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Find a monic quartic polynomial \(f(x)\) with rational coefficients whose roots include \(x=1-\sqrt{2}\) and \(x=2+\sqrt{5}\)

 Mar 24, 2020
 #1
avatar+111330 
+1

If  1 - √2  is  a root  then so is  1 + √2

 

Same for  2 + √5

 

So   we  can find this as

 

[  x - (1 -√2) ]  [x -(1 + √2) ] [ x - ( 2 + √5) ] [ x - (2 - √5) ]

 

 

Taking this in pieces, we have

 

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

 

[x^2  - x + √2x - x - √2x + 1 - 2]  =

 

[ x^2 -2x - 1]   (1)

 

And

 

[x- (2 + √5) ] [ x - (2 - √5) ]   =

 

 [ x^2   -(2 + √5)x - (2 - √5)x  + (2 + √5)(2 - √5)  ]  = 

 

[x^2  -2x - √5x - 2x + √5x  + 4 - 5]  =

 

[ x^2 - 4x  - 1 ]  (2)

 

Take  the product of (1) (2)  =

 

[ x^2 - 2x - 1]  [ x^2 - 4x - 1]  =

 

x^4  - 2x^3  - x^2

       -4x^3 + 8x^2  + 4x

                 -x^2     +2x     +  1

____________________________

x^4  - 6x^3  + 6x^2 + 6x  + 1

 

 

 

 

cool cool cool

 Mar 24, 2020
 #2
avatar+1970 
+2

Impressive, Chris!

CalTheGreat  Mar 24, 2020

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