Solve m^7-n^7

$$m ^ { 7 } - n ^ { 7 }$$

Factor

$$\left(m-n\right)\left(m^{6}+m^{3}n^{3}+m^{2}n^{4}+mn^{5}+n^{6}+n^{2}m^{4}+nm^{5}\right)$$

Solution Steps

Consider m^{7}-n^{7} as a polynomial over variable m. Find one factor of the form m^{k}+p, where m^{k} divides the monomial with the highest power m^{7} and p divides the constant factor -n^{7}. One such factor is m-n. Factor the polynomial by dividing it by this factor.

$$\left(m-n\right)\left(m^{6}+m^{3}n^{3}+m^{2}n^{4}+mn^{5}+n^{6}+n^{2}m^{4}+nm^{5}\right)$$

Evaluate

$$m^{7}-n^{7}$$