Root Theorem for Polynomials of Degree > 2

[hedlund]

What is the theorem that states if \Omega is a polynom with degree > 1 with real coefficients. If there exists a complex number z = a + bi such that \Omega(a+bi)=0 then \overline{z} = a - bi is also a root of \Omega? For \Omega(x) = x^2 + px + q with p and q real then if a+bi is a root then a-bi is also a root if b \neq 0, that one is easy but I don't think it's easy for degree > 2 to prove it that's why I'm search for it's name.

 

[matt grime]

it doesn't have a name, as far as i know, and it is easy to prove. if z is a root of P, then z* is a root of P*, where * denotes conjugation, and by P*, I mean the polynomial where you replace the coeffs with their conjugates. (You understand that (uv)*=u*v*?)

 

[TenaliRaman]

It does get mentioned along with FTA but i wouldn't bet on it having some special name.

-- AI