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p divides f(x)=x^5 +5x^4 +5x^3 +5x^2 +1, g(x)=x^5 +5x^4 +3x^3 -5x^2 -1

Source: Germany 1997 p6a

February 22, 2020
polynomialprimesdividesalgebra

Problem Statement

Let us define ff and gg by f(x)=x5+5x4+5x3+5x2+1f(x) = x^5 +5x^4 +5x^3 +5x^2 +1, g(x)=x5+5x4+3x3−5x2−1g(x) = x^5 +5x^4 +3x^3 -5x^2 -1. Determine all prime numbers pp such that, for at least one integer x,0≤x<p−1x, 0 \le x < p-1, both f(x)f(x) and g(x)g(x) are divisible by pp. For each such pp, find all xx with this property.
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