How many points do you need to determine the equation of a polynomial of degree n?
A polynomial of degree
You cannot force the value of more than one coefficient with one point.
#color(blue)(f(0)), f(1), f(2),...,f(n)#
Then write out a sequence of differences of that sequence:
#color(blue)((f(1) - f(0))), (f(2)-f(1)), ..., (f(n) - f(n-1))#
Then a sequence of differences of those differences:
Repeating until only one term is left.
Then the first terms of each of these sequences form the coefficients of a direct formula for
This is the unique polynomial of degree