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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1-sqrt2 and x=3+sqrt5. Give your answer in expanded form.

 Jul 9, 2021
 #1
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If  1 -sqrt 2 is  a root  then  so  is 1 + sqrt 2

 

So...partially,  we  have

 

(x  - (1  -sqrt 2)  )  ( x  - (1 + sqrt 2) )    =

x^2 - (1 -sqrt 2)x - (1 + sqrt 2)x   +  1  - 2)   =

x^2 - 2x  - 1

 

Likewise  if     3   = sqrt 5 is a root  then so is 3  -sqrt 5

 

So

(x - ( 3 + sqrt 5))  ( x - (3  -sqrt 5))  =

x^2 -(3 +sqrt 5) x  - (3  -sqrt 5)) x  + 9 - 5  =

x^2  -6x  +  4

 

The polynomial is

 

(x^2   - 2x  -1) ( x^2  - 6x  + 4)  =

 

x^4  -6x^3  + 4x^2

      -2x^3   + 12x^2 - 8x

                  -    x^2  + 6x  - 4       =

 

x^4  - 8x^3  + 15x^2  - 2x  -  4   

 

 

cool cool cool 

 Jul 9, 2021

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