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