# Discriminant of a Polynomial

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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.

Contributed by: Izidor Hafner (December 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

For the meaning of the matrix/grid, see Sylvester Matrix (Wolfram *MathWorld*).

Reference

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

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