Discriminant

In algebra, the discriminant of a polynomial is a function of its coefficients, typically denoted by a capital D or the capital Greek letter delta (Δ). It gives information about the nature of its roots. The discriminant is zero if and only if the polynomial has a multiple root. 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 two roots of the polynomial are complex conjugates.

The discriminant of the cubic polynomial

is

In particular, the discriminant of

is

For higher degrees, the discriminant is always a polynomial function of the coefficients. It becomes significantly longer for the higher degrees. The discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, and that of a sextic has 246 terms. This is OEIS sequence .

If the lead coefficient a equals 1, the discriminant equals the product of the squared differences of all pairs of roots of the polynomial. Thus 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.

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