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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) $.