Retrieving "Repeated Root" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Discriminant

   Linked via "repeated root"

   $$\text{Disc}(P) = (-1)^{n(n-1)/2} \prod{1 \le i < j \le n} (ri - r_j)^2$$
   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: