are given the polynomial:
[tex]\displaystyle P(x) = 3x^3 - kx^2 + 5x + k[/tex]
And we want to determine the value of k such that (x - 2) is a factor of the polynomial.
Recall that the Factor Theorem states that a binomial (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0.
Our binomial factor is (x - 2). Thus, a = 2.
Hence, by the Factor Theorem, P(2) must equal zero.
Find P(2):
[tex]\displaystyle \begin{aligned} P(2) &= 3(2)^3 - k(2)^2 + 5(2) + k \\ \\ &= 3(8) - 4k + 10 + k \\ \\ &= 34 - 3k \end{aligned}[/tex]
This must equal zero. Hence:
[tex]\displaystyle \begin{aligned} 34 - 3k &= 0 \\ \\ -3k &= -34 \\ \\ k = \frac{34}{3} \end{aligned}[/tex]
In conclusion, k = 34/3.
