## 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 form There is a very important theory which makes factorising much easier for many equations.

If with each real then if is a complex number that is a root of the above equation then the complex conjugate is also a root.

Proof

If is a root of then Taking complex conjugates of both sides gives is a root.

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

Example: has a factor Use this to factorise the cubic expression.

The coefficients of the cubic are real, so since is a root, so is hence and are factors. Hence is a factor.

Inspection of gives so the other factor is and the cubic expression factorises: Example: The quintic polynomial has a factor Use this to factorise the expression.

The coefficients of the cubic are real, so since is a root, so is hence and are factors. Hence is a factor and the quintic factorises into two quadratics.

Inspection of gives so the other quadratic expression is which has the roots and The full factorisation is  