Exercise: Write $x^5-1$ as product of real polynomials of rate (max) 2.
I would start as $(x-1)(x^4+ x^3+x^2+x+1)$, but then I have no idea how to continue. Till now I had only even terms in parenthesis and I managed. Now I cannot get the solution. Please help, because I really have no idea how to continue. Is there maybe an algorithm for solving this?
The roots of $x^5-1$ are $1$, $e^{\pm\frac{2\pi i}5}$, and $e^{\pm\frac{4\pi i}5}$. Therefore,\begin{align}x^5-1&=(x-1)\left(x-e^{\frac{2\pi i}5}\right)\left(x-e^{\frac{-2\pi i}5}\right)\left(x-e^{\frac{4\pi i}5}\right)\left(x-e^{\frac{-4\pi i}5}\right)\\&=(x-1)\left(x^2-2\cos\left(\frac{2\pi}5\right)x+1\right)\left(x^2-2\cos\left(\frac{4\pi}5\right)x+1\right).\end{align}