Discriminant of a Polynomial

​
m
2
3
4
5
new polynomial
grid/matrix
show determinant
show discriminant
p(x)2
3
x
-
2
x
+x
p'(x)6
2
x
-2x+1
A =
2
-1
1
0
0
0
2
-1
1
0
6
-2
1
0
0
0
6
-2
1
0
0
0
6
-2
1
det(A) = 14
D(p) = -7
This Demonstration shows the discriminant of the polynomial
p(x)=
a
n
n
x
+
a
n-1
n-1
x
+⋯+
a
1
x+
a
0
. The discriminant of a polynomial of degree
n
is the quantity
n(n-1)/2
(-1)
R(p,p')/
a
n
, where
p'
is the derivative of
p
and
R(p,q)
is the resultant of
p
and
q
. The resultant is equal to the determinant of the corresponding Sylvester matrix. The discriminant of
p
is 0 if and only if
p
has a multiple root.

Details

The discriminant of a polynomial with leading coefficient 1 is the product over all pairs of roots
x
i
,
x
j
of
2
(
x
i
-
x
j
)
.
The equation
n(n-1)/2
(-1)
a
n
D(f)=R(f,f')
relates the discriminant and resultant.
To calculate the discriminant, we use the built-in Mathematica function Discriminant. The other way is to calculate the resultant using the Sylvester matrix and then the discriminant from the above equation.
For the meaning of the matrix/grid, see Sylvester Matrix.

References

[1] E. J. Borowski and J. M. Borwein, Dictionary of Mathematics, London: Collins, 1989 p. 169.

External Links

Resultant (Wolfram MathWorld)
Sylvester Matrix (Wolfram MathWorld)
Polynomial Discriminant (Wolfram MathWorld)
Polynomial (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Discriminant of a Polynomial"​
​http://demonstrations.wolfram.com/DiscriminantOfAPolynomial/​
​Wolfram Demonstrations Project​
​Published: December 29, 2016