Question

Factorize:
(a − b)³ + (b − c)³ + (c − a)³

Answer 1

${\left(a-b\right)}^3+{\left(b-c\right)}^3+{\left(c-a\right)}^3$
$$\begin{gathered} \mathrm{Putting}\left(a-b\right)=x,\left(b-c\right)=y\mathrm{and}\left(c-a\right)=z,\mathrm{we}\mathrm{get}: \\ {\left(a-b\right)}^3+{\left(b-c\right)}^3+{\left(c-a\right)}^3 \\ =x^3+y^3+z^3[\mathrm{Where}\left(x+y+z\right)=\left(a-b\right)+\left(b-c\right)+\left(c-a\right)=0] \\ =3xyz\left[\left(x+y+z\right)=0\Rightarrow x^3+y^3+z^3=3xyz\right] \\ =3\left(a-b\right)\left(b-c\right)\left(c-a\right) \end{gathered}$$