Factorise: $x^{2} + 16 y^{2} + 64 z^{2} - 8 x y - 64 y z + 16 z x$

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Solution

$x^{2} + 16 y^{2} + 64 z^{2} - 8 x y - 64 y z + 16 z x$
$\Rightarrow$ Using ${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
$\Rightarrow$ Keeping $x = a,$ $- 4 y = b,$ $8 z = c$
We get, ${(x - 4 y + 8 z)}^{2} .$
Hence, the answer is ${(x - 4 y + 8 z)}^{2} .$

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