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 a_{n} is the leading coefficient and r_{1},...,r_{n} are the roots (counting multiplicity) of the polynomial in some splitting field. It is the square of the Vandermonde polynomial.
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