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Home » \[ A = \frac12ab \] - AssociationVoting

\[ A = \frac12ab \] - AssociationVoting

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 oxed{rac{16\pi}{3} - 6\sqrt{3}}

📌 Question: Find the remainder when \(x^4 + 3x^3 - 2x^2 + x + 5\) is divided by \(x^2 - x + 1\).

📌 Solution: We perform polynomial division or use the fact that the remainder when dividing by a quadratic is linear: let

📌 x^4 + 3x^3 - 2x^2 + x + 5 = (x^2 - x + 1)Q(x) + ax + b

📌 Let \(\omega\) be a root of \(x^2 - x + 1 = 0\). Then \(\omega^2 = \omega - 1\), and \(\omega^3 = \omega(\omega - 1) = \omega^2 - \omega = (\omega - 1) - \omega = -1\). So \(\omega^3 = -1\), and \(\omega^6 = 1\).

📌 Now evaluate at \(x = \omega\):

📌 \omega^4 + 3\omega^3 - 2\omega^2 + \omega + 5 = a\omega + b

📌 We compute each term:

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