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10th: 1.423828125 - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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Question: Find the remainder when $ (x^2 + 1)^{2025} + 1 $ is divided...
Let $ f(x) = (x^2 + 1)^{2025} + 1 $, and we are to find the remainder...
Let $ d(x) = x^4 + x^2 + 1 $. Note that
x^6 - 1 = (x^2 - 1)(x^4 + x^2 + 1),
so $ x^6 \equiv 1 \pmod{d(x)} $. Thus, modulo $ d(x) $, all powers of...
We reduce $ f(x) \equiv (x^2 + 1)^{2025} + 1 \pmod{d(x)} $.
Letâs analyze $ (x^2 + 1)^{2025} \mod d(x) $.
Let $ y = x^2 $, then $ d(x) = y^2 + y + 1 $. Since $ y^3 \equiv 1 \mo...
Now $ x^2 \equiv y $, so $ x^2 + 1 \equiv y + 1 $, and
We reduce $ (y + 1)^{2025} $ modulo $ y^2 + y + 1 $. Since $ y^2 \equi...

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By the **Remainder Theorem**, the remainder is simply $ f(1) $.
f(1) = 1^3 + 2(1)^2 - 5(1) + 6 = 1 + 2 - 5 + 6 = 4.
So the remainder is:
f(x + 1) - f(x) = 6x + 4.
Find $ f(x) $.
This is a finite difference equation. Since the difference is linear i...
f(x) = ax^2 + bx + c.
f(x + 1) = a(x + 1)^2 + b(x + 1) + c = a(x^2 + 2
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