Factor
x\left(x+1\right)\left(x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)
Evaluate
x^{8}+x
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x\left(x^{7}+1\right)
Factor out x.
\left(x+1\right)\left(x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)
Consider x^{7}+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.
x\left(x+1\right)\left(x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)
Rewrite the complete factored expression. Polynomial x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1 is not factored since it does not have any rational roots.
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