Factorize $(a - 2 b)^{3} - 512 b^{3}$

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Solution

$(a - 2 b)^{3} - 512 b^{3} = (a - 2 b)^{3} - (8 b)^{3} = (a - 2 b - 8 b) [(a - 2 b)^{2} + (a - 2 b) \times 8 b + (8 b)^{2}] = (a - 10 b) (a^{2} + 4 a b + 52 b^{2})$

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2 \sqrt{2} a^{3} + 16 \sqrt{2} b^{3} + c^{3} - 12 a b c

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