What does discriminant mean?

Definitions for discriminant
dɪˈskrɪm ə nəntdis·crim·i·nant

This dictionary definitions page includes all the possible meanings, example usage and translations of the word discriminant.


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Wiktionary

  1. discriminantnoun

    An expression that gives information about the roots of a polynomial; for example, the expression D = b - 4ac determines whether the roots of the quadratic equation ax + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).

  2. discriminantadjective

    Serving to discriminate.

Wikipedia

  1. Discriminant

    In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the 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 factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} is b 2 − 4 a c , {\displaystyle b^{2}-4ac,} the quantity which appears under the square root in the quadratic formula. If a ≠ 0 , {\displaystyle a\neq 0,} this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. 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 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 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, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially equivalent).

Webster Dictionary

  1. Discriminantnoun

    the eliminant of the n partial differentials of any homogenous function of n variables. See Eliminant

  2. Etymology: [L. discriminans, p. pr. of discriminare.]

Wikidata

  1. Discriminant

    In algebra, the discriminant of a polynomial is a function of its coefficients which gives information about the nature of its roots. For example, the discriminant of the quadratic polynomial is Here for real a, b and c, if Δ > 0, the polynomial has two real roots, if Δ = 0, the polynomial has one real double root, and if Δ < 0, the polynomial has no real roots. The discriminant of the cubic polynomial is For higher degrees, the discriminant is always a polynomial function of the coefficients. It is significantly longer: the discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, that of a 6th degree polynomial has 246 terms, and the number of terms increases exponentially with the degree. A polynomial has a multiple root in the complex numbers if and only if its discriminant is zero. The concept also applies if the polynomial has coefficients in a field which is not contained in the complex numbers. In this case, the discriminant vanishes if and only if the polynomial has a multiple root in its splitting field. As the discriminant is a polynomial function of the coefficients, it is defined as soon as the coefficients belong to an integral domain R and, in this case, the discriminant is in R. In particular, the discriminant of a polynomial with integer coefficients is always an integer. This property is widely used in number theory.

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Numerology

  1. Chaldean Numerology

    The numerical value of discriminant in Chaldean Numerology is: 7

  2. Pythagorean Numerology

    The numerical value of discriminant in Pythagorean Numerology is: 7

Examples of discriminant in a Sentence

  1. Rick Meckler:

    When you see selling this broad and non-discriminant, when everything is being sold, The Nasdaq tends to end when The Nasdaq just exhausts everyone who’s worried.

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"discriminant." Definitions.net. STANDS4 LLC, 2024. Web. 19 Jun 2024. <https://www.definitions.net/definition/discriminant>.

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