Determinant of a quadratic form

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial ${\displaystyle ax^{2}+bx+c}$ is ${\displaystyle b^{2}-4ac,}$ which is zero if and only if the polynomial has a double root, and (in the case of real coefficients) is positive if and only if the polynomial has two real roots.

More generally, for a polynomial of an arbitrary degree, the discriminant is zero if and only if it has a multiple root, and, in the case of real coefficients, it is positive if and only if the number of non-real roots is a multiple of 4 when the degree is 4 or higher.

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