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Home » Thus, \( D \equiv 1, 4, \text or 7 \pmod9 \) - High Altitude Science

Thus, \( D \equiv 1, 4, \text or 7 \pmod9 \) - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 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 div...
Solution: We perform polynomial division or use the fact that the rema...
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:
\(\omega^4 = \omega \cdot \omega^3 = \omega(-1) = -\omega\)
\(3\omega^3 = 3(-1) = -3\)

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