If g is a linear combination of powers of x, i.e. a polynomial of degree n, it is spanned by the first n+1 powers of x, and by the previous theorem, there are n+1 points such that g(x), evaluated at these n+1 points, completely defines the polynomial. We don't know what these n+1 points are - well - actually we do. Any n+1 points will work.
If the points are 1 through 5, for a quartic polynomial, build a term for each of these 5 points. The term associated with x=1 is:
g(1) × (x-2) × (x-3) × (x-4) × (x-5) over (1-2) × (1-3) × (1-4) × (1-5)
This term = g(1) when x = 1, and is 0 for x = {2,3,4,5}. build such a term for the other 4 points, then add all five terms together. The resulting quartic polynomial, call it h, agrees with g on all 5 points.
If h-g is nonzero somewhere, then we have a quartic polynomial with 5 roots. This is too many roots. Hence we have written g as a linear combination of powers of x; the polynomial is established.
Given any function f(x), we can approximate f with an nth degree polynomial g, such that g and f agree on any set of n+1 points. We usually select local minima and maxima, and points of inflection, so that g follows the path of f.