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There is a unique polynomial P(x) of degree 4 with rational coefficients and leading coefficient 1 which has sqrt(1 + sqrt(6)) as a root. What is P(1)?

 Dec 22, 2021
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Based on the information provided: P(x) = x^4 + ax^3 + bx^2 + cx + d

P(1) = 1 + a + b + c + d

 

After a bit of educated guessing, it seems this polynomial would be \(\left(x + \sqrt{1 + \sqrt6}\right) \left(x + \sqrt{1 - \sqrt6} \right) \left(x - \sqrt{1 + \sqrt{6}}\right) \left(x - \sqrt{1 - \sqrt6}\right)\)

Which simplifies to \(x^4-2x^2-5\)

I'm not entirely sure why this works, but I think it's because it can be turned into a quadratic by setting y=x^2 (and the square roots must cancel out). Anyone?

Aside from that, 1-2-5 = -6.

 Dec 22, 2021

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