Let
$f(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$
and let the roots of the polynomial be arranged from lowest to highest: $r_1 \leq r_2 \leq \cdots \leq r_n$.
Then the list of the all roots of polynomial $f(x)$ can be express in to new polynomial, let's say $g(x)= (1\text{st},r_1),(2\text{nd},r_2)\cdots(n\text{th},r_n).$ Example: Starting with $f(x)=x^2-8x+15$ or $(x-3)(x-5)$, then $r_1=3,r_2=5$ now when $x=1,g(x)=3$ and $x=2,g(x)=5$. Therefore $g(x)=2x+1$.
Practical application: If you find one of the roots of polynomial therefore the remaining roots of $f(x)$ can be solved. Find the general equation of $g(x)$ where $f(x)=0$ http://www.wolfram.com/technology/guide/GigaNumerics/