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 thatWe can write this as
If the roots of this expression are
and
then
so that
henceand
Suppose thatWe can write this as
If the roots of this expression are
and
then
so that
henceand
Suppose thatWe can write this as
If the roots of this expression are
and
then
so that
henceand
There is a pattern here. If the polynomial is of degreeand 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.