Question

Factorise:
${(x-y)}^3+{(y-z)}^3+{(z-x)}^3$

Answer 1

We know the corollary: if $a+b+c=0$ then $a^3+b^3+c^3=3abc$

Using the above identity taking $a=x-y$, $b=y-z$ and $c=z-x$, we have $a+b+c=x-y+y-z+z-x=0$ then the equation $(x-y{)}^3+(y-z{)}^3+(z-x{)}^3$ can be factorised as follows:

$(x-y{)}^3+(y-z{)}^3+(z-x{)}^3=3(x-y\left)\right(y-z\left)\right(z-x)$

Hence, $(x-y{)}^3+(y-z{)}^3+(z-x{)}^3=3(x-y\left)\right(y-z\left)\right(z-x)$