# Deriving Newton's divided differences from sub-interval interpolation

I am reading about Newton's divided differences and I am confused by the following derivation of the coefficients of the Newton's polynomial.

The Newton's polynomial is given as

$$P_n(x) = a_0 + a_1(x - x_0) + a_2(x-x_0)(x-x_1) + \dots + a_n(x-x_0)...(x-x_{n-1})$$ (1)

A polynomial of order $$n$$ can be constructed from two polynomials of order $$n-1$$ by dividing the support in two parts like this

$$P_{k,k+1,\dots,k+j}(x) = \dfrac{(x- x_k)P_{k+1,\dots,k+j}(x)-(x- x_{k+j})P_{k,\dots,k+j-1}(x)}{x_{k+j} - x_k}$$ (2)

This I understand.

The next statement is confusing me.

Because a polynomial of order $$n$$ is unique for a specific support, an immediate consequence of this is that the coefficients $$a_i$$ in (1) are Newton's divided differences, because $$a_j$$ is the highest coefficient of a polynomial $$P^*_{0,\dots,j}(x)$$ for the $$(j+1)$$ support points $$((x_0, y_0), \dots (x_j, y_j))$$.

This "immediate consequence" is not at all immediate to me. :)

I understand that the divided differences can be derived in different ways for the n-th coefficient, there are answers already available for this question, I don't see how the interpolation with sub-polynomials immediately leads to $$a_n = [y_0, \dots, y_n]$$.