Chapter 11.3, Problem 52PPE

To determine

The value of k .

Answer to Problem 52PPE

  $k=12$

Explanation of Solution

Given:

  $x+3$ is a factor of $x^2-x-k$ .

Concept Used:

If an expression can divide the another higher degree expression completely, then the divisor expression is a factor of the higher degree expression.

Calculation:

Let $x+3$ is a factor of $x^2-x-k$ , then $x+3$ must divide $x^2-x-k$ completely. That is, the remainder should be zero. So, on dividing the expressions using the long division, it gives

  $x+3\begin{array}{c}\hfill x-4\\ \hfill \overline{)\begin{array}{l}{x}^2-x-k\\ \underset{\_}{-\left(x^2+3x\right)}\\ -4x-k\\ \underset{\_}{-\left(-4x-12\right)}\\ 12-k\end{array}}\end{array}$

If $x+3$ is a factor of $x^2-x-k$ , the remainder should be zero. That is,

  $\begin{array}{l}12-k=0\\ \Rightarrow k=12\end{array}$

Thus, the value of k is $k=12$ .