Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.