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Home » We test each pair \((x, y)\) to see if it satisfies the equation. - Parker Core Knowledge

We test each pair \((x, y)\) to see if it satisfies the equation. - Parker Core Knowledge

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Thus, the value of $ c $ is $\boxed{7}$.Question: A quantum error correction code requires the evaluation of a cubic polynomial $ f(x) $ such that $ f(1) = 3 $, $ f(2) = -1 $, $ f(3) = 5 $, and $ f(4) = -7 $. Determine $ f(5) $.

📌 Solution: Let $ f(x) = ax^3 + bx^2 + cx + d $. We use the given values to form a system of equations:

📌 f(1) &= a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 3 \quad \text{(1)}\\

📌 f(2) &= 8a + 4b + 2c + d = -1 \quad \text{(2)}\\

📌 f(3) &= 27a + 9b + 3c + d = 5 \quad \text{(3)}\\

📌 f(4) &= 64a + 16b + 4c + d = -7 \quad \text{(4)}

📌 $ (8a + 4b + 2c + d) - (a + b + c + d) = -1 - 3 $

📌 $ \Rightarrow 7a + 3b + c = -4 \quad \text{(5)} $

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