Discriminant of a polynomial $;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0,$ is defined as
\begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align}
where $,r_1,\dots,r_n,$ are roots of $P\left(x\right)$ (counting multiplicity)
In algebra, the discriminant of a polynomial is typically denoted by a capital $D$, capital script $\mathscr D$, or the capital Greek letter Delta $\Delta$. It gives information about the nature of its roots. Typically, the discriminant is zero if and only if the polynomial has a multiple root.
For example, the discriminant of the quadratic polynomial $\;ax^2+bx+c\;$ is $\;\Delta = b^2-4ac.\,$ Here for real $a,\,b$ and $c$, if $\Delta > 0$, the polynomial has two real roots, if $\Delta = 0$, the polynomial has one real double root, and if $\Delta < 0$, the two roots of the polynomial are complex conjugates.
