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.
+128405 If 1 -sqrt 2 is a root then so is 1 + sqrt 2
So...partially, we have
(x - (1 -sqrt 2) ) ( x - (1 + sqrt 2) ) =
x^2 - (1 -sqrt 2)x - (1 + sqrt 2)x + 1 - 2) =
x^2 - 2x - 1
Likewise if 3 = sqrt 5 is a root then so is 3 -sqrt 5
So
(x - ( 3 + sqrt 5)) ( x - (3 -sqrt 5)) =
x^2 -(3 +sqrt 5) x - (3 -sqrt 5)) x + 9 - 5 =
x^2 -6x + 4
The polynomial is
(x^2 - 2x -1) ( x^2 - 6x + 4) =
x^4 -6x^3 + 4x^2
-2x^3 + 12x^2 - 8x
- x^2 + 6x - 4 =
x^4 - 8x^3 + 15x^2 - 2x - 4
