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

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

Factor

$$\left(x+1\right)x^{3}\left(x^{4}-x^{3}+x^{2}+1\right)$$

Solution Steps

Factor out x^{3}.

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

Consider x^{5}+x^{2}+x+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 x+1.

$$\left(x+1\right)\left(x^{4}-x^{3}+x^{2}+1\right)$$

Rewrite the complete factored expression. Polynomial x^{4}-x^{3}+x^{2}+1 is not factored since it does not have any rational roots.

$$x^{3}\left(x+1\right)\left(x^{4}-x^{3}+x^{2}+1\right)$$

Evaluate

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