Let $p(x)=\sum\limits_{i=0}^na_ix^i$ be a real $n-$degree ($a_n\neq0$) polynomial with $n$ real distinct roots, $x_1,\ldots,x_n$. Is there a simple relation between the areas between the roots and the roots/coefficients? More specifically, let the areas between the roots be $A_i = \int\limits_{x_{i}}^{x_{i+1}}p(x)dx, 1\leq i\leq n-1$. Is there a sequence of functions $f_j(A_1,\ldots,A_{n-1})$, $0\leq j\leq n$ such that $$ p(x) = \sum\limits_{i=0}^{n}f_i(A_1,\ldots,A_{n-1})x^i $$ Maybe a relation between the roots themselves and the areas?
The sequence $A_i$ provides $n-1$ independent numbers. The roots of a polynomial are determined by $n$ independent numbers, since $$ \sum\limits_{i=0}^{n}a_ix_0^i = 0 \Leftrightarrow \sum\limits_{i=0}^{n}\frac{a_i}{a_n}x_0^i=0 $$ so $A_i$ I guess alone cannot be enough to fully determine the roots. Is there a hidden property that can complete the set?