Solve x^10-a^5

$$x ^ { 10 } - a ^ { 5 }$$

Factor

$$\left(x^{2}-a\right)\left(x^{8}+ax^{6}+a^{2}x^{4}+x^{2}a^{3}+a^{4}\right)$$

Solution Steps

Consider x^{10}-a^{5} as a polynomial over variable x. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{10} and m divides the constant factor -a^{5}. One such factor is x^{2}-a. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-a\right)\left(x^{8}+ax^{6}+a^{2}x^{4}+x^{2}a^{3}+a^{4}\right)$$

Evaluate

$$x^{10}-a^{5}$$