0

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.

flag
1 
 any two adjacent integers? – darij grinberg Jan 31 2012 at 14:35
4 
 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 2012 at 14:45
1 
 @James: What about $n=3$? – Ramiro de la Vega Jan 31 2012 at 15:24
6 
 The cubic $x^3-7x-7$ has two roots between $-2$ and $-1$. – Richard Stanley Jan 31 2012 at 19:04
2 
 Richard Stanley's solution generalizes for all $n\geq 3$; that is, $x^{n-3}(x^3-7x-7)$. – To be cont'd Feb 1 2012 at 0:31
show 3 more comments

Your Answer

Get an OpenID
or

Browse other questions tagged or ask your own question.