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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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