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Home » $3^2$ (from 9), - ECD Germany

$3^2$ (from 9), - ECD Germany

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Solution: Let us compute $ (x^2 + x + 1)^3 \mod (x^2 - x + 1) $.

📌 Note that $ x^2 + x + 1 $ and $ x^2 - x + 1 $ are related via complex roots:

📌 Let $ \omega $ be a primitive cube root of unity, satisfying $ \omega^3 = 1 $, $ \omega

📌 e 1 $, and $ \omega^2 + \omega + 1 = 0 $.

📌 We evaluate $ (x^2 + x + 1)^3 $ at $ x = \omega $:

📌 \omega^2 + \omega + 1 = 0 \Rightarrow x^2 + x + 1 = 0 ext{ at } x = \omega

📌 Thus, $ (x^2 + x + 1)^3 = 0 $ at $ x = \omega $, so $ x^2 + x + 1 $ is divisible by $ x^2 - x + 1 $, since both are degree 2 and the first vanishes where the second does.

📌 Therefore, the remainder is 0.

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