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Write down a polynomial of degree 4 with integer coefficients, so that two of the roots are 1 - sqrt(2) and 3 + sqrt(3).

 Dec 10, 2020
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The conjugates of these are also roots

 

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

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

x^2  -2 x +  1 -2 =

(x^2 -2x -1)

 

And

 

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

x^2 -x ( 3 + sqrt (3)) - x (3 -sqrt(3))  +  (3 -sqrt (3)) (3 + sqrt (3))

x^2  - 6x  + 9 - 3  =

(x^2 - 6x + 6)

 

So

 

(x^2 - 2x  - 1) (x^2  - 6x + 6)  =

 

x^4 - 6x^3 + 6x^2

      -2x^3  + 12x^2   - 12x

                     -x^2     +6x   - 6

_________________________

x^4  - 8x^3 + 17x^2 -  6x  - 6   

 

 

 

cool cool cool

 Dec 11, 2020

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