Solve a^3+b^3+c^3-3abc=?

$$a ^ { 3 } + b ^ { 3 } + c ^ { 3 } - 3 a b c = ?$$

Factor

$$\left(a+b+c\right)\left(a^{2}-ab-ac+b^{2}-bc+c^{2}\right)$$

Solution Steps

Consider a^{3}+b^{3}+c^{3}-3abc as a polynomial over variable a.

$$a^{3}-3bca+b^{3}+c^{3}$$

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

$$\left(a+b+c\right)\left(a^{2}-ab-ac+b^{2}-bc+c^{2}\right)$$

Evaluate

$$a^{3}-3abc+b^{3}+c^{3}$$