Factorize: $(a+b)^3 + (c-b)^3 - (a+c)^3$

Given :

The given term is $(a+b)^3 + (c-b)^3 - (a+c)^3$.

To do :

We have to factorise the given term.

Solution :

$(a+b)^3 + (c-b)^3 - (a+c)^3 = (a+b)^3 + (c-b)^3 +[-(a+c)]^3$

We know that,

$a^3 + b^3 + c^3 -3abc = (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ac)$.

If $a+b+c = 0$, then $a^3 + b^3 + c^3 =3abc$

Therefore, 

If $(a+b) + (c-b) + (-a-c) = 0$ then,

$(a+b)^3 + (c-b)^3 + (-a-c)^3 = 3(a+b) (c-b) (-a-c) $

$(a+b) + (c-b) + (-a-c) = a + b + c - b - a - c = 0$.

 $(a+b)^3 + (c-b)^3 + (-a-c)^3 = 3(a+b) (c-b) (-a-c) $

                                                   $ =-3(a+b) (c-b) (a+c)   $

                                                   $ =3(a+b) (b - c) (a+c)   $.

Therefore, $(a+b)^3 + (c-b)^3 - (a+c)^3 = 3(a+b) (b - c) (a+c)   $.

Tutorialspoint

Updated on: 10-Oct-2022

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