Discussion
The value of
\[ \Delta^{5} [(1 - x)(1 - x^{2})(1 + x^{2})] \]
is :
Answer: A
The value is \(5!\), taking the usual unit step.
Multiply out the product first:
\[ \begin{aligned} (1 - x)(1 - x^{2})(1 + x^{2}) &= (1 - x)(1 - x^{4}) \\ &= 1 - x - x^{4} + x^{5} \end{aligned} \]
This is a polynomial of degree 5 with leading coefficient 1. With step \(h = 1\), the fifth difference of \(x^{5}\) is \(5!\,h^{5} = 120\), and the fifth difference of every term of lower degree is 0. So the answer is \(5! = 120\).
0 would be right for \(\Delta^{6}\), and 1 and 5 confuse the leading coefficient or the degree with the value of the difference.
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