Solve x^4+x^13

$${ x }^{ 4 } + { x }^{ 13 }$$

Factor

$$\left(x+1\right)\left(x^{2}-x+1\right)x^{4}\left(x^{6}-x^{3}+1\right)$$

Solution Steps

Factor out x^{4}.

$$x^{4}\left(1+x^{9}\right)$$

Consider 1+x^{9}. Rewrite 1+x^{9} as \left(x^{3}\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).

$$\left(x^{3}+1\right)\left(x^{6}-x^{3}+1\right)$$

Consider x^{3}+1. Rewrite x^{3}+1 as x^{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).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-x+1,x^{6}-x^{3}+1.

$$x^{4}\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{6}-x^{3}+1\right)$$

Evaluate

$$x^{13}+x^{4}$$