Solve 5x^6-5x^4+x^3-x

$$5 { x }^{ 6 } -5 { x }^{ 4 } + { x }^{ 3 } -x$$

Factor

$$x\left(x-1\right)\left(x+1\right)\left(5x^{3}+1\right)$$

Solution Steps

Factor out x.

$$x\left(5x^{5}-5x^{3}+x^{2}-1\right)$$

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

$$5x^{3}\left(x^{2}-1\right)+x^{2}-1$$

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

$$\left(x^{2}-1\right)\left(5x^{3}+1\right)$$

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).

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

Rewrite the complete factored expression. Polynomial 5x^{3}+1 is not factored since it does not have any rational roots.

$$x\left(x-1\right)\left(x+1\right)\left(5x^{3}+1\right)$$

Evaluate

$$x\left(x^{2}-1\right)\left(5x^{3}+1\right)$$