This paper presents a comprehensive extension of the differential and algebraic framework for solving polynomial equations from Clifford algebras to the more general setting of non-commutative C* algebras. We systematically address the fundamental challenges posed by non-commutativity and infinite dimensional structure through the rigorous introduction of a path-ordering operator P that projects onto symmetric subspaces while preserving essential C*-algebraic structure. We establish a complete theoretical foundation for solving systems of multivariate polynomial equations defined within a non-commutative C*-algebra A by constructing a non-commutative C*-algebraic closure VncA.We provide complete constructive proofs with full mathematical rigor, including detailed derivations of non-commutative Newton identities and convergence analysis in C*-algebraic settings. The algorithms developed achieve well-defined computational complexity with comprehensive error analysis. Numerical validation demonstrates precision across various test cases in finite-dimensional C*-algebras. The work reconciles with classical impossibility results while demonstrating that explicit analytic solutions exist in appropriately extended algebraic structures that incorporate differential and operator-theoretic operations within a non-commutative C*-algebraic framework.