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.
+128473 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
