  The Jacobi system on a full-line lattice is considered when it contains
additional weight factors. A factorization formula is derived expressing the
scattering from such a generalized Jacobi system in terms of the scattering
from its fragments. This is done by writing the transition matrix for the
generalized Jacobi system as an ordered matrix product of the transition
matrices corresponding to its fragments. The resulting factorization formula
resembles the factorization formula for the Schr\"odinger equation on the full
line.
