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

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

Factor

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

Solution Steps

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

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

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

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

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.

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

Evaluate

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