.
Let p(x) be a cubic polynomial. If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then compute p(4).
~~~~~~~~~~~~~~~~~~~~~~~~
It can be solved/worded/presented differently, in much shorter way.
Consider the polynomial p(x) - x.
It is easy to check that the polynomial p(x)-x takes the value of 0 at the points x= 0, 1, 2, 3.
In other words, the polynomial p(x)-x has 4 (four) different roots.
Since the polynomial P(x)-x has the degree not greater than 3, from Algebra,
it implies that the polynomial is identically equal to zero.
So, we proved that p(x) - x = 0.
It implies p(x) = x.
Then p(4) = 4.
ANSWER. We proved that under given condition, the polynomial p(x) is equal to x: p(x) = x, and p(4) = 4.
Solved.
--------------------------------
Strictly speaking, the problem formulation, as it is presented in the post, is INCORRECT.
A correct formulation should be in THIS form:
Polynomial p(x) has a degree less than 4.
If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then compute p(4).
As it is given in the post, it demonstrates
to all people around that the problem's creator does not know basics of Algebra
and can not express his thoughts in mathematically correct form.