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If the product (3x^2 - 5x + 4)(7 - x) can be written in the form ax^3 + bx^2 + cx + d, where a,b,c,d are real numbers, then find 8a + 4b + 2c + d.

 Jan 11, 2022
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Let \(f(x) = ax^3 + bx^2 + cx + d = (3x^2 - 5x + 4)(7 - x)\). Note that \(8a + 4b + 2c + d = f(2)\), so hence we can just plug \(x = 2\) into \((3x^2 - 5x + 4)(7 - x)\) for an answer of \[(3 \cdot 2^2 - 5 \cdot 2 + 4)(7 - 2) = (12 - 10 + 4)(7 - 2) = (6)(5) = \boxed{30}.\]

 Jan 11, 2022

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