On p.28 (here) Reid proves that the curve $y^2=x(x-1)(x-\lambda)$ for $\lambda\neq 0,1$ has no rational paramaterisation. At one point (on p.29), he has the equation $$r^2=ap(p-q)(p-\lambda q)$$ where $p$ and $q$ are polynomials over a field $k$, and $a$, $\lambda$ are constants in the field. He then says that "by considering factorisation into primes, there exist nonzero constants $b,c,d\in k$ such that $$bp,c(p-q),d(p-\lambda q)$$ are all squares in $k[t]$.
Edit: $p$ and $q$ are known to be coprime.
I feel like this should be obvious, but I don't follow his argument. Why do we know that those factors are squares (up to units)?