Question

Find k such that $f\left(x\right)=x^4-k{x}^3+k{x}^2+1$has the factor$x+2$.

Short answer

The required value of k is $-\frac{17}{12}$.

Step-by-step solution

Step 1. Concept Used 

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

Step 2. Form an equation 

As it is given that$x+2$ is a factor of $f\left(x\right)=x^4-k{x}^3+k{x}^2+1$.

So by the factor theorem we have $f(-2)=0$.

$\begin{array}{rcl}f(-2)& =& 0\\ {(-2)}^4-k{(-2)}^3+k{(-2)}^2+1& =& 0\\ 16+8k+4k+1& =& 0\\ 17+12k& =& 0\end{array}$

Step 3. Solve the equation 

In the equation subtract $17$from both sides

$\begin{array}{rcl}17+12k-17& =& 0-17\\ 12k& =& -17\end{array}$

Now divide both sides by $12$ and solve for k.

$\begin{array}{rcl}\frac{12k}{12}& =& \frac{-17}{12}\\ k& =& -\frac{17}{12}\end{array}$