Hi everyone!
Today I'd like to write yet another blog about polynomials. It's quite well-known that the system
has a unique solution $$$P(x)$$$ among polynomials of degree at most $$$n$$$. One of direct ways to prove that such a polynomial exists is through Lagrange's interpolation. To have a better grasp of it, let's recall that $$$P(x) \equiv P(x_0) \pmod{x-x_0}$$$, thus the given system of equations can be perceived as
It is known from the Chinese remainder theorem that $$$P(x)$$$ is unique modulo $$$Q(x) = (x-x_0)\dots(x-x_n)$$$ and can be explicitly computed as
where $$$Q_i(x) = \frac{Q(x)}{x-x_i}$$$. Noteworthy, $$$Q_i(x_i) = Q'(x_i)$$$, as $$$Q'(x) = Q_0(x) + \dots + Q_n(x)$$$.
