Every polynomial interpolates itself -- meaning that you can often apply this interpolation procedure to your favorite/nemesis polynomial or equivalently rewrite your polynomial of interest in this Lagrange basis, and see if the coefficients lead you anywhere. This is especially helpful in proving polynomial inequalities. For instance, Chebyshev polynomials T_n enjoy an alternation property over their extremal points -- so in the Lagrange basis, in many problems (e.g. Markov type inequalities) they emerge as the extremal case in the triangle inequality.
My beef with this approach is that it is a little unsatisfying in the sense that it sort of feels like we "got lucky". That is, it highlights this special feature (alternation) while burying the interesting structure that leads to such polynomials being extremal in these problems, as can be seen if you attempt certain seemingly trivial extensions of classical inequalities -- but nevertheless it's an important trick in extremal polynomial theory and approximation more broadly