Editing Finite difference (section)

===Application===
This identity can be used to find the lowest-degree polynomial that intercepts a number of points {{math|(''x'', ''y'')}} where the difference on the ''x''-axis from one point to the next is a constant {{math|''h'' &ne; 0}}. For example, given the following points:

{| class="wikitable"
|-
! ''x'' !! ''y''
|-
| 1|| 4
|-
| 4|| 109
|-
| 7|| 772
|-
| 10|| 2641
|-
| 13|| 6364
|}

We can use a differences table, where for all cells to the right of the first {{math|''y''}}, the following relation to the cells in the column immediately to the left exists for a cell {{math|(''a'' + 1, ''b'' + 1)}}, with the top-leftmost cell being at coordinate {{math|(0, 0)}}:
<math display="block">(a+1, b+1) = (a, b+1) - (a, b)</math>

To find the first term, the following table can be used:

{| class="wikitable"
|-
! {{math|''x''}} !! {{math|''y''}} !! {{math|&Delta;''y''}} !! {{math|&Delta;<sup>2</sup>''y''}} !! {{math|&Delta;<sup>3</sup>''y''}}
|-
! 1
| 4
|-
! 4
| 109|| 105
|-
! 7
| 772|| 663|| 558
|-
! 10
| 2641|| 1869|| 1206|| 648
|-
! 13
| 6364|| 3723|| 1854|| 648
|}

This arrives at a constant {{math|648}}. The arithmetic difference is {{math|1=''h'' = 3}}, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree {{math|3}}. Thus, using the identity above:
<math display="block">648 = a \cdot 3^3 \cdot 3! = a \cdot 27 \cdot 6 = a \cdot 162</math>

Solving for {{math|''a''}}, it can be found to have the value {{math|4}}. Thus, the first term of the polynomial is {{math|4''x''<sup>3</sup>}}.

Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again:

{| class="wikitable"
|-
! {{mvar|x}} !! {{mvar|y}} !! {{math|&Delta;''y''}} !! {{math|&Delta;<sup>2</sup>''y''}}
|-
! 1
| {{math|4 − 4(1)<sup>3</sup> {{=}} 4 − 4 {{=}} 0}}
|-
! 4
| {{math|109 − 4(4)<sup>3</sup> {{=}} 109 − 256 {{=}} −147}}|| −147
|-
! 7
| {{math|772 − 4(7)<sup>3</sup> {{=}} 772 − 1372 {{=}} −600}}|| −453|| −306
|-
! 10
| {{math|2641 − 4(10)<sup>3</sup> {{=}} 2641 − 4000 {{=}} −1359}}|| −759|| −306
|-
! 13
| {{math|6364 − 4(13)<sup>3</sup> {{=}} 6364 − 8788 {{=}} −2424}}|| −1065|| −306
|}

Here, the constant is achieved after only two pairwise differences, thus the following result:
<math display="block">-306 = a \cdot 3^2 \cdot 2! = a \cdot 18</math>

Solving for {{math|''a''}}, which is {{math|−17}}, the polynomial's second term is {{math|−17''x''<sup>2</sup>}}.

Moving on to the next term, by subtracting out the second term:

{| class="wikitable"
|-
! {{math|''x''}} !! {{math|''y''}} !! {{math|&Delta;''y''}}
|-
! 1
| {{math|0 − (−17(1)<sup>2</sup>) {{=}} 0 + 17 {{=}} 17}}
|-
! 4
| {{math|−147 − (−17(4)<sup>2</sup>) {{=}} −147 + 272 {{=}} 125}}|| 108
|-
! 7
| {{math|−600 − (−17(7)<sup>2</sup>) {{=}} −600 + 833 {{=}} 233 }}|| 108
|-
! 10
| {{math|−1359 − (−17(10)<sup>2</sup>) {{=}} −1359 + 1700 {{=}} 341 }}|| 108
|-
! 13
| {{math|−2424 − (−17(13)<sup>2</sup>) {{=}} −2424 + 2873 {{=}} 449 }}|| 108
|}

Thus the constant is achieved after only one pairwise difference:
<math display="block">108 = a \cdot 3^1 \cdot 1! = a \cdot 3</math>

It can be found that {{math|1=''a'' = 36}} and thus the third term of the polynomial is {{math|'''36''x'''''}}. Subtracting out the third term:

{| class="wikitable"
|-
! {{math|''x''}} !! {{math|''y''}}
|-
! 1
| {{math|17 − 36(1) {{=}} 17 − 36 {{=}} −19}}
|-
! 4
| {{math|125 − 36(4) {{=}} 125 − 144 {{=}} −19}}
|-
! 7
| {{math|233 − 36(7) {{=}} 233 − 252 {{=}} −19}} 
|-
! 10
| {{math|341 − 36(10) {{=}} 341 − 360 {{=}} −19}}
|-
! 13
| {{math|449 − 36(13) {{=}} 449 − 468 {{=}} −19}}
|}

Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant {{math|−19}}. Thus, the lowest-degree polynomial intercepting all the points in the first table is found:
<math display="block">4x^3 - 17x^2 + 36x - 19</math>