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Home » after 1 hr: 20 × 0.95 = 19 - RoadRUNNER Motorcycle Touring & Travel Magazine

after 1 hr: 20 × 0.95 = 19 - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 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.

📌 Hence, the final answer is:

📌 $$### Question 1

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