When looking at possible factors of a polynomialit is necessary to look at the coefficients of the highest and lowest powers of

If a factor of the polynomialisthenmust be a factor of the coefficient of the highest power ofandmust be a factor of the lowest power ofFor example ifthen 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 ofand the positive constant (the coefficient of) dictate factors of the formOnly the last four factors are possible therefore.

Now we can instead use the fact that ifis a factor thenis a root so

Ifis a factor then

Ifis a factor then

Ifis a factor then

Ifis a factor then

We can try each of these in turn though of course it being easing to work with integers, first findand

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 thatsoandis a factor. Long division ofbygives the quadraticwhich can be easily factorised asthen