IB MATHEMATICS Topic : Complex Numbers – 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