mixing forward, backward, and central differences. Polynomials For a given polynomial of degree , expressed in the function , with real numbers and and lower order terms (if any) marked as : After pairwise differences, the following result can be achieved, where is a real number marking the arithmetic difference: Only the coefficient of the highest-order term remains. As this result is constant with respect to , any further pairwise differences will have the value . Inductive proof Base case Let be a polynomial of degree : This proves it for the base case. Step case Let be a polynomial of degree where and the coefficient of the highest-order term be . Assuming the following holds true for all polynomials of degree : Let be a polynomial of degree . With one pairwise difference: As , this results in a polynomial of degree , with as the coefficient of the highest-order term. Given the assumption above and pairwise differences (resulting in a total of pairwise differences for ), it can be found that: This completes the proof. Application This identity can be used to find the lowest-degree polynomial that intercepts a number of points where the difference on the x-axis from one point to the next is a constant . For example, given the following points: We can use a differences table, where all cells to the right of the first , the following relation to the cells in the column immediately to the left exists for a cell , with the top-leftmost cell being at coordinate : To find the first term, the following table can be used: This arrives at a constant . The arithmetic difference is , as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree . Thus, using the identity above: Solving for , it can be found to have the value . Thus, the first term of the polynomial is . Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again: Here, the constant is achieved after only 2 pairwise differences, thus the following result: Solving for , which is , the polynomial's second term is . Moving on to the next term, by subtracting out the second term: Thus the constant is achieved after only 1 pairwise difference: It can be found