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

THEORY OF EQUATIONS

Multiply both sides of this identity by another factor x - -,,,+1
and we get
( - al) - ). (x - n)j(x - an+,) = Xn+l + (a -, +l)n
+ (a2 - a1C+)xn-1 +... - Ca(+a.
But a - a,+l= - ((ti + a(2 + ~ + a,) a- n+L,
a2- alc(t~, (taC2 4-+ a + - l + tn- ~) + (a1,+ c2
4- +. + 1)42+i,
a3 - aC2an+1 - (a12t3 + *) - (atla2 + (t1a, + *.*)a+ci,
- * * ~+l (- *)y *a 2 * * * 6a*1.
- nC(Il(t+l — (-)+ (t[)nl1(C"(3." (gn+l'
Hence, if the laws hold for n factors, they hold for n +1
factors. But from actual multiplication we know that the
laws hold wheln - = 4, therefore they must hold when n = 5.
Holding for n = 5, they must hold when n = 6, and so on for
any positive integral value of n.
14. It might appear that tile n distinct relations existing
between the coefficients and roots of an equation of the ottlb
degree should offer some advantage in the general solution of
the equation, that one of the t roots could be obtained by tlhe
elimination of the (n - 1) roots from the i equations. IBut
this process offers no advantage, for on performing this elilination we merely reproduce the proposed equation. Take, for
example, the cubic X3 + aCx2 + a2x + a3 = O.
We have
a2 =- aa2 + ac3 +- a2a3,
a3 - - a1(a2x3.
To eliminate a2 and a,, multiply both sides of the first equation by ta2, both sides of the second by al, and add the results
to the third equation.
We obtain a13 + al12 + a2al + a3 =0,