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 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
Ifis a root of
then
Taking complex conjugates of both sides gives
is a root.
This means thatand
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 sinceis a root, so is
hence
and
are factors. Hence
is a factor.
Inspection ofgives
so the other factor is
and the cubic expression factorises:
Example: The quintic polynomialhas a factor
Use this to factorise the expression.
The coefficients of the cubic are real, so sinceis a root, so is
hence
and
are factors. Hence
is a factor and the quintic factorises into two quadratics.
Inspection ofgives
so the other quadratic expression is
which has the roots
and
The full factorisation is