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

THEORY OF EQUATIONS

Thus, to find the resultant of
f(x) aot03 + a1x2 + aCx - a8 = 0,
and F(x) = Co2 + c1X + c2 = 0,
we have f (x) ao — 3 4 a 1 X2 Ct a2x + a3 = O,
xf(x) Cox4 + Cax3 + a2X2 + a3x = 0,
F(x) - + Cox2 + c1x + C2 = 0,
xF(x) = + Co3 + C1X2 + c2x = 0,
x2F(x) _ Cox4 + C1x3 + C2S2 = 0.
That the four unknowns x, x2, x2, x4, may satisfy the five
equations, it is necessary that
0 aO ac a2 a3
ao c a2 a 3 0,
R- 0 0 Co Ci c2 =0.
0 Co Ci C2 0
Co c1 C2 0 0
R is the resultant.
76. Discriminant of f(x) = 0. It was proved in ~ 21 that if
f(x) = 0 has a multiple root, that root satisfiesf '(x) = 0. The
condition that f(x) 0= and f'(x) = 0 have a root in common is
expressed by the vanishing of their resultant. The resultant
of f(x) = 0 and f'(x) = 0 is called the discriminant of f(x) = 0.
The discriminant of an equation f(x) = 0 may be otherwise
defined as the simplest function of the coeflicients, or of the roots,
whose vanishing signifies that the equation has equal roots.
If f(x) = 0 and f'(x) = 0 have a common root, this root will
satisfy also nf(x) -f'(x) = 0. Instead of finding the resultant
of f(x). and f'(x), we may therefore find the resultant of
nf(x) -f'(x) = 0 and f'(x) = O. The latter mode of procedure
is preferable, because it gives us the resultant clear of an
extraneous factor.