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

THEORY OF EQUATIONS

74. Euler's Method of Elimination. If (x) = 0 and F(x) = 0,
as defined in ~ 73, have a root a in common, we may write
f(x) -(x- a)f (X)
F(x) (x- a)F1(x),
where f/i(x) A= l a"-1 +-.A2l-2 + *** + Ai,
FI(x) ClXm-1 4+ C02x —2 -... -1- Cm
the coefficients A1, *, A,, and C,,.., C,, being undetermined
quantities.
We obtain easily the identical equation of the (mz + - - l)th
degree f(x). F,(x) F(x). f(x).
Performing the indicated multiplications and equating coefficients of like powers of x, we obtain nm + n homogeneous
equations. Eliminating the undetermined coefficients, we obtain the required resultant.
Thus, find the resultant of
aox2 + a1x + a2 = O, co2 + Cx + C2 = 0.
If they have a root in common, we obtain the identity
(Cx +.C)(Cx to 2 + ctl + a) = (A1x +4- ) (C0X2 + C1 + C2)
or (Cl7ao - A1i0o)3 + (aCll + Cao - Ale, - A2Co)x2
+ (CCa2 4+ C(a, - Al, - A2C1)x +2 C2- A2c2 0.
Equating coefficients,
CAao - Alo = 0,
Clal +- C2a0 - AC1 - Ao0 = 0,
C12 + C2ai - Ac2 - A9cl = 0,
C2a2 - A2c2 = 0.
In order that the four homogeneous equations I may be consistent with each other it is necessary that