Find the product using appropriate identity:
$(a - b - c) (a^{2} + b^{2} + c^{2} + a b - b c + c a)$

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Solution

We know the identity $(a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = a^{3} + b^{3} + c^{3} - 3 a b c$

Using the above identity taking $a = a$ , $b = - b$ and $c = - c$ , the product $(a - b - c) (a^{2} + b^{2} + c^{2} + a b - b c + c a)$ can be computed as follows:

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

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

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