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One must be careful to understand the exact question being asked, which is not made easy by the book, which is not very clear, despite the author's claim that it is easy.
The key question is "polynomial in what variable?". I have to assume that the intention is to write it as a polynomial in terms of [latex]x_r [/latex]. If that is the case, we can take your example and make it clear.
[latex] \Delta y_3=x_4^3-x_3^3=(x_3+\Delta x)^3-x_3^3 [/latex]
Hopefully, now it is clear because if one expands the [latex](x_3+\Delta x)^3 [/latex] using the Binomial theorem, it's obvious that the highest order term will subtract out and only powers of n-1 are left.