SECTION 3.7 Discriminant 179
R(g,h) = det
1 an− 1 ··· a 1 a 0 0 ··· 0
0
0
... S(g,h)
...
0
.
Finally, one checks that
S(x+a,h(x)) Z(^0) n,m+1 In
×
1 an− 1 ··· a 1 a 0 0 ··· 0
0
0
... S(g,h)
...
0
=S((x+a)g(x),h(x)),
and we’re done.
SinceR(x−a,x−b) =a−b,we immediately obtain the following:
Corollary 1. Let f(x), g(x) be polynomials with real coefficients,
and have leading coefficientsanandbm, respectively. Assumef(x)has
zerosα 1 , ..., αn, and thatg(x)has zerosβ 1 , ..., βm. Then
R(f,g) =amnbnm
∏n
i=1
∏m
j=1
(αi−βj).
Corollary 2. R(f,g) = 0if and only iff(x)andg(x)have a common
zero.
The following corollary will be quite important in the next section.
Corollary 3. Let f(x), g(x) be polynomials with real coefficients,
and have leading coefficientsanandbm, respectively. Assumef(x)has