# Thread: Need Assistance with proof of the existence of conjugate complex roots

1. ## Need Assistance with proof of the existence of conjugate complex roots

Prove:
given that f(x) = Anx^n + An-1X^n-1+ ....A1X + A0 , and An does not = 0
=> if f(x) has a root of the form (A+Bi),then it must have a root of the form
(A-Bi). (complex roots)

All I can figure out so far is to utilize the Conjugate Roots theorem but our professor gave us a hint that we should use the Remainder theorem? or the discriminate? I'm not quite sure but basically I need to Prove why the conjugate exists in the polynomial.

Regards, Chris

2. The way you have stated the question makes it false.
Consider $\displaystyle z^2 + \left( {2 - i} \right)z + 1$. There are two complex roots that are not conjugates.

The usual statement includes $\displaystyle A_n \in \Re .$ Now it is a theorem.
Here is an outline of a way to prove it.
If $\displaystyle w$ is a root of $\displaystyle f(z)$ then $\displaystyle Re(f(w)) = Re(f(\bar w))$ and $\displaystyle Im(f(w)) = - Im(f(\bar w)).$
Can you show that? If so it follows that $\displaystyle {\bar w}$ is also a root.

3. Appreciate the fast reply, But I'm not quite understanding your notations.
Re(f(w))...Im(f(w)), What does the Re, Im, W-bar mean?

Also, I'm assuming the coefficients are real too.

Need more of an step by step explanation over the computer.
Thanks for your time and effort.

Regards, Chris