How to prove that a given polynomial $P(x)$ has no interger roots. How to prove that a given polynomial $P(x)$ has no integer roots.

There are a few case, firstly, the easy case is when the polynomial is a constant, then there are either infinitely many, or none.
The second case is when we have a non-zero polynomial. $$a_nx^n+\dots+a_1x+a_0=0$$
where $a_n\neq0$. W.L.O.G, $a_0\neq0$ (just factorise $x$ so that $a_0\neq0$).
Then to see if there are integer roots, find the factorisation of $a_0$ and try solving the polynomial for each factor of $a_0$.
This works because the product of roots is $a_0$ up to sign.
