Solve x^10-y^10

$${ x }^{ 10 } - { y }^{ 10 }$$

Factor

$$\left(x-y\right)\left(x+y\right)\left(x^{4}+x^{2}y^{2}-xy^{3}+y^{4}-yx^{3}\right)\left(x^{4}+x^{2}y^{2}+xy^{3}+y^{4}+yx^{3}\right)$$

Solution Steps

Rewrite x^{10}-y^{10} as \left(x^{5}\right)^{2}-\left(y^{5}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{5}-y^{5}\right)\left(x^{5}+y^{5}\right)$$

Consider x^{5}-y^{5}. Consider x^{5}-y^{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^{5} and m divides the constant factor -y^{5}. One such factor is x-y. Factor the polynomial by dividing it by this factor.

$$\left(x-y\right)\left(x^{4}+x^{2}y^{2}+xy^{3}+y^{4}+yx^{3}\right)$$

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

$$\left(x+y\right)\left(x^{4}+x^{2}y^{2}-xy^{3}+y^{4}-yx^{3}\right)$$

Rewrite the complete factored expression.

$$\left(x-y\right)\left(x+y\right)\left(x^{4}+x^{2}y^{2}-xy^{3}+y^{4}-yx^{3}\right)\left(x^{4}+x^{2}y^{2}+xy^{3}+y^{4}+yx^{3}\right)$$

Evaluate

$$x^{10}-y^{10}$$