Solve x^16+x

$$x ^ { 16 } + x$$

Factor

$$x\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{10}-x^{5}+1\right)$$

Solution Steps

Factor out x.

$$x\left(x^{15}+1\right)$$

Consider x^{15}+1. Rewrite x^{15}+1 as \left(x^{5}\right)^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x^{5}+1\right)\left(x^{10}-x^{5}+1\right)$$

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.

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

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^{10}-x^{5}+1.

$$x\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{10}-x^{5}+1\right)$$

Evaluate

$$x^{16}+x$$