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the hidden dungeon only i can enter - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Find the remainder when \( u^5 - 3u^3 + 2u + 8 \) is divided by \( u^2 - 2u + 1 \).

📌 We are to divide \( u^5 - 3u^3 + 2u + 8 \) by \( u^2 - 2u + 1 \). Note that:

📌 \[ u^2 - 2u + 1 = (u - 1)^2 \]

📌 Let \( f(u) = u^5 - 3u^3 + 2u + 8 \). Since the divisor is quadratic, the remainder is linear: \( R(u) = au + b \).

📌 By the Remainder Theorem for polynomials, we use polynomial division or the method of undetermined coefficients. Alternatively, use substitution at the double root.

📌 \[ f(u) = (u - 1)^2 Q(u) + au + b \]

📌 \( f(1) = a(1) + b = a + b \)

📌 Differentiate: \( f'(u) = 2(u - 1)Q(u) + (u - 1)^2 Q'(u) + a \), so \( f'(1) = a \)

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