regarding Integer solutions

Question

Let $f(x)=x^7-105x+12.$ Show that $f(m)$ is not prime for any integer $m$.

Answer 1

We have $2\mid f(x)$ for all integers $x$. On the other hand, $f(x)=2$ is impossible over the integers, since $x^7-105x+10$ has no rational root by the rational root theorem.