Real roots strictly between two adjacent integers for monic polynomials in $\mathbb{Z}[x]$

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.

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any two adjacent integers? – darij grinberg Jan 31 '12 at 14:35
Why not choose two polynomials with roots between your favourite two consecutive integers, and then multiply them together? The product will have all the roots that the factors have. – James Cranch Jan 31 '12 at 14:45
@James: What about $n=3$? – Ramiro de la Vega Jan 31 '12 at 15:24
The cubic $x^3-7x-7$ has two roots between $-2$ and $-1$. – Richard Stanley Jan 31 '12 at 19:04
Richard Stanley's solution generalizes for all $n\geq 3$; that is, $x^{n-3}(x^3-7x-7)$. – Unknown Feb 1 '12 at 0:31