Answer
$k=-\frac{17}{12}$
Work Step by Step
The factor theorem says that if f(c)=0, then (x-c) is a factor of f(x) and if (x-c) is a factor of f(x), then f(c)=0.
Hence $f(-2)=(-2)^4-k(-2)^3+k(-2)^2+1=0\\16+8k+4k+1=0\\17+12k=0\\17=-12k\\k=-\frac{17}{12}$.
