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P(a^2+a+1) if P(x)=x^4+3x^2-9x+1

Source: 2006 VMEO III Shortlist SL A2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

October 3, 2021

algebrapolynomial

Problem Statement

Given a polynomial

P(x)=x^4+3x^2-9x+1

. Calculate

P(\alpha^2+\alpha+1)

where

\alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}}