Solve a^5-a^3b^2+a^2{b}^{3}-{a}^{3}{b}^{2}+a{b}^{4}-{b}^5+{a}^3-a{b}^2+{b}^3

$${ a }^{ 5 } - { a }^{ 3 } { b }^{ 2 } + { a }^{ 2 } { b }^{ 3 } - { a }^{ 3 } { b }^{ 2 } +a { b }^{ 4 } - { b }^{ 5 } + { a }^{ 3 } -a { b }^{ 2 } + { b }^{ 3 }$$

Factor

$$\left(a^{2}-b^{2}+1\right)\left(a^{3}-ab^{2}+b^{3}\right)$$

Solution Steps

Multiply and combine like terms.

$$a^{5}+a^{3}+a^{2}b^{3}+ab^{4}-ab^{2}-b^{5}+b^{3}-2b^{2}a^{3}$$

Consider a^{5}+a^{3}+a^{2}b^{3}+ab^{4}-ab^{2}-b^{5}+b^{3}-2b^{2}a^{3} as a polynomial over variable a.

$$a^{5}+\left(-2b^{2}+1\right)a^{3}+b^{3}a^{2}+\left(b^{4}-b^{2}\right)a-b^{5}+b^{3}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{5} and m divides the constant factor -b^{5}+b^{3}. One such factor is a^{2}-b^{2}+1. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}-b^{2}+1\right)\left(a^{3}-ab^{2}+b^{3}\right)$$

Evaluate

$$\left(a^{2}-b^{2}+1\right)\left(a^{3}-ab^{2}+b^{3}\right)$$