The answer is p(x) = x. The simplest (while not most elegant) way to solve this task is to compute directly.

First, lead it to a common denominator (a-b)(b-c)(c-a), denote this denominator as D. We must consider a sign - at each summand: p(x) = [-a(x-b)(x-c)(b-c) - b(x-a)(x-c)(c-a) - c(x-a)(x-b)(a-b)]/[D]

We see that D didn`t depend on the x, let`s calculate the numerator: