## 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