  We present an algorithm to compute all factorizations into linear factors of
univariate polynomials over the split quaternions, provided such a
factorization exists. Failure of the algorithm is equivalent to
non-factorizability for which we present also geometric interpretations in
terms of rulings on the quadric of non-invertible split quaternions. However,
suitable real polynomial multiples of split quaternion polynomials can still be
factorized and we describe how to find these real polynomials. Split quaternion
polynomials describe rational motions in the hyperbolic plane. Factorization
with linear factors corresponds to the decomposition of the rational motion
into hyperbolic rotations. Since multiplication with a real polynomial does not
change the motion, this decomposition is always possible. Some of our ideas can
be transferred to the factorization theory of motion polynomials. These are
polynomials over the dual quaternions with real norm polynomial and they
describe rational motions in Euclidean kinematics. We transfer techniques
developed for split quaternions to compute new factorizations of certain dual
quaternion polynomials.
