Question

Find $k$ so that $3x^4+8x^3-4kx+k$ may be divisible by $x-2$.

Answer 1

Determine the value of $k$.

Let $f\left(x\right)=3x^4+8x^3-4kx+k$

If $(x–a)$ is a factor of $p\left(x\right)$, then $p\left(a\right)=0$.

Since, $\left(x-2\right)$ is a factor of $f\left(x\right)$, then $f\left(2\right)=0$.

$\begin{array}{rcl}f\left(2\right)& =& 0\\ 3{\left(2\right)}^4+8{\left(2\right)}^3-4k\left(2\right)+k& =& 0\\ 48+64-8k+k& =& 0\\ 112-7k& =& 0\\ k& =& 16\end{array}$\

Hence, the value of $k$ is $16$.