Definition:Discriminant of Polynomial

Let $$f \left({x}\right)$$ be a polynomial in $$x$$ of degree $$n$$.

That is, let $$f \left({x}\right) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + a_n$$.

Let the roots of the equation $$f \left({x}\right) = 0$$ be $$x_1, x_2, \ldots, x_n$$.

Then the discriminant $$\Delta$$of the equation $$f \left({x}\right) = 0$$ is defined as:

$$\mathbf {Define:} \ \Delta \ \stackrel {\mathbf {def}} {=\!=} \ \prod_{1 \le i < j \le n} \left({x_i - x_j}\right)^2$$.