How do you show that this polyonomial is irreducible over Z?

[dlarahasle]

Let f be a polynomial with coefficients in Z, and with the degree d. Show that if there is more than 2d different n, such that, f(n) is a prime-number, f must be irreducible over Z.

I guess that you have to reason that if it is irreducible over Z it must also be irreducible over Q, and work from that. But what does it really mean that 2d different n, such as f(n) is a prime-number? How does that effect the polynomial? If f = g*h then either g(n) or h(n) must equal 1. The other one must equal a prime-number. But I don't know where to go from this.Should you use Eisensten's criterion?