Roots of Polynomial Equations with Real Coeffcients

If a polynomial equationhas real coefficients, and ifis a root, so thatthen the complex conjugate ofis also a root so that

Proof:

Ifis a root so thatthen taking the complex conjugate of both sides givessincefor henceis also a root.

This means that given a complex rootwe can write down two factorsandmultiply them together to getand this polynomial will have real coefficients. We can perform long division of the original polynomial by this to obtain a polynomial with degree two less than the original polynomial. We can do this repeatedly if we know several complex roots.

Example:has a rootSince the coefficients ofare real,is also a root. Thenis a factor the other factoris found by long division to give the factorisation :

Thenis a root and

The roots will be symmetrically distributed about the real axis when plotted on an Argand diagram.