In algebra, the discriminant of a polynomial with real or complex coefficients is a certain expression in the coefficients of the polynomial which is a homogeneous polynomial in the coefficients and gives information on the nature of the roots; in particular, it is equal to zero if and only if the polynomial has a multiple root (i.e. a root with multiplicity greater than one) in the complex numbers. For example, the discriminant of the quadratic polynomial
- is
The discriminant of the cubic polynomial
- is
The discriminant of a quartic is significantly longer, having 16 terms.
This 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.
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