Check the properties of given polynomial of order $7$

Question

Consider $$f(x)=x^{7}-105x+12$$ Then which of the following is\are correct?

• $f(x)$ is reducible over $\Bbb Q$
• There exists an integer $m$ such that $f(m)= 105$
• There exists an integer $m$ such that $f(m)=2$
• $f(m)$ is not a prime for any integer $m$

The function is continuous and attain every value on real plane but, the main problem here is about "$m$" which is integer.I only know how to check irreducibility of given polynomial but here it is asking about reducibility,one think i know about reducibility is if polynomial is reducible over $\Bbb Z$ then it is reducible over $\Bbb Q$.

Answer 1

By Eisenstein with $p=3$ the first is not true.

The second is not true because the equation $x^7-105x-93=0$ has no integer roots. We need to check only $\{\pm1,\pm3\pm31\pm93\}$

By the same way we can get that the third is wrong.

The fourth is true because $f(m)$ is an even number and the third is wrong.