I have the polynomial x^8+1, I know that there's no root for solve this in RR[x] but i want to factorize this to the minimal expression. This is possible or this is irreducible?
I have the polynomial , I know that there's no root for solve this in but i want to factorize this to the minimal expression. This is possible or this is irreducible?
Answer & Explanation
This can be derived by setting equal to a product of two monic quadratics with unknown coefficients and then solving for said coefficients little by little. As a consequence,
We can go further. For example, set
Rule out to obtain from the second equation, then plug those candidates into the first equation yielding with b=1. The second factor of that is written above is similarly reducible. I leave the details as an exercise. The full factorization is
over the real numbers R. With the quadratic formula applied to the above you can get the roots to exactly (they are precisely the primitive 16th roots of unity) in the form of nested radicals, and hence the full factorization in C.
By the way, I should mention that the only nonlinear polynomials over R that are irreducible are quadratics with negative discriminant. This is because the nonreal roots of any real-coefficient polynomial can be paired off into conjugates, and then each conjugate pair of linear factors can be put together to obtain a real quadratic, thus every real-coefficient polynomial can be factored into real linear and quadratic factors.