So lets suppose that we have the following
$x_1= 0, y_1=1$
and we know that all the above $x_i,y_i$ belong to $p_2(x)=x^2+1$ , and we want to add and extra $x_4=2$ where $y_4=-7$ and we want to find by using the Divided Differences method a new polynomial $p$ that includes $x_4$. My question is do i have to do all the calculations again from the beginning? Since it's only one extra $x_i$ i feel like i dont have to calculate all the divided differences again but i am not sure. Isn't there a smarter way to find the new polynomial?