Solve a^{3}b-ab^{3}+b^3c-bc^3+c^3a-ca^3

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

Factor

$$\left(a-b\right)\left(a-c\right)\left(b-c\right)\left(a+b+c\right)$$

Solution Steps

Consider a^{3}b-ab^{3}+b^{3}c-bc^{3}+c^{3}a-ca^{3} as a polynomial over variable a.

$$\left(b-c\right)a^{3}+\left(-b^{3}+c^{3}\right)a+b^{3}c-bc^{3}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power \left(b-c\right)a^{3} and n divides the constant factor -bc^{3}+cb^{3}. One such factor is a+b+c. Factor the polynomial by dividing it by this factor.

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

Consider -ab^{2}+ac^{2}+ba^{2}-bc^{2}-ca^{2}+cb^{2}. Consider -ab^{2}+ac^{2}+ba^{2}-bc^{2}-ca^{2}+cb^{2} as a polynomial over variable a.

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

Find one factor of the form pa^{q}+u, where pa^{q} divides the monomial with the highest power \left(b-c\right)a^{2} and u divides the constant factor -bc^{2}+cb^{2}. One such factor is a-c. Factor the polynomial by dividing it by this factor.

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

Consider ab-ac-b^{2}+bc. Do the grouping ab-ac-b^{2}+bc=\left(ab-ac\right)+\left(-b^{2}+bc\right), and factor out a in the first and -b in the second group.

$$a\left(b-c\right)-b\left(b-c\right)$$

Factor out common term b-c by using distributive property.

$$\left(b-c\right)\left(a-b\right)$$

Rewrite the complete factored expression.

$$\left(a-b\right)\left(a+b+c\right)\left(a-c\right)\left(b-c\right)$$

Evaluate

$$\left(a-b\right)\left(a-c\right)\left(b-c\right)\left(a+b+c\right)$$