Giải thích các bước giải:
$\begin{array}{l}
{\left( {x - y} \right)^3} + {\left( {y - z} \right)^3} + {\left( {z - x} \right)^3}\\
= \left[ {{{\left( {x - y} \right)}^3} + {{\left( {y - z} \right)}^3}} \right] + {\left( {z - x} \right)^3}\\
= \left( {\left( {x - y} \right) + \left( {y - z} \right)} \right)\left( {{{\left( {x - y} \right)}^2} - \left( {x - y} \right)\left( {y - z} \right) + {{\left( {y - z} \right)}^2}} \right) + {\left( {z - x} \right)^3}\\
= \left( {x - z} \right)\left( {{{\left( {x - y} \right)}^2} - \left( {x - y} \right)\left( {y - z} \right) + {{\left( {y - z} \right)}^2} - {{\left( {z - x} \right)}^2}} \right)\\
= \left( {x - z} \right)\left( {{{\left( {x - y} \right)}^2} - {{\left( {z - x} \right)}^2} - \left( {x - y} \right)\left( {y - z} \right) + {{\left( {y - z} \right)}^2}} \right)\\
= \left( {x - z} \right)\left( {\left( {x - y - \left( {z - x} \right)} \right)\left( {x - y + \left( {z - x} \right)} \right) + \left( {y - z} \right)\left( {y - z - \left( {x - y} \right)} \right)} \right)\\
= \left( {x - z} \right)\left( {\left( {z - y} \right)\left( {2x - y - z} \right) + \left( {y - z} \right)\left( {2y - x - z} \right)} \right)\\
= \left( {x - z} \right)\left( {y - z} \right)\left( { - 2x + y + z + 2y - x - z} \right)\\
= \left( {x - z} \right)\left( {y - z} \right)\left( { - 3x + 3y} \right)\\
= - 3\left( {x - z} \right)\left( {y - z} \right)\left( {x - y} \right)
\end{array}$