The derivative of a degree $5$ polynomial $p\in\mathbb{C}[z]$ is a degree four polynomial $p'\in\mathbb{C}[z]$, and as such, the zeros of $p'$ may be found explicitly using the quartic formulae.

One may think of a finite Blaschke product as playing the same role on the unit disk that a polynomial plays on the plane (in some ways!). In particular, a degree $5$ finite Blaschke product has $5$ zeros and $4$ critical points in the disk. However in general, a finite Blaschke product $B$ will have additional critical points outside the unit disk.

Suppose $B$ is a finite Blaschke product of degree $n=5$. Let $w_1,w_2,w_3,w_4\in\mathbb{D}$ be the critical points of $B$ which lie in the disk. Assume that no $w_i=0$. Due to the conjugate symmetry of $B$, the critical points of $B$ outside the unit disk are exactly $1/\bar{w_1},1/\bar{w_2},1/\bar{w_3},1/\bar{w_4}\in\mathbb{D}\setminus\overline{\mathbb{C}}$. Thus if $B'=P/Q$ for relatively prime polynomials $P$ and $Q$, then $\deg(P)=8$.

**Can the known conjugate symmetry of the degree $8$ polynomial $P$ across the unit circle allow us to find its zeros using the quartic formulae?**

NOTE 1: The above is a narrowing of this question on MSE, wherein I also asked about whether a degree $8$ polynomial $p(z)$ whose set of zeros is known to consist of four conjugate pairs could be factored somehow using the quartic formula.

NOTE 2: The same question above goes for degree $4$ finite Blaschke products. Of course for a degree $3$ or less Blaschke product, there are at most $4$ critical points anyway, so one may just use the quartic formula straightforwardly.