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Find a monic quartic polynomial with rational coefficients whose roots include x = 1 - sqrt(2) and x = 2 + sqrt(7).
 

 May 14, 2021
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Since one of the roots is  1 - sqrt(2),  another root will be  1 + sqrt(2).

 

Since one of the roots is  2 + sqrt(7),  another root will be  2 - sqrt(7).

 

One possible solution is:  ( x - [1 - sqrt(2)] ) · ( x - [1 + sqrt(2)] ) · ( x - [ 2 + sqrt(7) ] ) · ( x - [ 2 - sqrt(7) ] )  =  0

 

Other solutions can be found by multiplying that solution by an non-zero real number.

 May 14, 2021

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