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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 2 - 3(sqrt2) and x = 1 - sqrt3. Give your answer in expanded form.

 Feb 20, 2018
 #1
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If   2 - 3sqrt (2)  is a root so is its conjugate, 2 + 3sqrt(2)

And if 1 - sqrt (3)  is a root then so is  1 + sqrt(3).....so we have

 

[ x - (2 - 3sqrt (2) ]  [ x -  (2 + 3sqrt(2) ] [ x - ( 1 - sqrt(3) ] [ x - ( 1 + sqrt (3) ]     simplify

 

[ x^2 - x(2 - 3sqrt (2) - x (2 + 3sqrt (2) + 4 - 18 ]  [ x - ( 1 - sqrt(3) ] [ x - ( 1 + sqrt (3) ]    

 

[ x^2 - 4x - 14 ]   [  x^2 - x(1 - sqrt (3) - x (1 + sqrt (3) + 1 - 3 ]

 

  [ x^2 - 4x - 14 ] [ x^2 -2x - 2 ] =

 

x^4  - 6x^3 - 8x^2 + 36x + 28

 

 

 

cool cool cool

 Feb 20, 2018

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