Question

Find k such that$f\left(x\right)=x^3-k{x}^2+kx+2$ has the factor $x-2$.

Short answer

The required value of $k$is $5$.

Step-by-step solution

Step 1. Concept Used

The factor theorem states that if f is a polynomial function, then $x-c$is a factor of fif and only if$f\left(c\right)=0$.

Step 2. Form an equation

As it is given that $x-2$is a factor of $f\left(x\right)=x^3-k{x}^2+kx+2$.

So by the factor theorem we have $f\left(2\right)=0$.

$\begin{array}{rcl}f\left(2\right)& =& 0\\ 2^3-k·2^2+k·2+2& =& 0\\ 8-4k+2k+2& =& 0\\ 10-2k& =& 0\end{array}$

Step 3. Solve the equation

In the equation subtract $10$both sides

$\begin{array}{rcl}10-2k-10& =& 0-10\\ -2k& =& -10\end{array}$

Now divide both sides by $-2$ and solve for k.

$\begin{array}{rcl}\frac{-2k}{-2}& =& \frac{-10}{-2}\\ k& =& 5\end{array}$