Discussion
Let f(x) be a polynomial of degree n. Then the nth forward difference of f(x), that is, \(\Delta^{n}[f(x)]\) is :
Answer: B
The nth forward difference is constant.
Each application of \(\Delta\) lowers the degree of a polynomial by one: if \(f(x) = a x^{n} + \ldots\), then \(\Delta f(x) = f(x + h) - f(x) = a n h\, x^{n-1} + \ldots\). After n applications the result is the constant
\[ \Delta^{n} f(x) = a\, n!\, h^{n} \]
It is not always 0 (that is \(\Delta^{n+1}\)), nor always 1 (it depends on the leading coefficient a and the step h), and it can certainly be determined.
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