Solve x^225+1

$${ x }^{ 225 } +1$$

Factor

$$\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{20}-x^{15}+x^{10}-x^{5}+1\right)\left(x^{50}-x^{25}+1\right)\left(x^{150}-x^{75}+1\right)$$

Solution Steps

Rewrite x^{225}+1 as \left(x^{75}\right)^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x^{75}+1\right)\left(x^{150}-x^{75}+1\right)$$

Consider x^{75}+1. Rewrite x^{75}+1 as \left(x^{25}\right)^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x^{25}+1\right)\left(x^{50}-x^{25}+1\right)$$

Consider x^{25}+1. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{25} and m divides the constant factor 1. One such factor is x^{5}+1. Factor the polynomial by dividing it by this factor.

$$\left(x^{5}+1\right)\left(x^{20}-x^{15}+x^{10}-x^{5}+1\right)$$

Consider x^{5}+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.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{4}-x^{3}+x^{2}-x+1,x^{20}-x^{15}+x^{10}-x^{5}+1,x^{50}-x^{25}+1,x^{150}-x^{75}+1.

$$\left(x^{4}-x^{3}+x^{2}-x+1\right)\left(x+1\right)\left(x^{20}-x^{15}+x^{10}-x^{5}+1\right)\left(x^{50}-x^{25}+1\right)\left(x^{150}-x^{75}+1\right)$$

Evaluate

$$x^{225}+1$$