Factorise the following:
${(x + y + z)}^{3} - {(2 x - y)}^{3} - {(2 y - z)}^{3} - {(2 z - x)}^{3}$

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Solution

Let us consider $2 x - y + 2 y - z + 2 z - x = x + y + z$ , therefore, the given problem can be rewritten as:

$[(2 x - y) + (2 y - z) + (2 z - x)]^{3} - (2 x - y)^{3} - (2 y - z)^{3} - (2 z - x)^{3}$

We know the identity: $(a + b + c)^{3} - a^{3} - b^{3} - c^{3} = 3 (a + b) (b + c) (c + a)$

Using the above identity taking $a = (2 x - y)$ , $b = (2 y - z)$ and $c = (2 z - x)$ , the equation
$[(2 x - y) + (2 y - z) + (2 z - x)]^{3} - (2 x - y)^{3} - (2 y - z)^{3} - (2 z - x)^{3}$ can be factorised as follows:

$(2 x - y) + (2 y - z) + (2 z - x)]^{3} - (2 x - y)^{3} - (2 y - z)^{3} - (2 z - x)^{3}$
$= 3 ((2 x - y) + (2 y - z)) ((2 y - z) + (2 z - x)) ((2 x - y) + (2 z - x))$
$= 3 (2 x - y + 2 y - z) (2 y - z + 2 z - x) (2 x - y + 2 z - x) = 3 (2 x + y - z) (2 y + z - x) (x - y + 2 z)$
Hence, $(x + y + z)^{3} - (2 x - y)^{3} - (2 y - z)^{3} - (2 z - x)^{3} = 3 (2 x + y - z) (2 y + z - x) (x - y + 2 z)$

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