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P(X) is divisible by Q(X) iff b=1

Source: Romanian MO 2001

January 14, 2011
algebrapolynomialalgebra proposed

Problem Statement

Let n≥2n\ge 2 be an even integer and a,ba,b real numbers such that bn=3a+1b^n=3a+1. Show that the polynomial P(X)=(X2+X+1)n−Xn−aP(X)=(X^2+X+1)^n-X^n-a is divisible by Q(X)=X3+X2+X+bQ(X)=X^3+X^2+X+b if and only if b=1b=1.
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