# Thread: Complex roots in conjugate pairs

1. ## Complex roots in conjugate pairs

Given that $\displaystyle p+iq$, where p and q are real and $\displaystyle q\neq 0$, is a root of the equation
$\displaystyle a_0 z^n+a_1 z^{n-1}+...+a_n$
where $\displaystyle a_0, a_1, ..., a_n$ are all real, prove that $\displaystyle p-iq$ is also a root.

This is the first part of the question, its the harder part. Second part is about finding constants from roots, two of which are complex.
How do I start? I know its true, but how do I prove it?
Thanks

2. Originally Posted by arze
Given that $\displaystyle p+iq$, where p and q are real and $\displaystyle q\neq 0$, is a root of the equation
$\displaystyle a_0 z^n+a_1 z^{n-1}+...+a_n$
where $\displaystyle a_0, a_1, ..., a_n$ are all real, prove that $\displaystyle p-iq$ is also a root.

This is the first part of the question, its the harder part. Second part is about finding constants from roots, two of which are complex.
How do I start? I know its true, but how do I prove it?
Thanks
http://en.wikipedia.org/wiki/Complex...e_root_theorem