Here we will learn the
process of On Factorization of expressions of the Form a^{3} + b^{3} + c^{3} ,
where a + b + c = 0.

We have, a^{3} + b^{3} + c^{3} = a^{3} + b^{3} – (-c)^{3}

=
a^{3} + b^{3} – (a + b)^{3}, [Since, a + b + c
= 0]

=
a^{3} + b^{3} – {a^{3} + b^{3} + 3ab(a + b)}

= -3ab(a + b)

= -3ab(-c)

= 3abc

Therefore, a + b + c = 0, a^{3} + b^{3} + c^{3} = 3abc.

Solved example on factorization of expressions of the form
a^{3} + b^{3} + c^{3}, where a + b + c = 0:

Factorize: (a + b)^{3} + (c – b)^{3} – (a + c)^{3}.

**Solution:**

Here, given expression = (a + b)^{3} + (c – b)^{3} – (a + c)^{3}.

=
(a + b)^{3} + (c – b)^{3} +{– (a + c)}^{3}, Where a + b + c – b + {-(a + c)} = 0.

Therefore the given expression = 3(a + b)(c – b){-(a + c)} = 3(a + b)(b – c)(c + a).

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