Motion polynomials (polynomials over the dual quaternions with the non-zero real norm) describe rational motions. We will present a necessary and sufficient condition for reduced bounded monic motion polynomials to admit factorizations into linear factors, and we give algorithms 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. In addition, bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial.