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

(x + 2) (x^{2} + 7 y^{2} - 5 y)

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

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

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(x - 9) (x^{2} + 7 y^{2} - 5 x y)

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Solution

The correct option is C $(3 x - 9) (9 x^{2} + 7 y^{2} - 15 x y)$
$27 x^{3} - 54 x^{2} y + 36 x y^{2} - 7 y^{3}$
Using,
$(a - b)^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$
$= (3 x)^{3} - 3 (3 x)^{2} (2 y) + 3 (3 x) (2 y)^{2} - (2 y)^{3} + y^{3}$
$= {(3 x - 2 y)}^{3} + y^{3}$
Using,
$(a^{3} + b^{3}) = (a + b) (a^{2} - a b + b^{2})$
$= (3 x - 2 y + y) [(3 x - 2 y)^{2} - (3 x - 2 y) (y) + (y)^{2}]$
$= (3 x - y) (9 x^{2} + 4 y^{2} - 12 x y - 3 x y + 2 y^{2} + y^{2})$
$= (3 x - y) (9 x^{2} + 7 y^{2} - 15 x y)$

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