Solve (x-y)(x^{2}+xy+y^{2})+(y-z)(y^2+yz+z^2)+(z-x)(z^2+zx+x^2)

$$( x - y ) ( x ^ { 2 } + x y + y ^ { 2 } ) + ( y - z ) ( y ^ { 2 } + y z + z ^ { 2 } ) + ( z - x ) ( z ^ { 2 } + z x + x ^ { 2 } )$$

Evaluate

$$0$$

Solution Steps

Use the distributive property to multiply x-y by x^{2}+xy+y^{2} and combine like terms.

$$x^{3}-y^{3}+\left(y-z\right)\left(y^{2}+yz+z^{2}\right)+\left(z-x\right)\left(z^{2}+zx+x^{2}\right)$$

Use the distributive property to multiply y-z by y^{2}+yz+z^{2} and combine like terms.

$$x^{3}-y^{3}+y^{3}-z^{3}+\left(z-x\right)\left(z^{2}+zx+x^{2}\right)$$

Combine -y^{3} and y^{3} to get 0.

$$x^{3}-z^{3}+\left(z-x\right)\left(z^{2}+zx+x^{2}\right)$$

Use the distributive property to multiply z-x by z^{2}+zx+x^{2} and combine like terms.

$$x^{3}-z^{3}+z^{3}-x^{3}$$

Combine -z^{3} and z^{3} to get 0.

$$x^{3}-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$0$$

Factor

$$0$$