Resolve into factors the expression ${(a + b + c)}^{3} - a^{3} - b^{3} - c^{3}$

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Solution

The given expression becomes 0 when either $a + b = 0 o r b + c = 0 o r c + a = 0$
and hence all these are its factors of 3rd degree.
Now assume ${(a + b + c)}^{3} - a^{3} - b^{3} - c^{3} = λ (a + b) (b + c) (c + a)$
Choose $a = 0, b = 1, c = 2$ ,
we get $λ = 3.$

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