# Delimiting the roots of a polynomial

Given a polynomial of degree $n$, namely, $y=a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$, with $a_i\in \mathbb{R}$, I would to know if it is possible to compute a rational number $K$ such that the distance between each of the (at most $n$) real roots of the polynomial is $K$.

• The question is ambiguous in several ways, please revise and clarify. – MCH Jan 24 '13 at 3:57
• At most $K$ or at least $K$? – arnab Jan 24 '13 at 8:32

You can (without too much work) find a number $K$ such that if $|x| > K$ then $|a_n x^n| > \sum_{k<n} |a_k x^k|$, and so all roots are inside $(-K,K)$ and the distance between them is smaller than $2K$.