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if P(r) = 0 then Q(r^2) = 0, P(x) = x^3 - 2x + 1, Q(x) = x^3 - 4x^2 + 4x - 1

Source: 2020 New Zealand MO Round 2 p1 NZMO

September 22, 2021
algebrapolynomial

Problem Statement

Let P(x)=x3−2x+1P(x) = x^3 - 2x + 1 and let Q(x)=x3−4x2+4x−1Q(x) = x^3 - 4x^2 + 4x - 1. Show that if P(r)=0P(r) = 0 then Q(r2)=0Q(r^2) = 0.
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