Solve x^9+x^8+x^7+x^6

$${ x }^{ 9 } + { x }^{ 8 } + { x }^{ 7 } + { x }^{ 6 }$$

Factor

$$\left(x+1\right)\left(x^{2}+1\right)x^{6}$$

Solution Steps

Factor out x^{6}.

$$x^{6}\left(x^{3}+x^{2}+x+1\right)$$

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

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

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

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

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

$$x^{6}\left(x+1\right)\left(x^{2}+1\right)$$

Evaluate

$$\left(x+1\right)\left(x^{2}+1\right)x^{6}$$