As you noted it is the difference of two squares:
However these are the difference and sum respectively of two cubes:
only factorises over the complex numbers.
Consider the polynomial .
(a) Find all monic irreducible factors of over .
(b) Find all monic irreducible factors of over .
(c) Find all monic irreducible factors of over .
I'm really not sure what to do, I know that , so the the roots of p(x) are ±1 & (x+1) & (x-1) are two linear factors. Anyway the first question here asks for monic (degree 1) irreducible (so that it cannot be factorized as P(x)=a(x)b(x) where ) factors of p(x) over Q (rationals). Can anyone show me the method for solving one of them, so that I'll be able to try solving the rest of it on my own. Thanks.
I'm still a little confused...
(a) So all monic irreducible factors of p(x) over Q are:
(b) All monic irreducible factors of p(x) over R are:
(c) Now to find monic irreducible factors of p(x) over C are we use the quadratic formula for :
, , (x+1), (x-1)
are all the monic irreducible factors of p(x) over complex numbers.
Are my answers correct? Did I list all the factors correctly for each question?