An introduction to the modern theory of equations, by Florian Cajori.

SOME ELEMENTARY PROPERTIES OF EQUATIONS 11

The coeficient a, of the third term is equal to the sum of the
')rocldcts of the roots taken two by two.
The coefficient a, of the fourth term, with its sign changed, is
equal to the sum of the products of the roots taken three by three;
and so on, the signs of the coefficients being taken alternately!negative and positive, and the number of roots taken in each prodact increasing by unity every time we advance to a new coefficient, tutil finally the last term in the equation is reached,
which is numerically equal to the product of all the roots andl
which is positive or negative according as n, the degree of the
equation, is even or odd. In symbols, these laws may be expressed as follows:
C1 = - (1C + C + (3 + *.* + C,),
a, = () + ""3 + 1 t2a3 + ** + -. - l n-) I
a3 - ((1t((t3 + aiC 4 + ** + on-2.n- l.).. I
a = (- 1)(1(t23 an. n
When in the equation f(x) = 0 the coefficient cO of the term
x" is not unity, we must divide each term of the equation by a0.
''he sum of the roots is then equal to-al, the sum of their
products, two by two, is a, and so on.
a0
The laws expressing the relations between the coefficients
of an equation and the roots were obtained above by observing the relations existing in the three products obtained by
actual multiplication. To remove any doubt which may be
entertained as to the generality of these laws we proceed as
follows. Suppose these laws to hold when n factors are multiplied together; that is, suppose that
(x - t,)(x - ta)... (x - an) = x" + a"-1l +...+ a,,,
where a,aC2, *., aq,
have the values shown in I.