Solve m^7-a^7x^7

$$m ^ { 7 } - a ^ { 7 } x ^ { 7 }$$

Factor

$$\left(m-ax\right)\left(a^{6}x^{6}+ma^{5}x^{5}+m^{2}a^{4}x^{4}+a^{3}m^{3}x^{3}+a^{2}x^{2}m^{4}+axm^{5}+m^{6}\right)$$

Solution Steps

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

$$\left(-ax+m\right)\left(a^{6}x^{6}+ma^{5}x^{5}+m^{2}a^{4}x^{4}+a^{3}m^{3}x^{3}+a^{2}x^{2}m^{4}+axm^{5}+m^{6}\right)$$

Evaluate

$$m^{7}-\left(ax\right)^{7}$$