A secret sharing scheme is a method of distributing a secret among a set of participants in a such way such that certain subset of participants are qualified to compute the secret by combining their shares. Shamir [1] used a polynomial f (x)=∑ t-1 i= 0 aixi over finite field GF (q) to construct his scheme. In this paper we propose to construct a secret sharing scheme by using a more general polynomial than Chun [4] and Shamir used. Our proposed polynomial is of the form f (x)=∑ t-1 i= 0 ai (xi+ xqi) over a finite field GF (q). We use an algebraic approach to reconstruct the secret. Some problems related to solving the system of linear equations generated by pulling the shares are investigated.