Solve a^2+a^3-2ab+b^2-b^3

$$a ^ { 2 } + a ^ { 3 } - 2 a b + b ^ { 2 } - b ^ { 3 }$$

Factor

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

Solution Steps

Consider a^{2}+a^{3}-2ab+b^{2}-b^{3} as a polynomial over variable a.

$$a^{3}+a^{2}-2ba+b^{2}-b^{3}$$

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}+b^{2}. One such factor is a-b. Factor the polynomial by dividing it by this factor.

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

Evaluate

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