Putnam 2002 B6

Let $p$ be a prime number. Prove that the determinant of the matrix $$\begin{pmatrix} x & y & z \\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{pmatrix}$$ is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a,b,c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)