BUILT-IN MATHEMATICA SYMBOL

Discriminant

computes the discriminant of the polynomial poly with respect to the variable var.

Discriminant[poly, var, Modulus->p]
computes the discriminant modulo

.

The discriminant of a polynomial with leading coefficient one is the product over all pairs of roots , of .

A Method option can be given, with typical possible values being Automatic, , , , and .

Discriminant of a quadratic:

Out[1]=

Discriminant of a polynomial with numeric coefficients:

Discriminant of a general cubic:

Discriminant of a general quintic:

Discriminants are squares of differences of roots:

This compares timings of the available methods of discriminant computation:

By default the discriminant is computed over the rational numbers:

Compute the discriminant of the same polynomial over the integers modulo 2:

Compute the discriminant of the same polynomial over the integers modulo 3:

Decide whether a polynomial has multiple roots:

Find the condition for a cubic to have multiple roots:

The discriminant is zero if and only if the polynomial has multiple roots:

The discriminant can be represented in terms of roots as

:

Equation

Using exact coefficients, this indicates no common root:

With approximate coefficients, this does indicate a common root:

in this case, using higher precision resolves the problem: