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 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 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 (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.
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.
Read more about Discriminant: Formula, Quadratic Formula, Discriminant of A Polynomial, Nature of The Roots, Discriminant of A Polynomial Over A Commutative Ring, Discriminant of A Conic Section, Discriminant of A Quadratic Form, Discriminant of A Differentiable Function, Alternating Polynomials