Factorize: $27 x^{3} - 125 y^{3}$

(3 x + 5 y) (9 x^{2} + 25 y^{2} + 15 x y)

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(3 x - 5 y) (9 x^{2} + 25 y^{2} - 15 x y)

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(3 x + 5 y) (9 x^{2} + 25 y^{2} - 15 x y)

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(3 x - 5 y) (9 x^{2} + 25 y^{2} + 15 x y)

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Solution

The correct option is D $(3 x - 5 y) (9 x^{2} + 25 y^{2} + 15 x y)$
$27 x^{3} - 125 y^{3}$
$\Rightarrow$ Using $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$\Rightarrow {(3 x)}^{3} - {(5 y)}^{3} = (3 x - 5 y) ({(3 x)}^{2} + {(5 y)}^{2} + (3 x) (5 y))$
$= (3 x - 5 y) (9 x^{2} + 15 x y + 25 y^{2})$
Hence, the answer is $(3 x - 5 y) (9 x^{2} + 15 x y + 25 y^{2}) .$

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(i) $x^{2} + y^{2} + z^{2} - x y + x + y z by x + y - z$

(ii) $x^{2} + 4 y^{2} + z^{2} + 2 x y + x z - 2 y z by x - 2 y - z$

(iii) $x^{2} + 4 y^{2} + 2 x y - 3 x + 6 y + 9 by x - 2 y + 3$

(iv) $9 x^{2} + 25 y^{2} + 15 x y + 2 x - 20 y + 16 by 3 x - 5 y + 4$

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