# where does this comes from: $D=\prod _{i<j} \left(x_i-x_j\right)^2$

The background is about the relation of roots and coefficients, and the symmetric polynomial.

My book directly write (1), can you tell me where and how does this come from?

I know $\prod \left(x_i-x_j\right)$ is the result of Vandermonde Determinant, so is (1) related with Det?

\begin{align*}D=\prod _{i<j} \left(x_i-x_j\right)^2\tag{1}\end{align*}

\begin{align*}x^3+a_1x^2+a_2x+a_3\tag{2}\end{align*}

\begin{align*}D=a_1^2a_2^2-4a_2^3-4a_1^3a_3-27a_3^2+18a_1a_2a_3.\tag{3}\end{align*}

Can you show me how to make (1) becomes (3)?

The discriminant $D$ is a symmetric version of the Vandermonde determinant.
• Why $D=\prod _{i<j} \left(x_i-x_j\right)^2$=0, means the monic polynomial has multiple root over C? – User19912312 Aug 11 '13 at 14:17