Solve a-b-a^3+b^3

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

Factor

$$\left(a-b\right)\left(1-b^{2}-ab-a^{2}\right)$$

Solution Steps

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

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

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

$$\left(a-b\right)\left(-a^{2}-ab-b^{2}+1\right)$$

Evaluate

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