Chapter 8.3, Problem 32WE

To determine

To find : the value of $k$

Answer to Problem 32WE

The value of $k$ is $-1,3$

Explanation of Solution

Given information : When $x^3+k{x}^2-k^{2}x+14$ is divided by $x+2$ , the remainder is $0$

Calculation :

  $\begin{array}{l}\frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}\\ \frac{x^3+k{x}^2-k^{2}x+14}{x+2}=x^2+\left(k-2\right)x+\left(k^2-2k+4\right)-\frac{2k^2+4k+6}{x+2}\\ \frac{-2k^2+4k+6}{x+2}\\ -2k^2+4k+6=0\\ k^2-2k-3=0\\ \left(k-3\right)\left(k+1\right)\\ k=-1,3\end{array}$