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).

Read more about this topic: Discriminant

Famous quotes containing the word formula:

“Every formula which expresses a law of nature is a hymn of praise to God.”
—Maria Mitchell (1818–1889)

“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)

“The formula for achieving a successful relationship is simple: you should treat all disasters as if they were trivialities but never treat a triviality as if it were a disaster.”
—Quentin Crisp (b. 1908)