Factorize the following expression:

$a^{3} - b^{3} - c^{3} - 3 abc$

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(a - b - c) (a^{2} + b^{2} + c^{2} + ab - bc + ca)

(a - b + c) (a^{2} - b^{2} + c^{2} + ab - bc + ca)

(a + b - c) (a^{2} + b^{2} + c^{2} - ab - 3 bc + 3 ca)

(a - b - c) (a^{2} - b^{2} + c^{2} + 3 ab - bc + ca)

answer is A.

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Detailed Solution

Given expression is,

a^{3} - b^{3} - c^{3} - 3 abc

It is known that,

x^{3} + y^{3} + z^{3} - 3 xyz = (x + y + z) (x^{2} + y^{2} + z^{2} - zy - yz - zx)

Thus, the given expression using above identity can be written as,

a^{3} - b^{3} - c^{3} - 3 abc

= (a^{3}) + (- b)^{3} + (- c)^{3} - 3 (a) (- b) (c)

= [a + (- b) + (- c)] [a^{2} + (- b)^{2} + (- c)^{2} - (a \times - b) - (- b \times - c) - (- c \times a)]

= (a - b - c) (a^{2} + b^{2} + c^{2} + ab - bc + ca)

Thus,

a^{3} - b^{3} - c^{3} - 3 abc = (a - b - c) (a^{2} + b^{2} + c^{2} + ab - bc + ca)

So, option 1 is correct.

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