Solve acb+bac+cba-a^3-b^3-c^3

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

Factor

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

Solution Steps

Multiply and combine like terms.

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

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

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

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power -a^{3} and n 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

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