Question 19
(IRE 1) Let $f(x)=x^8+4x^6+2x^4+28x^2+1$. Let $p\gt 3$ be a prime and suppose there exists an integer $z$ such that $p$ divides $f(z)$. Prove that there exist integers $z_1,z_2,\ldots,z_8$ such that if
$$g(x)=(x-z_1)(x-z_2)\cdots(x-z_8),$$
then all coefficients of $f(x)-g(x)$ are divisible by $p$.
$$g(x)=(x-z_1)(x-z_2)\cdots(x-z_8),$$
then all coefficients of $f(x)-g(x)$ are divisible by $p$.