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

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

Factor

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

Solution Steps

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

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

Factor out common term x^{3}-1 by using distributive property.

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

Consider x^{3}-1. Rewrite x^{3}-1 as x^{3}-1^{3}. The difference 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.

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

Evaluate

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