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Find a monic quartic polynomial $f(x)$ with rational coefficients whose roots include $x=1-\sqrt 2$ and $x=2+\sqrt 5$.

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Mar 31, 2021
edited by Dennis070sinneD  Mar 31, 2021

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If   1 - sqrt 2  is a root....so is  1 + sqrt 2

And

If  2 + sqrt 5  is a root.....so is 2 - sqrt 5

We have

[ x - ( 1 - sqrt 2) ] [ x - (1 + sqrt 2) ] [ x - (2 +sqrt 5) ] [ x - ( 2 - sqrt 5)]

This  is  a little tedious  to expand  (but not impossible )....I'll  use WolramAlpha here to help out

f(x)   =  x^4 - 6 x^3 + 6 x^2 + 6 x + 1

Mar 31, 2021