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a common factor and their greatest common divisor is not constant. Theorem 10.4.3
is proved.
10.4.4. Corollary.
Two polynomials f(X) and g(X) defined over an algebraically
closed field have a common root if and only if R(f,g) = 0.
Proof.
Obvious.
The resultant can be used to check for multiple roots of a polynomial.
10.4.5. Corollary.
A polynomial f(X) has a nonconstant factor of multiplicity larger
than 1 if and only if R(f,f ¢) = 0. In particular, f(X) has a multiple root if and only if
R(f,f ¢) = 0.
Proof.
By Theorem B.8.10, f(X) has a nonconstant factor of multiplicity larger than
1 if and only if that factor also divides f ¢(X). Now use Theorem 10.4.3.
Definition.
Let f(X) be a nonconstant polynomial. The resultant R(f,f¢) is called the
resultant of f
.
Here are two useful formulas for resultants that are worth stating explicitly.
10.4.6. Example.
The resultant of the polynomials a
1
X + a
0
and b
1
X + b
0
is
aa
bb
1
0
.
1
0
The resultant of the polynomials a
2
X
2
+ a
1
X + a
0
and b
2
X
2
10.4.7. Example.
+ b
1
X
+ b
0
is
aaa
aaa
bbb
bbb
0
aaa
bbb
aaa
bbb
0
0
2
1
0
2
1
0
2
0
aa
bb
aa
bb
aa
bb
2
1
0
2
1
0
2
0
2
1
1
0
=-
=
-
.
0
0
0
2
1
0
2
1
0
2
0
2
1
1
0
0
2
1
0
2
1
0
10.4.8. Theorem.
Let
()
=
m
m
-
1
fX
a X
+
a
X
+
...
+
a
,
m
m
-
1
0
()
=
n
n
-
1
gX
b X
+
b
X
+
...
+
b
n
n
-
1
0
be two polynomials where the a
i
and b
j
are homogeneous polynomials of degree
m - i and n - j, respectively, in the variables X
1
, X
2
,..., X
r
and a
m
b
n
π 0. Then the
resultant R(X
1
,X
2
,...,X
r
) of f and g (with respect to X) is either identically equal to
0 or a homogeneous polynomial of degree mn.
