Discriminant
A Discriminant is a algebraic expression that can be used to find roots of a function.
- AKA: Function Discriminant.
- …
- Example(s):
- The discriminant of the quadratic equation [math]\displaystyle{ ax^2+bx+c=0 }[/math] is [math]\displaystyle{ b^2-4ac }[/math] and roots of [math]\displaystyle{ x }[/math] can be found by solving [math]\displaystyle{ x=\frac{-b\pm \sqrt{ b^2-4ac}}{2a} }[/math] (Quadratic Formula).
- Polynomial Discriminant
- Elliptic Discriminant.
- Counter-Example(s):
- See: Discriminant Analysis, Characteristic Equation, Algebra, Polynomial, Polynomial Function, Root of a Function, Quadratic Polynomial, Double Root, Real Number, Degree of a Polynomial, Multiple Root, Number Theory.
References
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Discriminant Retrieved:2017-6-24.
- In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial [math]\displaystyle{ ax^2+bx+c }[/math] is [math]\displaystyle{ b^2-4ac, }[/math] which is zero if and only if the polynomial has a double root, and (in the case of real coefficients) is positive if and only if the polynomial has two real roots.
More generally, for a polynomial of an arbitrary degree, the discriminant is zero if and only if it has a multiple root, and, in the case of real coefficients, it is positive if and only if the number of non-real roots is a multiple of 4.
The discriminant is widely used in number theory, either directly or through its generalization as the discriminant of a number field. For factoring a polynomial with integer coefficients, the standard method consists in factoring first its reduction modulo a prime number not dividing the discriminant (and not dividing the leading coefficient). In algebraic geometry, the discriminant with respect to one of the variables characterizes the points of a hypersurface where the implicit function theorem does not apply.
The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.
- In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial [math]\displaystyle{ ax^2+bx+c }[/math] is [math]\displaystyle{ b^2-4ac, }[/math] which is zero if and only if the polynomial has a double root, and (in the case of real coefficients) is positive if and only if the polynomial has two real roots.
2017b
- (MathWorld, 2017) ⇒ Weisstein, Eric W. "Discriminant." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Discriminant.html
- A discriminant is a quantity (usually invariant under certain classes of transformations) which characterizes certain properties of a quantity's roots. The concept of the discriminant is used for binary quadratic forms, elliptic curves, metrics, modules, polynomials, quadratic curves, quadratic fields, quadratic forms, and in the second derivative test.
SEE ALSO: Binary Quadratic Form Discriminant, Circle Discriminant, Conic Section Discriminant, Determinant, Discriminant Analysis, Elliptic Discriminant, Metric Discriminant, Modular Discriminant, Polynomial Discriminant, Quadratic Curve Discriminant, Second Derivative Test Discriminant
- A discriminant is a quantity (usually invariant under certain classes of transformations) which characterizes certain properties of a quantity's roots. The concept of the discriminant is used for binary quadratic forms, elliptic curves, metrics, modules, polynomials, quadratic curves, quadratic fields, quadratic forms, and in the second derivative test.