I guess you don't know what a tensor product is.
If you want tensor products you first need vector spaces.
The idea is that you create a new space, called the tensor space. This space fullfilled all vector space axioms (abelian group for the inner composition, outer operation for the scalar multiplication and the structure like a monoid (but now real monoid) and a structure which looks like distributive laws but like the monoid no real distributive laws. People often says that a vector space has distributive laws but this is not true.) So a tensor is only an element of a tensor space. If you have polynomials, you have to interpretate them as vectors in the spaces of polynomials.
If you take vectorspaces in the cartesian product (thats the reason why one say tensor to the -th) and map them into a new space with the map (like tensor).
1.) has to be multi linear, which means linear in each argument. Think about the determinant, which is an alternating multilinear form. Its the same thing
2.) is universal. This means, that you have one unity map to an aribratary space such you can build as the composition of this.