My questions are about a polynomial with the following form: $$q(x) = 1 + \alpha p(x)p(-x),~ \alpha \in \mathbb{R}^+$$ where $p(x)$ is a polynomial with real coefficients and degree $n$: $$ p(x) = \sum_{i=0}^{n} c_i x^i,~ c_i\in \mathbb{R}$$
I would like to factorize $q(x)$ in the following form: $$ q(x) = \prod_{i=1}^{2n}(x - r_{q,i})$$
If the roots of $p(x)$ are known:
- Is there a simple relationship between $r_q$ and the roots of $p$, denoted $r_p$ ?
I observed $q(x)$ presents only even powers of $x$ and that its roots come in the form $\pm r_q$. (see Properties of roots of an even polynomial). In order to group the roots $r_q$ by type (complex conjugate pairs or reals), I'm wondering:
- Is there a way to know how much roots of each type there will be, depending of the degree $2 n$ of $q(x)$ ?
Thank you for your help!