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).

9th Grade Math

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