There is an intimate relationship between the coefficients and the roots of polynomial equations, best illustrated by writing a polynomial in terms of its roots and expanding.
Suppose that
We can write this as
If the roots of this expression are
and
then
so that
hence
and
Suppose that
We can write this as
If the roots of this expression are
and
then
so that
hence
and
Suppose that
We can write this as
If the roots of this expression are
and
then
so that
hence
and
There is a pattern here. If the polynomial is of degree
and is divided throughout by the coefficient of
then the coefficients when written in terms of the roots alternate in sign and consist of all possible products of the roots taken 1, 2, 3,..,n at a time.