Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 225: 107

$k=5$

The factor theorem states that when $f(a)=0$, then we have $(x-a)$ as a factor of $f(x)$ and when $(x-a)$ is a factor of $f(x)$, then $f(a)=0$. We simplify the given equation as follows: $f(2)=2^3-k(2)^2+(2)k+2=0 \\ 8-4k+2k+2=0\\ 10-2k=0\\2k =10 \\ k=5$