Discriminant of a Polynomial
This Demonstration shows the discriminant of the polynomial . The discriminant of a polynomial of degree is the quantity , where is the derivative of and is the resultant of and . The resultant is equal to the determinant of the corresponding Sylvester matrix. The discriminant of is 0 if and only if has a multiple root.
The discriminant of a polynomial with leading coefficient 1 is the product over all pairs of roots , of .
The equation 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.
 E. J. Borowski and J. M. Borwein, Dictionary of Mathematics, London: Collins, 1989 p. 169.