Why can't $x^k+5x^{k-1}+3$ be factored? I have a polynomial $P(x)=x^k+5x^{k-1}+3$, where $k\in\mathbb{Z}$ and $k>1$. Now I have to show that you can't factor $P(x)$ into two polynomials with degree $\ge1$ and only integer coefficients. How can I show this?  

Assume $k\ge2$.
By the rational root theorem, only $\pm1$ and $\pm3$ are candidates for rational roots - and by inspection are not roots. Therefore,
if $P(x)=Q(x)R(x)$ with $q:=\deg Q>0$, $r:=\deg R>0$, we conclude that $q\ge2$ and $r\ge 2$. Modulo $3$ we have $(x-1)x^{k-1}=x^k-x^{k-1}$, hence wlog. $Q(x)\equiv (x-1)x^{q-1}\pmod 3$, $R(x)\equiv x^r\pmod 3$.This implies that the constant terms of both $Q$ ans $R$ are multiples of $3$, hence the constant term of $P$ would have to be a multiple of $9$ - contradiction
