+1725 Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1 - \sqrt{2} and x = 2 and x = 7. Give your answer in expanded form.
+130466 If 1 -sqrt 2 is a root, so is 1 + sqrt 2
So
(x - (1 -sqrt 2)) ( x - (1 + sqrt 2)) (x - 2) ( x -7)
Taking this by parts
(x - (1 - sqrt 2)) ( x -(1 + sqrt 2) =
x^2 - (1-sqrt 2)x - (1 + sqrt 2)x + ( 1-sqrt 2)(1 + sqrt 2) =
x^2 -2x + 1 - 2 =
x^2 -2x - 1
And ( x -2) (x -7) = x^2 - 9x + 14
So
x^2 - 2x -1
x^2 -9x + 14
___________ =
x^4 -9x^3 + 14x^2
-2x^3 + 18x^2 - 28x
- x^2 + 9x - 14 =
x^4 -11x^3 + 31x^2 - 19x - 14
