Function of the coefficients of a polynomial that gives information on its roots

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.

This results from the expression of the discriminant in terms of the roots
This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.

As the discriminant is defined in terms of a determinant, this property results immediately from the similar property of determinants.

When one is only interested in knowing whether a discriminant is zero (as is generally the case in algebraic geometry), these properties may be summarised as:

This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.

This restricts the possible terms in the discriminant. For the general quadratic polynomial there are only two possibilities and two terms in the discriminant, while the general homogeneous polynomial of degree two in three variables has 6 terms. For the general cubic polynomial, there are five possibilities and five terms in the discriminant, while the general homogeneous polynomial of degree 4 in 5 variables has 70 terms

Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots collapse. Most of the cases, where such a generalized discriminant is defined, are instances of the following.

Two quadratic forms, and thus two discriminants may be associated to a conic section.

It is zero if the conic section degenerates into two lines, a double line or a single point.