In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. For example, the discriminant of the quadratic polynomial Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

ax² + bx + c       is       b² − 4ac.

The discriminant of the cubic polynomial

ax³ + bx² + cx + d       is       b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd.

This 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 no multiple roots in its splitting field. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

The concept of discrimant has been generalized to other algebraic structures besides polynomials, including conic sections, quadratic forms, and algebraic number fields. Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications. In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes. ... This article or section does not cite its references or sources. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ...

Formulas for the discriminant

The quadratic polynomial

ax² + bx + c       has discriminant       b² − 4ac.

The cubic polynomial

ax³ + bx² + cx + d       has discriminant       b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd.

Simpler polynomials have simpler expressions for their discrminants. For example the monic quadratic polynomial In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

x² + bx + c       has discrminant       b² − 4c.

The monic cubic polynomial

x³ + bx² + cx + d       has discriminant       b²c² − 4c³ − 4b³d − 27d² + 18bcd.

The monic cubic without quatratic term

x³ + cx + d       has discriminant       − 4c³ − 27d².

The discriminant in the quadratic formula

The quadratic polynomial P(x) = ax² + bx + c has discriminant D = b² − 4ac, which is the quantity under the square root sign in the quadratic formula. For real numbers a, b, c, one has: f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

• When D > 0 , P(x) has two distinct real roots , and its graph crosses the x-axis twice.
• When D = 0, P(x) has two coincident real roots , and its graph is tangent to the x-axis.
• When D < 0 , P(x) has no real roots, and its graph lies strictly above or below the x-axis. In this case, P(x) has two distinct complex roots.

Discriminant of a polynomial

The discriminant of the general polynomial

is, up to a factor, equal to the determinant of the (2n − 1)×(2n − 1) matrix (see Sylvester matrix) In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a Sylvester matrix is a matrix associated to two polynomials that gives us some information about those polynomials. ...

The determinant of this matrix is known as the resultant of p(x) and p'(x), notation R(p,p'). The discriminant D(p) of p(x) is now given by the formula In mathematics, the resultant of two monic polynomials and over a field is defined as the product of the differences of their roots, where and take on values in the algebraic closure of . ...

.

For example, in the case n = 4, the above determinant is

The discriminant of the degree 4 polynomial is then obtained from this determinant upon dividing by a₄.

Equivalently, the discriminant is equal to

where r₁, ..., r_n are the complex roots (counting multiplicity) of the polynomial p(x): In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

This second expression makes it clear that, p has a multiple root if and only if the discriminant is zero. (This multiple root can be complex.) It has been suggested that this article or section be merged with Logical biconditional. ...

The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots r_i remains valid; the roots have to be taken in some splitting field of the polynomial. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

Discriminant of a conic section

For a conic section defined by the real polynomial: Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

ax² + bxy + cy² + dx + ey + f= 0,

the discriminant is equal to

b² − 4ac,

and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will not work for degenerate cases (when the polynomial factorises). In geometry, two sets have the same shape if one can be transformed to another by a combination of translations, rotations and uniform scalings. ... The ellipse and some of its mathematical properties. ... Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... A graph of a hyperbola. ...

Discriminant of a quadratic form

There is a substantive generalisation to quadratic forms Q over any field K of characteristic ≠ 2. These can be written as a sum of terms In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Look up Characteristic in Wiktionary, the free dictionary. ...

a_iL_i²

where the L_i are linear forms and 1 ≤ i ≤ n where n is the number of variables. Then the discriminant is the product of the a_i, taken in K/K², and is then well defined (i.e., up to squares). A more invariant way to say this is as (the class of) the determinant of a symmetric matrix for Q. In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

Discriminant of an algebraic number field

See main article, Discriminant of an algebraic number field. In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes. ...