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

THEORY OF EQUATIONS

is n(n - 1). This product is homogeneous, for if in any other
term, say (n - 1)?'L0?la(l"((tal.( an)'-2, we substitute for the coefficients their equivalents in terms of the roots, by the relations
of ~ 13, say ai + (-4 + *.. +- a, for - Cl, we see that this term
aC
likewise is of the degree n(Q - 1) in the roots. Hence the
product I(ac - a2)2 is of the same degree in the roots as the
resultant of f(x) =0 and f '(x) = 0, and, therefore, as the discriminant of f(x)= 0. Consequently, this product can differ
from the discriminant by a nzmericca factor only.
Ex. 1. Show that the resultant of x2 - x - 42 = 0 and x2 - 4 x - 77 = 0
is zero, proving that the left members of the equations have a common
factor.
Ex. 2. Find the resultant of
aox3+ + ax 0 axnd Cox3 + C X2 + CX + C3 = 0 by Euler's method.
Ex. 3. For what value of a will the two equations x3 + ax2 - - 1 0
and x2 + 3 x - 7 = 0 have a root in common?
Ex. 4. Using Sylvester's method of elimination, find the discriminant.
of bo3 + 3 blx2+ 3 b2+ b3 = 0.
Ex. 5. Find the discriminant of x" - 1 = 0. Has the equation equal
roots?
Ex. 6. Find the discriminant of xn+l - a - x + 1 = 0.