In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers
D
\sqrt {D}
is the solution to Pell’s equation for
D
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
∈
Z
(u_n,v_n)_{n\in \mathbb {Z}}
. 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
)
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
)
v_n(t)
has a repeated root. Restricting our analysis to
Q
[
t
]
\mathbb {Q}[t]
, we give an upper bound on the number of “new” factors of
v
n
(
t
)
v_n(t)
of degree at most
N
N
. Furthermore, we show that all “new” linear rational factors of
v
n
(
t
)
v_n(t)
can be found when
n
≤
3
n \leq 3
, and all “new” quadratic rational factors when
n
≤
6
n \leq 6
.