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.1
You are probably aware of the well-known formula of the discriminant for the quadratic polynomial , which is , and use this formula to compute the roots.
However, the discriminant actually allows us to deduce some properties of the roots without computing them. In the case of a quadratic polynomial, it is zero if – and only if – the polynomial has a double root. It is positive if the polynomial has two real roots, and it is negative if roots are complex.
The calculator below computes the discriminant, and you can find a bit more theory on discriminants immediately underneath it.
The discriminant for a polynomial of degree n: can be defined either in terms of the quotient of the resultant or in terms of the roots.
In terms of the roots, the discriminant is equal to
Technically, one can derive the formula for the quadratic equation without knowing anything about the discriminant. Then, if you plug derived expressions for the roots into the definition above, you will end up with the .
In terms of the quotient of the resultant, the discriminant is equal to
where Res is the resultant of A and the first derivation of A'. The resultant, in short, is the determinant of the Sylvester matrix of A and A'.
In the case of a quadratic polynomial, the A is and the A' is . If you write down the Sylvester matrix for these two polynomials and derive the determinant, you will again end up with the .
Higher degree discriminant computation
Using the second definition, you can derive formulas for a polynomial of higher degrees (the link below has formulas for degree 3 and degree 4), but they are quite complex.
OEIS sequence A007878 lists 5 terms for polynomials of a degree of 3; 16 terms for a degree of 4; 59 terms for a degree of 5; and finally 3,815,311 terms for polynomials of a degree of 12.
The calculator below computes the discriminant of a higher degree polynomial from the resultant of a polynomial and its derivative