Solve a^6-a^5-a^3+a

$$a ^ { 6 } - a ^ { 5 } - a ^ { 3 } + a$$

Factor

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

Solution Steps

Factor out a.

$$a\left(a^{5}-a^{4}-a^{2}+1\right)$$

Consider a^{5}-a^{4}-a^{2}+1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by a-1.

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

Rewrite the complete factored expression. Polynomial a^{4}-a-1 is not factored since it does not have any rational roots.

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

Evaluate

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