a(n!) = Δ
a: leading coefficient n: degree of polynomial Δ: Finite Differences
Example
we have this table of values. lets find the leading coefficient from it.
| x | f(x) | F.D | S.D | T.D | Q.D |
|---|---|---|---|---|---|
| -2 | -54 | - | - | - | - |
| -1 | -8 | 46 | - | - | |
| 0 | 0 | 8 | -38 | - | - |
| 1 | 6 | 6 | -2 | 36 | - |
| 2 | 22 | 16 | 10 | 12 | -24 |
| 3 | 36 | 14 | -2 | -12 | -24 |
| 4 | 12 | -24 | -38 | -36 | -24 |
| It is fourth differences. | |||||
| Also, If the differences are negative, then the leading coefficient is negative. | |||||
| Why? because Factorial of the degrees are based off polynomial. |
So, use the formula again a(n!) = Δ a(4!) = -24 a(24) = -24 a = -1