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Home » The vertex is \(\boxed(3,\ 0)\). - RoadRUNNER Motorcycle Touring & Travel Magazine

The vertex is \(\boxed(3,\ 0)\). - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 1, 2026 👤 scraface

May 01, 2026

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h(x^3 + 1) = (x^3 + 1)^2 + 2 \Rightarrow h(u) = u^2 + 2
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 $...
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...
Let $ \omega $ be a primitive cube root of unity, satisfying $ \omega^...
e 1 $, and $ \omega^2 + \omega + 1 = 0 $.
We evaluate $ (x^2 + x + 1)^3 $ at $ x = \omega $:

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