Solve x^10-1

$$x ^ { 10 } - 1$$

Factor

$$\left(x-1\right)\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)$$

Solution Steps

Rewrite x^{10}-1 as \left(x^{5}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{5}-1\right)\left(x^{5}+1\right)$$

Consider x^{5}-1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)$$

Consider x^{5}+1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{4}-x^{3}+x^{2}-x+1,x^{4}+x^{3}+x^{2}+x+1.

$$\left(x-1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x+1\right)\left(x^{4}+x^{3}+x^{2}+x+1\right)$$

Evaluate

$$x^{10}-1$$