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Topic: Factorization of a^3 + b^3 + c^3 - 3abc
Replies: 0

 Dave L. Renfro Posts: 4,792 Registered: 12/3/04
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