Given f(x)=x^3-x+k, and x-2 is a factor of f (x), then what are the roots of f(x).
Using synthetic division
2 [ 1 0 -1 k ]
2 4 6
_________________
1 2 3 k + 6
k + 6 = 0
k = -6
So
2 [ 1 0 -1 -6 ]
2 4 6
____________________
1 2 3 0
The remaining polynomial is
x^2 + 2x + 3 = 0 complete the square
x^2 + 2x = -3
x^2 + 2x + 1 = -3 + 1
(x + 1)^2 = -2 take both roots
x + 1 = sqrt (-2) x + 1 = -sqrt (-2)
x = -1 + sqrt ( 2) i x = -1 - sqrt (2) i ⇒ the other two roots