Could you EASILY decompose (a-b)^3 + (b-c)^ + (c-a)^3 into the product of linear factors?

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Aslanbek Naziev

Note that X^3 + Y^3 = (X+Y)^3 - 3XY(X+Y). Apply this identity, and the answer will follow like magic!

I'm sorry that the problem with such beautiful solution did not attract attention.

Solution.

(a-b)^3 + (b-c)^ + (c-a)^3 = ((a-b)+(b-c))^3 - 3(a-b)(b-c)((a-b)+(b-c)) + (c-a)^3 = (a-c)^3 - 3(a-b)(b-c)(a-c) - (a-c)^3 = 3(a-b)(b-c)(c-a)!