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 \( | \alpha | \).

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