Shew that $(a^{2} + b^{2} + c^{2})^{3} + 2 (b c + c a + a b)^{3} - 3 (a^{2} + b^{2} + c^{2}) (b c + c a + a b)^{2} = (a^{3} + b^{3} + c^{3} - 3 a b c)^{2}$ .

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Solution

Let $X = a^{2} + b^{2} + c^{2}, Y = b c + c a + a b$ , Then

$L.H.S. = X^{3} + 2 Y^{3} - 3 X Y^{2} = (X - Y)^{2} (X + 2 Y)$

$= (X - Y)^{2} (a + b + c)^{2} = {(X - Y) (a + b + c)}^{2}$

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

$= (a^{3} + b^{3} + c^{3} - 3 a b c)^{2} = R . H . S .$

Hence, proved.

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