Discussion
The value of
\[ \Delta^{10}[(1 - x)(1 - 2x^{2})(1 - 3x^{3})(1 - 4x^{4})] \]
is :
Answer: A
The value is 24 × 10!.
The product is a polynomial of degree \(1 + 2 + 3 + 4 = 10\). Its leading term comes from multiplying the leading terms:
\[ (-x)(-2x^{2})(-3x^{3})(-4x^{4}) = 24x^{10} \]
The tenth forward difference of a polynomial of degree 10 is constant and depends only on the leading term: with unit step, \(\Delta^{10}x^{10} = 10!\). The lower-degree terms vanish under \(\Delta^{10}\). So
\[ \Delta^{10}[\ldots] = 24 \times 10! \]
12 would come from a sign or product slip in the leading coefficient, and 8! from the wrong degree.
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