Question 5
Simplify:
$(2 x - 5 y)^{3} - (2 x + 5 y)^{3}$

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Solution

$(2 x - 5 y)^{3} - (2 x + 5 y)^{3} = [(2 x)^{3} - (5 y)^{3} - 3 (2 x) (5 y) (2 x - 5 y)] - [(2 x)^{3} + (5 y)^{3} + 3 (2 x) (5 y) (2 x + 5 y)]$
$[U s i n g t h e i d e n t i t y, (a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b)$ and $(a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b)]$
$= (2 x)^{3} - (5 y)^{3} - 30 x y (2 x - 5 y) - (2 x)^{3} - (5 y) 3 - 30 x y (2 x + 5 y)$
$= - 2 (5 y)^{3} - 30 x y (2 x - 5 y + 2 x + 5 y)$
$= - 2 \times 125 y^{3} - 30 x y (4 x) = - 250 y^{3} - 120 x^{2} y$

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