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.

  • 1
    $\begingroup$ any two adjacent integers? $\endgroup$ – darij grinberg Jan 31 '12 at 14:35
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    $\begingroup$ 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. $\endgroup$ – James Cranch Jan 31 '12 at 14:45
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    $\begingroup$ @James: What about $n=3$? $\endgroup$ – Ramiro de la Vega Jan 31 '12 at 15:24
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    $\begingroup$ The cubic $x^3-7x-7$ has two roots between $-2$ and $-1$. $\endgroup$ – Richard Stanley Jan 31 '12 at 19:04
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    $\begingroup$ Richard Stanley's solution generalizes for all $n\geq 3$; that is, $x^{n-3}(x^3-7x-7)$. $\endgroup$ – Unknown Feb 1 '12 at 0:31

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