Solve x^11-y^11

$${ x }^{ 11 } - { y }^{ 11 }$$

Factor

$$\left(x-y\right)\left(x^{10}+x^{5}y^{5}+x^{4}y^{6}+x^{3}y^{7}+x^{2}y^{8}+xy^{9}+y^{10}+y^{4}x^{6}+y^{3}x^{7}+y^{2}x^{8}+yx^{9}\right)$$

Solution Steps

Consider x^{11}-y^{11} 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^{11} and m divides the constant factor -y^{11}. One such factor is x-y. Factor the polynomial by dividing it by this factor.

$$\left(x-y\right)\left(x^{10}+x^{5}y^{5}+x^{4}y^{6}+x^{3}y^{7}+x^{2}y^{8}+xy^{9}+y^{10}+y^{4}x^{6}+y^{3}x^{7}+y^{2}x^{8}+yx^{9}\right)$$

Evaluate

$$x^{11}-y^{11}$$