The Invariant Root

Algebra Level 5

Consider all fifth degree polynomials $p(x)$ with integer coefficients, such that $p(x)$ has at least 1 integral root, $p(2) = 13$ and $p(10 ) = 5$ .

There is a complex number $\alpha$ such that $p( \alpha ) = 0$ for all such polynomials. Find $| \alpha |$ .