Need Assistance with proof of the existence of conjugate complex roots

• Nov 27th 2006, 12:07 PM
type_speed
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
• Nov 27th 2006, 12:59 PM
Plato
The way you have stated the question makes it false.
Consider $z^2 + \left( {2 - i} \right)z + 1$. There are two complex roots that are not conjugates.

The usual statement includes $A_n \in \Re .$ Now it is a theorem.
Here is an outline of a way to prove it.
If $w$ is a root of $f(z)$ then $Re(f(w)) = Re(f(\bar w))$ and $Im(f(w)) = - Im(f(\bar w)).$
Can you show that? If so it follows that ${\bar w}$ is also a root.
• Nov 27th 2006, 08:28 PM
type_speed