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Home » Thus, the length is \(2 \times 40 = 80\). - IX Labs

Thus, the length is \(2 \times 40 = 80\). - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Then, substitute $ u = x^3 - 1 $:

📌 h(x^3 - 1) = (x^3 - 1)^2 + 2 = x^6 - 2x^3 + 1 + 2 = x^6 - 2x^3 + 3

📌 oxed{x^6 - 2x^3 + 3}

📌 Question: Find the remainder when $ (x^2 + x + 1)^3 $ is divided by $ x^2 - x + 1 $.

📌 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 $.

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