+288 Find a monic quartic polynomial \(f(x)\) with rational coefficients whose roots include \(x=1-\sqrt{2}\) and \(x=2+\sqrt{5}\).
+128406 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
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x^4 - 6x^3 + 6x^2 + 6x + 1
