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If \(x = \dfrac{1 + \sqrt{3}}{2}\) then compute \(4x^3 + 2x^2 - 8x + 7\)

 Jul 13, 2020
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Observation: What's the minimal polynomial of \(x\)?

 

\(2x = 1 + \sqrt 3\\ 2x - 1 = \sqrt 3\\ (2x - 1)^2 = 3\\ 4x^2 - 4x - 2 = 0\\ 2x^2 - 2x - 1 = 0\)

 

Having that in mind:

 

\(\quad 4x^3 + 2x^2 - 8x + 7\\ = 4x^3 - 4x^2 - 2x + 6x^2 - 6x + 7\\ =2x(2x^2 - 2x - 1) + 3(2x^2 - 2x - 1) + 10\\ = 2x(0) + 3(0) + 10\\ = \boxed{10}\)

 Jul 13, 2020

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