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Home » They Thought They Had Everything—Then They Received This One Gift, and Life Changed Forever! - Midis

They Thought They Had Everything—Then They Received This One Gift, and Life Changed Forever! - Midis

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 $ 7 - 5 = 2 $,

📌 $ 9 - 7 = 2 $.

📌 Constant second differences of 2 indicate a quadratic polynomial. But the problem states $ f(x) $ models a cubic trendâthis suggests possible higher-order behavior or a cubic polynomial with leading coefficient zero in effective growth, yet we proceed with the given data.

📌 Assume $ f(x) = ax^3 + bx^2 + cx + d $. Use the system:

📌 $ a + b + c + d = 3 $

📌 For $ x = 2 $:

📌 $ 8a + 4b + 2c + d = 8 $

📌 For $ x = 3 $:

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We use the **Remainder Theorem**:
The remainder when a polynomial $ f(u) $ is divided by $ u - c $ is $...
Here, divide $ f(u) = u^3 + 2 $ by $ u + 1 = u - (-1) $.
So, compute $ f(-1) $:
f(-1) = (-1)^3 + 2 = -1 + 2 = 1
Question: Let $ a $ and $ b $ be complex numbers such that
\frac{a}{b} + \frac{b}{a} = 3.
Find the value of $ \frac{a^3}{b^3} + \frac{b^3}{a^3} $.
Let $ x = \frac{a}{b} \Rightarrow \frac{b}{a} = \frac{1}{x} $, so:
x + \frac{1}{x} = 3

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