There
is a very important theory which makes factorising much easier for
many equations.
A Level Maths Notes: FP1 – 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.
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
Ifis
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 ofby
these in turn or use some other method to factorise out the
quadratics hence factorisinginto
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
