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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Thus, $ f(5) = \boxed{-65} $

📌 Question: An anthropologist models cultural adaptation index $ A(x) $ over time $ x $ (in decades) using a cubic polynomial satisfying $ A(1) = 2 $, $ A(2) = -4 $, $ A(3) = 6 $, and $ A(4) = -8 $. Find the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $.

📌 Solution: By the Remainder Theorem, the remainder when $ A(x)^3 + 2 $ is divided by $ x - 5 $ is $ A(5)^3 + 2 $. So we must compute $ A(5) $.

📌 Let $ A(x) = ax^3 + bx^2 + cx + d $. Use the given values:

📌 A(1) &= a + b + c + d = 2 \quad \text{(1)}\\

📌 A(2) &= 8a + 4b + 2c + d = -4 \quad \text{(2)}\\

📌 A(3) &= 27a + 9b + 3c + d = 6 \quad \text{(3)}\\

📌 A(4) &= 64a + 16b + 4c + d = -8 \quad \text{(4)}

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