hence, by multiplication, we have
.
Equating the coefficients of like powers of x in this identity, we have
in which S_1, stands for the sum of the roots a, b, c, \ldots k; S_2
stands for the sum of the products of the roots taken two
at a time, and so on to S_n, which equals the continued
product of all the roots. That is:
(1) The coefficient of the second term with its sign changed equals the sum of the roots.
(2) The coefficient of the third term equals the sum of all the products of the roots taken two at a time.
(8) The coefficient of the fourth term with its sign changed equals the sum of all the products of the roots taken three at a time, and so on.
(4) The last term equals the continued product of all the roots, the sign being + or - according as n is even or odd.
571. It follows that if the equation is in the general form:
(1) The sum of the roots is zero if the second term is wanting.
(2) One root, at least, is zero if the last term is wanting.
572. The student might suppose that the relations established in the preceding article would enable him to solve any proposed equation; for the number of the relations is equal to the number of the roots. A little reflection will show that this is not the case; for suppose we eliminate any n-1 of the quantities a, b, c, \ldots k, and so obtain an equation to determine the remaining one; then since these quantities are involved symmetrically in each of the equations,
