Editing Discriminant (section)

==Low degrees==
The discriminant of a [[linear polynomial]] (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (using the usual conventions for the [[empty product]] and considering that one of the two blocks of the [[Sylvester matrix]] is [[empty matrix|empty]]). There is no common convention for the discriminant of a constant polynomial (i.e., polynomial of degree 0).

For small degrees, the discriminant is rather simple (see below), but for higher degrees, it may become unwieldy. For example, the discriminant of a [[generic polynomial|general]] [[Quartic function|quartic]] has 16 terms,<ref>{{cite book
|title=Elimination practice: software tools and applications
|first1=Dongming
|last1=Wang
|publisher=[[Imperial College Press]]
|year=2004
|isbn=1-86094-438-8
|url=https://books.google.com/books?id=ucpk6oO5GN0C&pg=PA180
|at=ch. 10 p. 180}}
</ref> that of a [[Quintic function|quintic]] has 59 terms,<ref>{{cite book
|title=Discriminants, resultants and multidimensional determinants
|first1=Israel M.
|last1=Gelfand
|author1-link=Israel Gelfand
|first2=Mikhail M.
|last2=Kapranov
|author2-link=Mikhail Kapranov
|first3=Andrei V.
|last3=Zelevinsky
|author3-link=Andrei Zelevinsky
|publisher=[[Birkhäuser]]
|year=1994
|isbn=3-7643-3660-9
|page=1
|url=http://blms.oxfordjournals.org/cgi/reprint/28/1/96|archive-url=https://archive.today/20130113052223/http://blms.oxfordjournals.org/cgi/reprint/28/1/96|url-status=dead|archive-date=2013-01-13}}
</ref> and that of a [[sextic equation|sextic]] has 246 terms.<ref>{{cite book
|title=Solving polynomial equations: foundations, algorithms, and applications
|first1=Alicia
|last1=Dickenstein
|author1-link=Alicia Dickenstein
|first2=Ioannis Z.
|last2=Emiris
|publisher=[[Springer Science+Business Media|Springer]]
|year=2005
|isbn=3-540-24326-7
|url=https://books.google.com/books?id=rSs-pQNrO_YC&pg=PA26
|at=ch. 1 p. 26}}
</ref>
This is [[OEIS]] sequence {{OEIS link|A007878}}.
<!-- Please don't add numbers of terms of higher degrees (like 7/1103, 8/5247 and others of http://oeis.org/A007878) without providing proper sources. Thanks -->

===Degree 2===
{{see also|Quadratic equation#Discriminant}}

The quadratic polynomial <math>ax^2+bx+c \,</math> has discriminant
:<math>b^2-4ac\,.</math>

The square root of the discriminant appears in the [[quadratic formula]] for the roots of the quadratic polynomial:
:<math>x_{1,2}=\frac{-b \pm \sqrt {b^2-4ac}}{2a}.</math>

where the discriminant is zero if and only if the two roots are equal. If {{math|''a'', ''b'', ''c''}} are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two [[complex conjugate]] roots if it is negative.<ref>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=0-387-40397-3
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=ch. 10.3 pp. 153–154}}</ref>

The discriminant is the product of {{math|''a''{{sup|2}}}} and the square of the difference of the roots.

If {{math|''a'', ''b'', ''c''}} are [[rational number]]s, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.

===Degree 3===
{{seealso|Cubic equation#Discriminant}}
[[File:Discriminant of cubic polynomials..png|thumb|The zero set of discriminant of the cubic {{math|''x''<sup>3</sup> + ''bx''<sup>2</sup> + ''cx'' + ''d''}}, i.e. points satisfying {{math|1=''b''<sup>2</sup>''c''<sup>2</sup> − 4''c''<sup>3</sup> − 4''b''<sup>3</sup>''d'' − 27''d''<sup>2</sup> + 18''bcd'' = 0}}.]]
The cubic polynomial <math>ax^3+bx^2+cx+d \,</math> has discriminant
:<math>b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd\,.</math><ref>{{cite web| url=https://brilliant.org/wiki/cubic-discriminant/| title=Cubic Discriminant {{!}} Brilliant Math & Science Wiki| access-date=2023-03-21}}</ref><ref>{{cite web| url=https://www.johndcook.com/blog/2019/07/14/discriminant-of-a-cubic/| title=Discriminant of a cubic equation| date=14 July 2019| access-date=2023-03-21}}</ref>

In the special case of a [[Depressed cubic#Depressed cubic|depressed cubic]] polynomial <math>x^3+px+q</math>, the discriminant simplifies to
:<math> -4p^3-27q^2\,.</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.<ref>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=0-387-40397-3
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156}}</ref>

The square root of a quantity strongly related to the discriminant appears in the [[Cubic equation#General cubic formula|formulas for the roots of a cubic polynomial]]. Specifically, this quantity can be {{math|−3}} times the discriminant, or its product with the square of a rational number; for example, the square of {{math|1/18}} in the case of [[Cardano formula]].

If the polynomial is irreducible and its coefficients are rational numbers (or belong to a [[number field]]), then the discriminant is a square of a rational number (or a number from the number field) if and only if the [[Galois group]] of the cubic equation is the [[cyclic group]] of [[order (group theory)|order]] three.

===Degree 4===
[[File:Quartic Discriminant.png|thumb|The discriminant of the quartic polynomial {{math|''x''<sup>4</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e''}}. The surface represents points ({{math|''c'', ''d'', ''e''}}) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.]]
The [[quartic polynomial]]
<math> ax^4+bx^3+cx^2+dx+e\,</math>
has discriminant
:<math>\begin{align}
{} & 256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e \\[4pt]
& {} -27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de \\[4pt]
& {} +18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde \\[4pt]
& {} -4b^3d^3-4b^2c^3e+b^2c^2d^2\,.
\end{align}</math>

The depressed quartic polynomial
<math> x^4+cx^2+dx+e\,</math>
has discriminant
:<math>\begin{align}
{} & 16c^4e -4c^3d^2 -128c^2e^2+144cd^2e -27d^4 + 256e^3\,.
\end{align}</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.