When looking at possible factors of a polynomial
it is necessary to look at the coefficients of the highest and lowest powers of
If a factor of the polynomialis
then
must be a factor of the coefficient of the highest power of
and
must be a factor of the lowest power ofFor example if
then possible factors are
We can however cut down the number of possible options by looking at the coefficients. The negative coefficient ofand the positive coefficient of
and the positive constant (the coefficient of
) dictate factors of the form
Only the last four factors are possible therefore.
Now we can instead use the fact that if
is a factor then
is a root so
If
is a factor then
If
is a factor then
If
is a factor then
If
is a factor then
We can try each of these in turn though of course it being easing to work with integers, first find
and
sois not a factor.
sois a factor.
sois a factor.
Sinceis a quadratic it only has two factors and
If p(x) is a quadratic or polynomial of higher degree, this method is probably the best method that can be used with pencil and paper. Suppose that
Possible factors are
Running though the possible roots gives eventually that
so
and
is a factor. Long division ofby
gives the quadratic
which can be easily factorised as
then