## 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

and

Sinceandare complex.

and

Obviouslyand

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.

(3) and(4)

Taking the complex conjugate of the first one gives(5)

(5) – (4) gives

Taking the complex conjugate gives

Adding gives(6)

Divide byand add to the complex conjugate. Do this repeatedly until you end up with

Repeat this process for decreasing powers ofto obtainfor all