Solve 5x^65+5=

$$5 { x }^{ 65 } +5=$$

Factor

$$5\left(x+1\right)\left(x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{52}-x^{39}+x^{26}-x^{13}+1\right)$$

Solution Steps

Factor out 5.

$$5\left(x^{65}+1\right)$$

Consider x^{65}+1. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{65} and m divides the constant factor 1. One such factor is x^{13}+1. Factor the polynomial by dividing it by this factor.

$$\left(x^{13}+1\right)\left(x^{52}-x^{39}+x^{26}-x^{13}+1\right)$$

Consider x^{13}+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.

$$\left(x+1\right)\left(x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1,x^{52}-x^{39}+x^{26}-x^{13}+1.

$$5\left(x+1\right)\left(x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{52}-x^{39}+x^{26}-x^{13}+1\right)$$

Evaluate

$$5\left(x^{65}+1\right)$$