I tried to find an answer for some time, the answer is simple probably ,say I have a function given by set of points and their values like ${(4,1),(5,0),(6,0)}$, how do I calculate Lagrange interpolation polynomial for them? We know that $$w(x) = 1*l_0(x) + 0*l_1(x)+0*l_2(x)$$ But I think we are obviously losing information here calculating only for the first term the $w(x)$. I can't find a clue.
How to proceed?
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, $(4,1),(6,0),(7,0)$.
