Factorise: $27 x^{3} - 54 x^{2} y + 36 x y^{2} - 7 y^{3}$

(3 x - y) (9 x^{2} + 7 y^{2} - 15 x y)

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(x - y) (9 x^{2} + 7 y^{2} - 25 x y)

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(3 x - y) (9 x^{2} + 7 y^{2} - 25 x y)

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(x + y) (9 x^{2} + 7 y^{2} - 15 x y)

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Solution

The correct option is A $(3 x - y) (9 x^{2} + 7 y^{2} - 15 x y)$
$27 x^{3} - 54 x^{2} y + 36 x y^{2} - 7 y^{3}$
$= 27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3} + y^{3}$
$= (3 x)^{3} - 3 (3 x)^{3} (2 y) + 3 (3 x) (2 y)^{2} - (2 y)^{3} + y^{3}$
$= (3 x - 2 y)^{3} + y^{3}$
$= (3 x - 2 y + y) ((3 x - 2 y)^{2} - (3 x - 2 y) y + y^{2})$
$= (3 x - y) (9 x^{2} - 2 (3 x) (2 y) + (2 y)^{2} - 3 x y + 2 y^{2} + y^{2})$
$= (3 x - y) (9 x^{2} + 7 y^{2} - 15 x y)$

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