Solve a^7-b^7

$$a ^ { 7 } - b ^ { 7 }$$

Factor

$$\left(a-b\right)\left(a^{6}+a^{3}b^{3}+a^{2}b^{4}+ab^{5}+b^{6}+b^{2}a^{4}+ba^{5}\right)$$

Solution Steps

Consider a^{7}-b^{7} 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^{7} and m divides the constant factor -b^{7}. One such factor is a-b. Factor the polynomial by dividing it by this factor.

$$\left(a-b\right)\left(a^{6}+a^{3}b^{3}+a^{2}b^{4}+ab^{5}+b^{6}+b^{2}a^{4}+ba^{5}\right)$$

Evaluate

$$a^{7}-b^{7}$$