Solve {c}{x^3+y^3+z^3}{-3xyz}

$$\left. \begin{array} { c } { x ^ { 3 } + y ^ { 3 } + z ^ { 3 } } \\ { - 3 x y z } \end{array} \right.$$

Factor

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

Solution Steps

Consider x^{3}+y^{3}+z^{3}-3xyz as a polynomial over variable x.

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

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{3} and m divides the constant factor y^{3}+z^{3}. One such factor is x+y+z. Factor the polynomial by dividing it by this factor.

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

Evaluate

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