Past exam of the algebra course of the University of Oxford / 2007 / A / 4 / iii / Solution

For $d(X)=X^n+a_{n-1}{X}^{n-1}+\cdots +a_0$, the determinant of its companion matrix is $(-1{)}^n{a}_0$. ThereforeThus invertibility is readable from the constant coefficient, with no factorization or elimination.