Solve x^4+x^3+x+1

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

Factor

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

Solution Steps

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

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

Factor out common term x+1 by using distributive property.

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

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. Polynomial x^{2}-x+1 is not factored since it does not have any rational roots.

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

Evaluate

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