algebra

prove
(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3=3(a^2-b^2)(b^2-c^2)(c^2-a^2)

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  1. The following replacement may not always work, but it works in this case, and simplifies the problem.

    Try replacing a² by A, b² by B and c² by C in the given expression.

    Expand the left-hand side:

    (A-B)³ + (B-c)³ + (C-A)³
    =A³-3A²B+3AB²-B³+
    B³-3B²C+3BC²-C³+
    C³-3C²A+3CA²-A³
    =-3A²B+3AB²-3B²C+3BC²-3C²A+3CA²
    =3(-A²B+AB²-B²C+BC²-C²A+CA²)
    =3(A-B)(B-C)(C-A)

    The last factorization is not obvious, but can be inspired by expanding the right-hand side.

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