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1. Discriminant

   Linked via "resultant"

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

2. Discriminant

   Linked via "resultant"

   Relation to Resultants
   The discriminant of a polynomial} $P(x)$ is intimately connected to the resultant/) of $P(x)$ and its derivative/) $P'(x)$. The resultant, $\text{Res}(P, P')$, is a determinant whose vanishing indicates the presence of a common root between $P$ and $P'$. The relationship is given by:
   $$\text{Disc}(P) = \frac{(-1)^{n(n-1)/2}}{a_n} \text{Res}(P, P')$$

3. Discriminant

   Linked via "resultant"

   $$\text{Disc}(P) = \frac{(-1)^{n(n-1)/2}}{a_n} \text{Res}(P, P')$$
   where $a_n$ is the leading coefficient of $P$. This reliance on the resultant/) provides a practical algorithmic method for calculating the discriminant for high-degree polynomials without explicitly solving for the roots/) [5].
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