Solve x^6-x^5+8x^4-8x^3

$${ x }^{ 6 } - { x }^{ 5 } +8 { x }^{ 4 } -8 { x }^{ 3 }$$

Factor

$$\left(x-1\right)\left(x^{2}+8\right)x^{3}$$

Solution Steps

Factor out x^{3}.

$$x^{3}\left(x^{3}-x^{2}+8x-8\right)$$

Consider x^{3}-x^{2}+8x-8. Do the grouping x^{3}-x^{2}+8x-8=\left(x^{3}-x^{2}\right)+\left(8x-8\right), and factor out x^{2} in the first and 8 in the second group.

$$x^{2}\left(x-1\right)+8\left(x-1\right)$$

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

$$\left(x-1\right)\left(x^{2}+8\right)$$

Rewrite the complete factored expression. Polynomial x^{2}+8 is not factored since it does not have any rational roots.

$$x^{3}\left(x-1\right)\left(x^{2}+8\right)$$

Evaluate

$$\left(x-1\right)\left(x^{2}+8\right)x^{3}$$