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Find a monic quartic polynomial \(f(x)\) with rational coefficients whose roots include \(x = 1 -sqrt(2)\)  and \(x = 2+sqrt(5)\). Give your answer in expanded form.

 May 29, 2020
 #1
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f(x) = x^4 - 2x^3 + 3x^2 - x + 4.

 May 29, 2020
 #2
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If  1 - sqrt(2)  is a solution, so is  1 + sqrt(2).

If  2 + sqrt(5)  is a solution, so is  2 - sqrt(5).

 

The factors will be:  [ x - ( 1 - sqrt(2) ) ] · [ x - ( 1 + sqrt(2) ) ] · [ x - ( 2 + sqrt(5) ) ] · [ x - ( 2 - sqrt(5) ) ] 

 

Multiplying out  [ x - ( 1 - sqrt(2) ) ] · [ x - ( 1 + sqrt(2) ) ]    --->   [ x2 - 2x - 1 ]

 

Multiplying out  [ x - ( 2 + sqrt(5) ) ] · [ x - ( 2 - sqrt(5) ) ]    --->   [ x2 - 4x - 1 ]

 

Multiplying out  [ x2 - 2x - 1 ] · [ x2 - 4x - 1 ]   --->   x4 - 6x3 + 6x2 + 6x + 1

 May 29, 2020

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