Discriminant

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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 ${\displaystyle ax^{2}+bx+c}$ is

${\displaystyle b^{2}-4ac,}$

the quantity which appears under the square root in the quadratic formula. If ${\displaystyle a\neq 0,}$ 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.^[1] 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).