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📅 May 4, 2026 👤 scraface

May 04, 2026

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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 = \om...
Now evaluate at \(x = \omega\):
\omega^4 + 3\omega^3 - 2\omega^2 + \omega + 5 = a\omega + b
We compute each term:
\(\omega^4 = \omega \cdot \omega^3 = \omega(-1) = -\omega\)
\(3\omega^3 = 3(-1) = -3\)
\(-2\omega^2 = -2(\omega - 1) = -2\omega + 2\)
\(\omega\)
So \(a\omega + b = 3\omega + 4\), hence \(a = 3\), \(b = 4\).

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