Solve 1/7x^6+6/7x^4-8/7x^5+1/7x

$$\frac { 1 } { 7 } x ^ { 6 } + \frac { 6 } { 7 } x ^ { 4 } - \frac { 8 } { 7 } x ^ { 5 } + \frac { 1 } { 7 } x$$

Factor

$$\frac{x\left(x-1\right)\left(x^{4}-7x^{3}-x^{2}-x-1\right)}{7}$$

Solution Steps

Factor out \frac{1}{7}.

$$\frac{x^{6}+6x^{4}-8x^{5}+x}{7}$$

Consider x^{6}+6x^{4}-8x^{5}+x. Factor out x.

$$x\left(x^{5}+6x^{3}-8x^{4}+1\right)$$

Consider x^{5}+6x^{3}-8x^{4}+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.

$$\left(x-1\right)\left(x^{4}-7x^{3}-x^{2}-x-1\right)$$

Rewrite the complete factored expression. Polynomial x^{4}-7x^{3}-x^{2}-x-1 is not factored since it does not have any rational roots.

$$\frac{x\left(x-1\right)\left(x^{4}-7x^{3}-x^{2}-x-1\right)}{7}$$

Evaluate

$$\frac{x\left(x^{5}-8x^{4}+6x^{3}+1\right)}{7}$$