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Algebraic Identities

Solve: i 8x...

Question

Solve:
(i) $8 x^{3} + y^{3}$
(ii) $16 (x + y)^{3} - 27$

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Solution

(i) $8 x^{3} + y^{3} = (2 x)^{3} + y^{3}$ $(∵ a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
$= (2 x + y) ((2 x)^{2} - (2 x) y + y^{2})$
$= (2 x + y) (4 x^{2} - 2 x y + y^{2})$
(ii) $16 (x + y)^{3} - 27$
$= 16 [x^{3} + 3 x^{2} y + 3 y^{2} x + y^{3}] - 27$
$= 16 x^{3} + 8 x^{2} y + 48 x y^{2} + 16 y^{3} - 27$ .

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