Solve m+m^6

$$m + m ^ { 6 }$$

Factor

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

Solution Steps

Factor out m.

$$m\left(1+m^{5}\right)$$

Consider 1+m^{5}. 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 m+1.

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

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

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

Evaluate

$$m^{6}+m$$