Knowing that $x_1 = 1-i$ is one of the roots of $P(x) = x^3 - (1-i)x^2 + x + (-1+i)$, find all the roots of P(x).
I only ever solved such problems for polynomials with real coefficients - then I could apply the complex conjugate root theorem and I also could do long division after finding successive roots to simplify the main polynomial. I don't know how to approach this problem.
Can I do long division of $P(x)$ by $(x-(1-i))$ somehow?
I'm looking more for universal solutions for approaching this kind of problems rather than tricks like factoring the polynomial because it may not work for similar polynomials.
The answer is: $x_1 = 1-i$, $x_2 = i$, $x_3 = -i$.