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