Ta có:
\(x^2.\left(y-z\right)+y^2.\left(z-x\right)+z^2.\left(x-y\right)\)
= \(x^2.\left(y-z\right)+y^2z-y^2x+z^2x-z^2y\)
= \(x^2.\left(y-z\right)+\left(y^2z-z^2y\right)-\left(y^2x-z^2x\right)\)
= \(x^2.\left(y-z\right)+yz\left(y-z\right)-x\left(y^2-z^2\right)\)
= \(x^2.\left(y-z\right)+yz\left(y-z\right)-x\left(y-z\right)\left(y+z\right)\)
= \(\left(y-z\right)\left[x^2+yz-x\left(y+z\right)\right]\)
= \(\left(y-z\right)\left[x^2+yz-xy-xz\right]\)
= \(\left(y-z\right)\left[\left(x^2-xy\right)-\left(xz-yz\right)\right]\)
= \(\left(y-z\right)\left[x\left(x^{ }-y\right)-z\left(x-y\right)\right]\)
= \(\left(y-z\right)\left(x-y\right)\left(x-z\right)\)
\(x^2\left(y-z\right)+y^2\left(z+x\right)+z^2\left(x-y\right)\)
\(\Leftrightarrow x^2\left(y-z\right)+y^2\left(z-x \right)+z^2\left[-\left(y-z\right)-\left(z-x\right)\right]\)\(\Leftrightarrow x^2\left(y-z\right)+y^2\left(z-x\right)-z^2\left(y-z\right)-z^2\left(z-x\right)\)\(\Leftrightarrow\left(y-z\right)\left(x-z\right)\left(x+z\right)+\left(z-x\right)\left(y-z\right)\left(y+z\right)\)\(\Leftrightarrow\left(y-z\right)\left(x-z\right)\left(x+z\right)-\left(y-z\right)\left(x-z\right)\left(z+y\right)\)\(\Leftrightarrow\left(y-z\right)\left(x-z\right)\left(x+z-y-z\right)=\left(y-z\right)\left(x-z\right)\left(x-y\right)\)
