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

ELIMINATION

93

coefficients of the equations, the expression R is called the
eliminant or resultant of the given equations.
In the above example the elimination was performed with
the aid of symmetric functions. This method generalized is
as follows:
73. Elimination by Symmetric Functions. To find the conditions that the two equations
f(x) _ax + alx-' + a2'-2 + *.. + can = 0,
F(x) E-XM Co Cm + cl C2- + C- + c = 0,
shall have a common root. For this purpose it is necessary
and sufficient that some one of the roots f/31, 'n, /.m of F(x) = 0
shall satisfy f(x) = 0, in which case the product
f(/1). f(#2) e ' f(PM)
must vanish.
We have f(f31) - ca0ft + allib-1 +... + an,
f (/2) - aof"2 + alf2t-1 + ~... + a,
f (Pm) E=- alom" + CT/- + + aI.
Multiplying these together, we obtain, after substituting for
the symmetric functions of f1, /2, * -, f3, which occur in the
product their values in terms of co,c, *, c,, and after clearing of
fractions, = c. (). )
-R = Co',.;(31) -.f()...f(,.).
Here R is the eliminant and is a rational integral function of
the coefficients of f(x) and F(x). Its vanishing is the condition
that the two givein equations hace a root in, common. The degree
of the resultant in the coefficients of the given equations is in
general m + n.
It is easy to see that we obtain the same eliminant by substituting the roots a1,a2, *, an of f(x) = O, in succession, for x
in the polynomial F(x).