The tutor gives brief coverage of a curiosity with polynomials.

Imagine expanding the product

(1-x)(1+x+x^2+x^3)

by multiplying, in turn, both terms from the first bracket by each one in the second:

(1-x)(1+x+x^2+x^3)=1-x+x-x^2+x^2-x^3+x^3-x^4

Note, in the expanded expression, the cancellation between the second and third terms, then between the fourth and fifth, and so on. The result is

(1-x)(1+x+x^2+x^3)=1-x^4

The cancellation between consecutive middle terms of the expansion makes it a telescoping sum. I’ll be talking about implications of telescoping sums in future posts.:)

Source:

Grimaldi, Ralph P. Discrete and Combinatorial Mathematics. Addison-Wesley: Toronto,
   1994.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.