  In this paper, we study the minimal free resolution of non-ACM divisors $X$
of a smooth rational normal surface scroll $S=S(a_1 ,a_2 ) \subset
\mathbb{P}^r$. Our main result shows that for $a_2 \geq 2a_1 -1$, there exists
a nice decomposition of the Betti table of $X$ as a sum of much simpler Betti
tables. As a by-product of our results, we obtain a complete description of the
graded Betti numbers of $X$ for the cases where $S=S(1,r-2)$ for some $r \geq
3$ and $S=S(2,r-3)$ for some $r \geq 6$.
