## Factorising Polynomials with Complex Roots

The Fundamental Theorem of Algebra states that a polynomial of degree n has n (not necessarily distinct) roots. This means that a polynomial of degree n may be factorised into n linear factors, each factor being of the formThere is a very important theory which makes factorising much easier for many equations.

Ifwith eachreal then ifis a complex number that is a root of the above equation then the complex conjugateis also a root.

Proof

Ifis a root ofthen

Taking complex conjugates of both sides gives

is a root.

This means thatandare both factors hence so isThis expression will have real coefficients we can possibly find expressions of this sort one by one and perform long division ofby these in turn or use some other method to factorise out the quadratics hence factorisinginto quadratics then linear factors.

Example:has a factorUse this to factorise the cubic expression.

The coefficients of the cubic are real, so sinceis a root, so ishenceandare factors. Henceis a factor.

Inspection ofgivesso the other factor isand the cubic expression factorises:

Example: The quintic polynomialhas a factorUse this to factorise the expression.

The coefficients of the cubic are real, so sinceis a root, so ishenceandare factors. Henceis a factor and the quintic factorises into two quadratics.

Inspection ofgivesso the other quadratic expression iswhich has the rootsand

The full factorisation is