Solve: $(a + b)^{3} - (a - b)^{3}$ .

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Answer 1

We know, $(a + b)^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$⟹$ $(a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b)$ .
Also, $(a - b)^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$

$⟹$ $(a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b)$ .

Then, $(a + b)^{3} -$ $(a - b)^{3}$
$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$ $- (a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2})$

$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$ $- a^{3} + b^{3} + 3 a^{2} b - 3 a b^{2}$

$= b^{3} + 3 a^{2} b$ $+ b^{3} + 3 a^{2} b$

$= 2 b^{3} + 6 a^{2} b$

$= 2 b (b^{2} + 3 a^{2})$ .

Therefore, option $B$ is correct.

Answer 2

We know, $(a + b)^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$

$⟹$ $(a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b)$ .

Also, $(a - b)^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$

$⟹$ $(a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b)$ .

Then, $(a + b)^{3} -$ $(a - b)^{3}$

$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$ $- (a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2})$

$= a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$ $- a^{3} + b^{3} + 3 a^{2} b - 3 a b^{2}$

$= b^{3} + 3 a^{2} b$ $+ b^{3} + 3 a^{2} b$

$= 2 b^{3} + 6 a^{2} b$

$= 2 b (b^{2} + 3 a^{2})$ .

Therefore, option $B$ is correct.

Solve: $(a + b)^{3} - (a - b)^{3}$ .

$2 a (3 a^{2} + b^{2})$

$2 b (3 a^{2} + b^{2})$

$2 (a^{3} + b^{3})$

None

Solution

$2 b (3 a^{2} + b^{2})$

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