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# need help with algebra

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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