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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.

 Feb 10, 2022
 #1
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Note that   (1 + sqrt2 )     and ( 3 -sqrt 5)  are also  roots

 

We have

 

(x - (1-sqrt 2) )  ( x - ( 1 + sqrt 2) )   ( x - (3 + sqrt 5) )  ( x - ( 3 + sqrt 5)

 

Simplify in steps

 

( x - ( 1 -sqrt2) )  ( x - (1 + sqrt 2))  =  

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

x^2 - [ 2 ] x  + 1  - 2  =

x^2 - 2x - 1

 

Similarly

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

x^2  - [( 3-sqrt 5) + (3 + sqrt 5)]x   +  (3 +sqrt 5) (3 - sqrt 5)  =

x^2 - 6x + 9 - 5  =

x^2 - 6x + 4

 

So

 

(x^2 - 2x -1) ( x^2 - 6x + 4)  =

 

    ( x^4 -2x^3 - x^2 )

 +        ( -6x^3 + 12x^2 + 6x)

 +                    (    4x^2  -8x  - 4)   =

 

x^4  - 8x^3 + 15x^2  -2x   - 4

 

 Feb 10, 2022

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