Factoring polynomials

Ivan Cnop
Vrije universiteit Brussel
icnop@vub.ac.be

Factoring polynomials over integers

Is fast and reliable

Factor[
      4 (x^2) - 12 x y  +  25 (z^2) - 30 y z  +  20 x z  +  
        9 (y^2)]

Factor[(x^4) + 4]

and allows experimenting

Table[Length[Factor[(x^n) - 1]], {n, 100}]

Factoring polynomials over other number systems

Problem

Sometimes does not yield the desired answer in Mathematica V4.

Factor[(x^4) + 1]

This is because the computer cannot guess the number system in which to perform

Solution

We can work around this problem

Factor[((((Sqrt[2]) x))^4) + 1] /. x -> x/(Sqrt[2])

Factor[(((a x))^4) + 1] /. x -> x/a

Expand[%]

Partial fractions

Are not fully expanded if the result of the integral is known

Apart[1/((x^4) + 1)]

Apart[1/((x^4) + 4)]