Editing Discriminant (section)

{{short description|Function of the coefficients of a polynomial that gives information on its roots}}
{{distinguish|Determinant}}
{{Other uses}}
{{more citations needed|date=November 2011}}

In [[mathematics]], the '''discriminant''' of a [[polynomial]] is a quantity that depends on the [[coefficient]]s and allows deducing some properties of the [[zero of a function|roots]] without computing them. More precisely, it is a [[polynomial function]] of the coefficients of the original polynomial. The discriminant is widely used in [[polynomial factorization|polynomial factoring]], [[number theory]], and [[algebraic geometry]].

The discriminant of the [[quadratic polynomial]] <math>ax^2+bx+c</math> is
:<math>b^2-4ac,</math>
the quantity which appears under the [[square root]] in the [[quadratic formula]]. If <math>a\ne 0,</math> this discriminant is zero [[if and only if]] the polynomial has a [[double root]]. In the case of [[real number|real]] coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct [[complex conjugate]] roots.<ref>{{Cite web|title=Discriminant {{!}} mathematics|url=https://www.britannica.com/science/discriminant|access-date=2020-08-09|website=Encyclopedia Britannica|language=en}}</ref> Similarly, the discriminant of a [[cubic polynomial]] is zero if and only if the polynomial has a [[multiple root]]. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots. 

More generally, the discriminant of a univariate polynomial of positive [[degree of a polynomial|degree]] is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non-real roots is a [[Multiple (mathematics)|multiple]] of 4 (including none), and negative otherwise.

Several generalizations are also called discriminant: the ''[[discriminant of an algebraic number field]]''; the ''discriminant of a [[quadratic form]]''; and more generally, the ''discriminant'' of a [[form (mathematics)|form]], of a [[homogeneous polynomial]], or of a [[projective hypersurface]] (these three concepts are essentially equivalent).