I am stuck trying to prove a problem that seems obvious: namely, given a monic polynomial with integral coefficients $$f(x)= x^n +a_1x^{n-1}+\ldots+a_1x+a_0, $$ $a_i\in \mathbb{Z}$, is it true that no pair of real roots lie strictly between any two adjacent integers?

I was able to prove it when $n=2$ and I have also sketched a proof for some $n$ that if such roots exist, then they will not be rational. Does this boil down to a nontrivial question or am I missing something?

Thank you.

adjacentintegers? – darij grinberg Jan 31 '12 at 14:35