  Factorization -- a simple form of standardization -- is concerned with
reduction strategies, i.e. how a result is computed. We present a new technique
for proving factorization theorems for compound rewriting systems in a modular
way, which is inspired by the Hindley-Rosen technique for confluence.
Specifically, our technique is well adapted to deal with extensions of the
call-by-name and call-by-value lambda-calculi. The technique is first developed
abstractly. We isolate a sufficient condition (called linear swap) for lifting
factorization from components to the compound system, and which is compatible
with beta-reduction. We then closely analyze some common factorization schemas
for the lambda-calculus. Concretely, we apply our technique to diverse
extensions of the lambda-calculus, among which de' Liguoro and Piperno's
non-deterministic lambda-calculus and -- for call-by-value -- Carraro and
Guerrieri's shuffling calculus. For both calculi the literature contains
factorization theorems. In both cases, we give a new proof which is neat,
simpler than the original, and strikingly shorter.
