Lagrange interpolation at 0's
Solution 1:
If you have points like $(4,1),(5,0),(6,0)$, it is true that only one of the Lagrange polynomials for those points will actually contribute to the final interpolating polynomial. However, all of the Lagrange polynomials in the basis "know" where the other nodes are located. For example here the Lagrange polynomial that is involved is $\frac{(x-5)(x-6)}{2}$, so it "knows" where those other zeros were, in the sense that you'd get a different result out of, say, $(4,1),(6,0),(7,0)$.