example of using Eisenstein criterion

For showing the irreducibility (http://planetmath.org/IrreduciblePolynomial2) of the polynomial

$$P(x):=x^5+5x+11$$

one would need a prime number dividing its other coefficients except the first one, but there is no such prime.  However, a suitable  $x:=y+a$  may change the situation.  Since  the binomial coefficients of${(y-1)}^5$ except the first and the last one are divisible by 5 and $11\equiv 1(mod5)$,  we try

$$x:=y-1.$$

Then

$$P(y-1)=y^5-5y^4+10y^3-10y^2+10y+5.$$

Thus the prime 5 divides other coefficients except the first one and the square of 5 does not divide the constant term of this polynomial in $y$, whence the Eisenstein criterion says that $P(y-1)$ is irreducible (in the field $\mathbb{Q}$ of its coefficients).  Apparently, also $P(x)$ must be irreducible.

It would be easy also to see that $P(x)$ does not have rational zeroes (http://planetmath.org/RationalRootTheorem).