Solve a^10+b^10

$$a ^ { 10 } + b ^ { 10 }$$

Factor

$$\left(a^{2}+b^{2}\right)\left(a^{8}+a^{4}b^{4}-a^{2}b^{6}+b^{8}-b^{2}a^{6}\right)$$

Solution Steps

Consider a^{10}+b^{10} as a polynomial over variable a. Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{10} and m divides the constant factor b^{10}. One such factor is a^{2}+b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}+b^{2}\right)\left(a^{8}+a^{4}b^{4}-a^{2}b^{6}+b^{8}-b^{2}a^{6}\right)$$

Evaluate

$$a^{10}+b^{10}$$