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 asIf the roots of this expression areandthenso that

henceand

Suppose thatWe can write this asIf the roots of this expression areand thenso that

henceand

Suppose thatWe can write this asIf the roots of this expression areand thenso that

henceand

There is a pattern here. If the polynomial is of degreeand is divided throughout by the coefficient ofthen 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.