Recursive Algorithm for Polynomial Division with Remainder

This paper presents a novel recursive algorithm for polynomial division over arbitrary fields [Formula: see text], generalizing Horner’s method to arbitrary-degree polynomials. Unlike classical approaches, our method employs backward recurrence relations and table-based computations that operate solely on coefficients, thereby eliminating the need for direct variable power manipulation. This coefficient-focused approach significantly simplifies the division process while maintaining the mathematical rigor. We establish theoretical foundations by proving the correctness and uniqueness of the algorithm through inductive arguments. This recursive framework enables an efficient implementation in both symbolic and numerical computing environments. The algorithm demonstrates a particular strength in finite field arithmetic ([Formula: see text] and [Formula: see text]), where its structured approach aligns perfectly with modular computation requirements. Comparative analysis shows that our method offers substantial pedagogical advantages through its transparent tabular representation, while maintaining computational efficiency comparable to that of conventional approaches. The algorithm’s clarity in coefficient manipulation makes it valuable for cryptographic applications, computer algebra systems, and educational contexts where understanding the internal structure of polynomial operations is essential. This study bridges the theoretical mathematics using practical computations. This provide an efficient computational tool. This method represents a significant advancement in terms of the polynomial division. It has applications across the computational domains.