Solve x^5+x^4+x^3+x^2+x+1

$$x ^ { 5 } + x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x + 1$$

Factor

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

Solution Steps

Do the grouping x^{5}+x^{4}+x^{3}+x^{2}+x+1=\left(x^{5}+x^{4}+x^{3}\right)+\left(x^{2}+x+1\right), and factor out x^{3} in x^{5}+x^{4}+x^{3}.

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

Factor out common term x^{2}+x+1 by using distributive property.

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

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^{2}+x+1.

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

Evaluate

$$x^{5}+x^{4}+x^{3}+x^{2}+x+1$$