  In the classical theory, a famous by-product of the continued fraction
expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell's
equation for $D$. It is well-known that, once an integer solution to Pell's
equation exists, we can use it to generate all other solutions
$(u_n,v_n)_{n\in\Zee}$. Our object of interest is the polynomial version of
Pell's equation, where the integers are replaced by polynomials with complex
coefficients. We then investigate the factors of $v_n(t)$. In particular, we
show that over the complex polynomials, there are only finitely many values of
$n$ for which $v_n(t)$ has a repeated root. Restricting our analysis to
$\Qee[t]$, we give an upper bound on the number of "new" factors of $v_n(t)$ of
degree at most $N$. Furthermore, we show that all "new" linear rational factors
of $v_n(t)$ can be found when $n\leq 3$, and all "new" quadratic rational
factors when $n\leq 6$.
