If $a + b + c = 0$ , then the value of $a^{2} (b + c) + b^{2} (c + a) + c^{2} (a + b)$ is:

a b c

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3 a b c

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- 3 a b c

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Solution

The correct option is C $- 3 a b c$
$a + b + c = 0$ $\Rightarrow$ $b + c = - a$ , $c + a = - b$
$a + b = - c$
$a^{2} (b + c) + b^{2} (c + a) + c^{2} (a + b)$
$= a^{2} \times - a + b^{2} \times - b + c^{2} \times - c$
$= - a^{3} - b^{3} - c^{3} = - (a^{3} + b^{3} + c^{3})$
If $a + b + c = 0$ then $a^{3} + b^{3} + c^{3} = 3 a b c$
$\Rightarrow$ $- (a^{3} + b^{3} + c^{3}) = - 3 a b c$

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