The differentiable manifold with f - structure were studied by many authors, for example: K. Yano [7], Ishihara [8], Das [4] among others but thus far we do not know the geometry of manifolds which are endowed with special polynomial $F_{a(j){ imes}(j)$-structure satisfying $$prodlimits_{j=1}^{k};[F^2+a(j)F+lambda^2(j)I];=;0$$ However, special quadratic structure manifold have been defined and studied by Sinha and Sharma [8]. The purpose of this paper is to study the geometry of differentiable manifolds equipped with such structures and define special polynomial structures for all values of j = 1, 2,$ldots$,$K;in;N$, and obtain integrability conditions of the distributions $pi_m^j$ and ${pilimits^{sim}}_m^j$.