Determining that the given integer is greater than all roots of given polynomial Given the integer $k$ and the polynomial $p(x)$ with all real roots and integer coefficients, how can I determine that $k$ is greater than all roots of $p(x)$, without calculating the exact values of its roots? For example, $k=6$ is greater than all roots of polynomial $p(x)=x^2-2x-8$, whose roots are -2 and 4.
(This is a part of some algorithm problem.)

Edit I missed one condition: $p(x)$ has exactly $n$ distinct real roots where $n$ is the degree of $p(x)$.

For the given example, let $\,x = t+6\,$ then $\,q(t)=p(x-6)=t^3 + 18 t^2 + 106 t + 196\,$. It follows by Descartes'  rule of signs that $\,q(t)\,$ has no positive roots, therefore $\,p(x)\,$ has no root larger than $6$.
In general, the location of the real roots can be algorithmically determined using Sturm sequences.
