We present a complete derivation of coprime factorizations for a class of multi-dimensional systems containing linear parameter-varying and uncertain systems. A generalization of coprime factors model reduction using a balanced truncation approach is then given, with error bounds on the factorized mapping in the l2-induced norm. The proposed reduction method is thus applicable to linear parameter-varying and uncertain systems that do not satisfy the usual l2-induced stability constraint required by the standard non-factored truncation methods.