Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 2 / Solution

Expanding the determinant along its first row givesThe required factorization isEach factor is a symmetric polynomial. The second is $\frac{1}{2}[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$, making its permutation symmetry especially transparent. This factorization holds over both real and complex coefficients.