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Home » The Hidden Power of ‘Thought’ Thoughts You Never Knew Exist! - Dachbleche24

The Hidden Power of ‘Thought’ Thoughts You Never Knew Exist! - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question: Let $ h(x) $ be a polynomial such that $ h(x^3 + 1) = x^6 + 3x^3 + 3 $. Find $ h(x^3 - 1) $.

📌 Solution: Observe that the right-hand side is $ (x^3 + 1)^2 + 2 $. However, we can directly write:

📌 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 $ x^2 - x + 1 $.

📌 Solution: Let us compute $ (x^2 + x + 1)^3 \mod (x^2 - x + 1) $.

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