Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 18J / a / Solution

For a monic polynomial with roots $r_1,\dots ,r_n$, its discriminant isHere $\mathrm{Disc}(g)=(u^3-v^3{)}^2$. Both $u^3-v^3$ and the Vandermonde product in the ${\alpha }_i$ are alternating homogeneous cubics. Evaluation at $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(0,1,-1)$ fixes the constant and gives

Using ${\alpha }_1+{\alpha }_2+{\alpha }_3=0$, ${\sum }_{i<j}{\alpha }_i{\alpha }_j=a$, and ${\alpha }_1{\alpha }_2{\alpha }_3=-b$, direct expansion givesConsequentlyandThus