Discriminant - Formula

Formula

The quadratic polynomial

has discriminant

The cubic polynomial

has discriminant

The quartic polynomial

has discriminant

$\Delta=256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de+ 18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2.\,$

These are homogeneous polynomials in the coefficients, respectively of degree 2, 4 and 6. They are also homogeneous in term of the roots, of respective degrees 2, 6 and 12.

Simpler polynomials have simpler expressions for their discriminants. For example, the monic quadratic polynomial has discriminant

The monic cubic polynomial without quadratic term has discriminant

In terms of the roots, these are homogeneous polynomials of degree 2 (quadratic) and 6 (cubic).

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