For S contained in one test scope, let 1_T set precisely the incidences in T to one. The coefficient of z_S in the multilinearized factor equals sum_{T subset S} (-1)^(|S|-|T|) B_i(1_T,V), computed in B_(n+1,n+1). At each 1_T, monomials contribute exactly when their selected variables lie in T; alternating subset sums cancel every coefficient except S. This identity uses only addition, subtraction and multiplication, so it still holds over this ring with zero divisors: no inverse or division is needed. For S empty, the coefficient is simply B_i(0,V).