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.

Guest Feb 20, 2018

#1**+1 **

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

CPhill
Feb 20, 2018