Roots and Complex Conjugates
If a polynomial has real coefficients, and some of the roots are conplex, then the complex conjugate of each root is also a root.
To illustrate this for quadratis, notice that the roots ofare
This observation can be generalised to polynomials of any order with real coefficients. If is a root of a polynomial then so is
Proof: Let(1) withreal.
Letbe a root, then(1)
Taking the complex conjugate gives(2) (ince all the coefficients are real).
Obviouslyis then also a root.
Conversely supposeandare distinct roots.
Taking the complex conjugate of the first one gives(5)
(5) – (4) gives
Taking the complex conjugate gives
Divide byand add to the complex conjugate. Do this repeatedly until you end up with
Repeat this process for decreasing powers ofto obtainfor all