Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $Δ(f_n)$ denote the discriminant of $f_n$, that is $Δ(f_n) = A^{2n-2}\prod_{i < j} (α_j - α_i)^2$ where $A$ is the leading coefficient of $f_n$ and $α_1,\ldotsα_n$ are its roots. We prove that with high probability $$|Δ(f_n)| = n^{2n} e^{-{\sf D}_\ast n(1+o(1))}$$ as $n\to \infty$, for some explicit universal constant ${\sf D}_\ast>0$. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.