Simplify:
${(2 x + y)}^{3} + {(2 x - y)}^{3} .$

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Solution

Consider the given expression ${(2 x + y)}^{3} + {(2 x - y)}^{3}$

We know that $A^{3} + B^{3} = (A + B) (A^{2} + B^{2} - A B)$

$∴ A = 2 x + y, B = 2 x - y$

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

$= 4 x [4 x^{2} + y^{2} + 4 x y + 4 x^{2} + y^{2} - 4 x y - (4 x^{2} - 2 x y + 2 x y - y^{2})]$

$= 4 x [8 x^{2} + 2 y^{2} - 4 x^{2} - y^{2}] = 4 x (4 x^{2} - y^{2})$

$= 4 x (4 x^{2} - y^{2})$

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