Chapter 1.3, Problem 64E

Solution

a.

To write: The two equations that $m$ and $n$ must satisfy.

The two equations are $m+n=0\text{ , }mn=16$

Given:

Expression is $x^2+16$

Calculations:

Consider the sum of two squares $x^2+16$ .

If the above sum is factorable then there are integers $m$ and $n$ such that

  $\begin{array}{c}{x}^2+16=\left(x+m\right)\left(x+n\right)\\ x^2+16=x^2+mx+nx+mn\\ x^2+16=x^2+\left(m+n\right)x+mn\end{array}$

By comparing the coefficients, $m+n=0\text{ , }mn=16$

b.

To show: There are no integers $m$ and $n$ that satisfy both equations you wrote in part (a).

Given:

Expression is $x^2+16$

Explanation:

The two conditions $m+n=0,mn=16$ cannot be satisfied by any pair of integers $m,n$ .

Because,

  $\begin{array}{ccccccc}m,n& 1,16& -1,-16& 2,8& -2,-8& 4,4& -4,-4\\ m+n& 17& -17& 10& -10& 8& -8\end{array}$

If $mn=16$

Then $m+n$ can never be zero.

Therefore, the sum of the squares $x^2+16$ is not factorable.