Discriminant (polynomial)

A polynomial’s discriminant is a quantity computed from its coefficients that vanishes exactly when the polynomial has a repeated root, over an algebraic closure.

Polynomial derivative: The polynomial obtained by differentiating each term according to the power rule. A repeated root is shared by a polynomial and its derivative.
Quadratic formula: A formula giving the roots of a quadratic equation from its coefficients. Its square-root term contains the quadratic discriminant, distinguishing real, repeated, and complex roots.
Repeated root: A root of a polynomial whose multiplicity is greater than one. Its presence is precisely the condition detected by a vanishing discriminant.
Polynomial discriminant: A coefficient-derived invariant that detects repeated roots of a polynomial. This broader name covers the same construction without specifying a particular polynomial context.
Vieta's formulas: Identities relating a polynomial’s coefficients to symmetric sums and products of its roots. They turn the root-difference formula for the discriminant into a coefficient expression.
Cubic equation: A polynomial equation of degree three. The cubic discriminant distinguishes one-real-root cases from three-real-root cases and identifies repeated roots.
Multiplicity of a root: The number of times a linear factor occurs in a polynomial’s factorization. The discriminant vanishes when at least one root has multiplicity greater than one.
Field discriminant: An invariant of a number field defined from the determinant of a trace pairing on an integral basis. Despite the shared name, it measures arithmetic properties of a number field rather than repeated polynomial roots.
Vandermonde determinant: The determinant of a matrix of powers of variables, equal to a product of their pairwise differences. Its square gives the product of squared root differences in the discriminant formula.
Discriminant locus: The set of parameter values for which a polynomial or algebraic object becomes singular or degenerate. For a family of polynomials, its discriminant vanishes exactly on parameter values with repeated roots.

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