My task is to find the gcd of $\displaystyle nx^{n+1}-(n+1)x+1$ and $\displaystyle x^{n}-nx-1$.

Long division isn't giving me a clue here. My hunch is that the gcd is one, but I'm not sure I can prove that by showing they are irreducible (over what ring would I do that anyway? the question does not specify) or something like that.