The Invariant Root

Algebra Level 5

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

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

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