You mean decomposition of fractions?Originally Posted by guess
Such as,
I think you are referring to your integration thread when was factorized to by TD.
you can do this since if you have
it can be expressed in the form if you expand the brackets you will see that it is indeed
now how would you think to do this?
well it comes from the difference of two squares where if you have this is
if we think of our x in the difference of two squares as (a+b) then (a+b)^2 is going to come out in the result somewhere, and (a+b)^2 = a^2 + b^2 + 2ab
now this has a^2+b^2 in it but it is 2ab too much.
however if we choose the right y we can get the 2ab to disappear. for instance
this expands to
so if we choose such that:
then:
so
we get:
so from the original
if we say a = 1 and b = x^2 it factorizes to:
it is worth noting however that
is difficult to factorize because it is in the form
factorizing things in the form for all odd k
so where p is a prime number not equal to 2 is factorizable into the form above. as the rest of the prime numbers are odd.
so for even k, if k = rp where p is a prime factor other than 2 then:
and since p is odd, this factorizes also, to:
so this only leaves things in the form
which don't factorize nicely.
does not factorize
if
then
so does not factorize either.
this is useful in searching for fermat primes, so we only know to look for primes in the form since all other indices factorize.