ok, there's a problem with the names you gave your subspaces: you already assumed V is the vector sapce. so the subspaces should be named, say, U, W, and Z.

as a vector space over let and let then but

let with a common root (in your problem ) then but and2. Suppose that p_0,...p_m are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).

the set has m + 1 elements. so it's linearly dependent, i.e. there exist not all zero such that

now multiply both sides of (1) by to get thus are linearly dependent. Q.E.D.