Chapter 8.6, Problem 47HP

To determine

To find: All the values of k so that the polynomial can be factored using integers.

Answer to Problem 47HP

  $7,12$

Explanation of Solution

Given:

  $x^2-8x+k,k>0$

Calculation:

The given polynomial is $x^2-8x+k,k>0$

Comparing this with $a{x}^2+bx+c$ ,

  $a=1,b=-8,c=k$

The value of $ac$ is $k$

From ac method the middle term b should be equal to the sum/difference of factors pairs of ac .

So, $-8$ can be written as

  $\begin{array}{l}-8=-1-7\\ -8=-2-6\end{array}$

The value of ac should be the product of these numbers. So, the values of ac are

  $\begin{array}{l}\left(-1\right)\left(-7\right)=7\\ \left(-2\right)\left(-6\right)=12\end{array}$

Hence, the values of k are $7,12$