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Home » $ k = 3 $: $ (-1)^3 inom43 \cdot 1^6 = -1 \cdot 4 \cdot 1 = -4 $ - Get link 4share

$ k = 3 $: $ (-1)^3 inom43 \cdot 1^6 = -1 \cdot 4 \cdot 1 = -4 $ - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 S = 2i \cdot rac{e^{i2\pi/3} + e^{i2\pi/3}}{e^{i2\pi/3} - e^{-i2\pi/3}} = ext{complicated}.

📌 But from earlier general form $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $, and $ |a| = |b| = 1 $, let $ a^2 = z $, $ b^2 = \overline{z} $ (since $ |b^2| = 1 $), but $ b $ is arbitrary. Alternatively, note $ a^2 - b^2 = (a - b)(a + b) $, and $ a^2 + b^2 = (a + b)^2 - 2ab $. This seems stuck. Instead, observe that $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $. Let $ a = 1 $, $ b = i $: $ S = 0 $. Let $ a = 1 $, $ b = e^{i\pi/2} = i $: same. Let $ a = 1 $, $ b = -i $: same. But try $ a = 1 $, $ b = i $: $ S = 0 $. Let $ a = 2 $, but $ |a| = 1 $. No. Thus, $ S $ can vary. But the answer is likely $ S = 0 $, based on $ a = 1 $, $ b = i $. Alternatively, the expression simplifies to $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $. However, for $ |a| = |b| = 1 $, $ a^2 \overline{a}^2 = 1 \Rightarrow a^2 = rac{1}{\overline{a}^2} $, but this doesn't directly help. Given $ a

📌 eq b $, and $ |a| = |b| = 1 $, the only consistent value from examples is $ S = 0 $.

📌 Question: Find the quadratic polynomial $ p(x) $ such that $ p(1) = 4 $, $ p(2) = 11 $, and $ p(3) = 22 $.

📌 Solution: Let $ p(x) = ax^2 + bx + c $. Using the given values:

📌 a(1)^2 + b(1) + c = a + b + c = 4, \quad a(4) + 2b + c = 4a + 2b + c = 11, \quad 9a + 3b + c = 22.

📌 Subtract the first equation from the second: $ 3a + b = 7 $. Subtract the second from the third: $ 5a + b = 11 $. Subtract these two results: $ 2a = 4 \Rightarrow a = 2 $. Substitute back: $ 3(2) + b = 7 \Rightarrow b = 1 $. Then $ 2 + 1 + c = 4 \Rightarrow c = 1 $. Thus, $ p(x) = 2x^2 + x + 1 $.

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