TITLE: Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$
QUESTION [17 upvotes]: So first of all, I used the following from my lecture notes: 
If $f \in \mathbb{Z}[x]$ is primitive (gcd of all the coefficients is 1) - $f$ irreducible in $\mathbb{Z}[x] \Leftrightarrow$ f irreducible in $\mathbb{Q}[x]$. 
So now I only have to show that $f_n = x^n + x + 3$ is irreducible in $\mathbb{Z}[x]$ for all $n \geq 2. $
Is there a pattern I should be appealing to in some way?

REPLY [4 votes]: This can be also shown by referring to irreducibility criterion as given by Osada, see for example Theorem 2.2.7 in Prasolov's book Polynomials: 

Let $f(x)=x^n+a_1x^{n-1}+\dots+a_{n-1}x\pm p$ be a polynomial with integer coefficients, where $p$ is a prime. If $p>1+|a_1|+\dots+|a_{n-1}|$, then $f$ is irreducible.

Since $3>1+1$, the irreducibility follows.