GCD of odd and even polynomial proof elaboration

Question

From my understanding of the Routh table, we are using Euclidean division of the odd and even parts of the characteristic equation to find the number of sign changes. When the whole row is zero, we see that the GCD is an even polynomial, which implies that the characteristic equation has opposite roots.

I am trying to understand the proof right here. From what I noticed, if the polynomial $p\left(z\right)$ has opposite roots, i.e. $\pm z_0$, then the greatest common divisor $d\left(z\right)$ of the even part $p_e$ and the odd part $p_o$ of the polynomial is a polynomial of all the opposite roots. In other words, $p\left(z\right) = d\left(z\right)p^*\left(z\right)$. The proof is to show that $p^*\left(z\right)$ does not have the opposite roots.

Here are my interpretations and questions:

1. In the first case where $p$ has no “opposite roots”, can someone give a breakdown of why the logic is ok? Why is it ok to say that $p$ has no “opposite roots” then continue with the opposite roots for $d$, $p_e$, and $p_o$ (in other words, what’s the logic)? Also, what makes one infer that $d\left(z\right) = z^k$ from this scenario?
2. Here, $d$ is broken up into $e$ and $f$, where $e = \prod\left(z^2-z_i^2\right)^{\alpha_i}$ and $f\left(z\right)$ is defined as $\prod\left(z + z_i\right)^{\beta_i}\prod\left(z - z_i\right)^{\gamma_i}$, where at most one of each pair $\left(\beta_i, \gamma_i\right)$ is nonzero. Here, $f\left(z\right)$ is the “mutually exclusive or” product such that it either contains $z_i$ or $-z_i$ (counting multiplicities), but not both. Am I right? (I have to comment, if the root is 0 with multiplicity, only the even number of zeros is taken, and either one 0 root or none is left)
3. Here, he says $d$ is the GCD of $e\cdot q_e = p_e$ and $e\cdot q_o = p_o$, which is $e$. Doesn’t this imply $d = e\cdot f = e \rightarrow f = 1$, which contradicts with the definition of $f$ being the “mutually exclusive or” product of the opposite roots?

Can someone elaborate/verify and/or give a better proof? Also, the link is the only thing I find when I google “GCD of odd and even part of polynomial”. There’s no other reliable scholarly source of the proof aside from the StackExchange question.

Addition: I know that Edward Routh’s A Treatise on the Stability of Motion, Particularly Steady State Motion page 33 and The Advanced Part of a Treatise on The Dynamics of A System of Rigid Bodies, Being Part II of A Treatise On The Whole Subject page 229 mention that the GCD is necessarily an even polynomial, while Felix R. Gantmacher’s The Theory of Matrices, Vol. 2 page 185 mentions that the “$p^*\left(z\right)$” has no opposite roots. Those three books are scholarly sources, but do not prove that the other factor has no opposite roots.