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1. Discriminant

   Linked via "Sylvester matrix"

   $$\text{Disc}(P) = (-1)^{n(n-1)/2} \prod{1 \le i < j \le n} (ri - r_j)^2$$
   This formula immediately shows that $\text{Disc}(P) = 0$ if and only if the polynomial has at least one repeated root. The discriminant can also be expressed in terms of the Sylvester matrix and the resultant/) of $P$ and its derivative/) $P'$.
   For cubic polynomials}, $x^3 + px + q = 0$, the discriminant simplifies substantially: