# Factorization of Expressions of the Form a$$^{3}$$ + b$$^{3}$$ + c$$^{3}$$, a + b + c=0

Here we will learn the process of On Factorization of expressions of the Form a3 + b3 + c3 , where a + b + c = 0.

We have, a3 + b3 + c3 = a3 + b3 – (-c)3

= a3 + b3 – (a + b)3, [Since, a + b  + c = 0]

= a3 + b3 – {a3 + b3 + 3ab(a + b)}

= -3ab(a + b)

= -3ab(-c)

= 3abc

Therefore, a + b + c = 0, a3 + b3 + c3 = 3abc.

Solved example on factorization of expressions of the form a3 + b3 + c3, 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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