Complex number polynomial, with no root given

[Jarfi]

Homework Statement

z^3 + (-5+2i)z^2 + (11-5i)z -10+2i =0 has a real root, find all the solutions to this equation.

The Attempt at a Solution

I have only solved imaginary number polynomials with a given root, but this has no given root, how do I find the real solution? that I can then use to factor it and find the rest of the solutions.

 

[SteamKing]

If you have no other techniques available, you can always assume a real solution and see if it satisfies the equation.

 

[Jarfi]

If you have no other techniques available, you can always assume a real solution and see if it satisfies the equation.

Oh nevermind, I found out how you do it, you just define a as a real number, put it in and get a real and imaginary part, both real and imaginary part are suppost to be equal to zero, so their mutual solution is the available real solution, this gave me the answer of two.

 

[Jarfi]

If you have no other techniques available, you can always assume a real solution and see if it satisfies the equation.

I already knew the 2 was an answer but my teacher would frown upon me if I'd assume a solution.