Discriminant

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In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomial

is

Here, if Δ > 0, the polynomial has two real roots, if Δ = 0, the polynomial has one real root, and if Δ < 0, the polynomial has no real roots. The discriminant of the cubic polynomial

is

The discriminants of higher order polynominals are significantly longer: the discriminant of a quartic has 16 terms,[1] that of a quintic has 59 terms,[2] and that of a 6th order polynominal has 246 terms.[3]


A polynomial has a multiple root (i.e. a root with multiplicity greater than one) 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.

Contents

Definition

Formula

In terms of the roots, the discriminant is given by

where an is the leading coefficient and r1,...,rn are the roots (counting multiplicity) of the polynomial in some splitting field. It is the square of the Vandermonde polynomial.

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