Question : What is the value of (a + b + c) {(a - b)2 + (b - c)2 + (c - a)2}?
Option 1: $2 a^3+2 b^3+2 c^3$
Option 2: $2a^3+2b^3+2c^3-6abc$
Option 3: $3abc$
Option 4: $6abc$
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Correct Answer: $2a^3+2b^3+2c^3-6abc$
Solution :
$(a+b+c)\left\{(a-b)^2+(b-c)^2+(c-a)^2\right\}$
$=((a+b+c)\left\{(a^2+b^2-2ab+b^2+c^2-2bc+c^2+a^2-2ac)\right\}$
$=((a+b+c)\left\{(2a^2+2b^2+2c^2-2ab-2bc-2ac)\right\}$
$=(2a^3+2ab^2+2ac^2-2a^2b-2abc-2a^2c+2a^2b+2b^3+2c^2b-2ab^2-2b^2c-2abc+2a^2c+2b^2c+2c^3-2abc-2bc^2-2ac^2)$
$=2a^3-2abc+2b^3-2abc+2c^3-2abc$
$=2a^3+2b^3+2c^3-6abc$
Hence, the correct answer is '$2a^3+2b^3+2c^3-6abc$'.
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