yet another important algebric identity

Several blogs ago,  I had suggested the use of a powerful fundamental algebraic identity:

a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca).

Now, this identity becomes a basic applied tool in Algebra and Trigonometry and Number theory, perhaps.

A salient feature of mathematics is that mathematicians always try to generalize a technique or concept.

Can you generalize the above identity as follows (a proof is required please):

Let there be quantities a,b,c,d…then prove that

a^{3}+b^{3}+c^{3}+d^{3}+\ldots -3(abc+bcd+abd+\ldots) is divisible by (a+b+c+d+ \ldots) and also  find the quotient.

Please do send your questions, comments, suggestions …

More later…

Nalin Pithwa

 

 

 

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