Solve a^5-a^4-a+1

$$a ^ { 5 } - a ^ { 4 } - a + 1$$

Factor

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

Solution Steps

Do the grouping a^{5}-a^{4}-a+1=\left(a^{5}-a^{4}\right)+\left(-a+1\right), and factor out a^{4} in the first and -1 in the second group.

$$a^{4}\left(a-1\right)-\left(a-1\right)$$

Factor out common term a-1 by using distributive property.

$$\left(a-1\right)\left(a^{4}-1\right)$$

Consider a^{4}-1. Rewrite a^{4}-1 as \left(a^{2}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

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

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

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

Rewrite the complete factored expression. Polynomial a^{2}+1 is not factored since it does not have any rational roots.

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

Evaluate

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