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Factorization of a^3 + b^3 + c^3  3abc
Posted:
Apr 23, 2009 1:03 PM


I've seen the identity
a^3 + b^3 + c^3  3abc
= (a + b + c)(a^2 + b^2 + c^2  ab  bc  ac)
applied in many ways (solving cubic equations, obtaining the cubic arithmetic and geometric mean inequality, showing that the collection of nonzero rational linear combinations of {1, 2^(1/3), 2^(2/3)} is closed under division, etc.), and it also makes frequent appearances in advanced algebra texts from the 1800s to the early 1900s.
Does anyone know of some early appearances and uses of this identity? By "early", I mean before the early 1800s. Also, has this identity, or something very closely connected to it, been given a name at some time in the literature (regardless of whether the name was widely known at the time or whether the name survived to the present time)?
Dave L. Renfro



