Solve x^2-x^200=

$$x ^ { 2 } - x ^ { 200 } =$$

Factor

$$\left(x-1\right)\left(x+1\right)x^{2}\left(x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{22}-x^{11}+1\right)\left(x^{22}+x^{11}+1\right)\left(-x^{66}+x^{33}-1\right)\left(x^{66}+x^{33}+1\right)$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(1-x^{198}\right)$$

Consider 1-x^{198}. Rewrite 1-x^{198} as 1^{2}-\left(-x^{99}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(1+x^{99}\right)\left(1-x^{99}\right)$$

Reorder the terms.

$$\left(x^{99}+1\right)\left(-x^{99}+1\right)$$

Consider x^{99}+1. Rewrite x^{99}+1 as \left(x^{33}\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).

$$\left(x^{33}+1\right)\left(x^{66}-x^{33}+1\right)$$

Consider x^{33}+1. Rewrite x^{33}+1 as \left(x^{11}\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).

$$\left(x^{11}+1\right)\left(x^{22}-x^{11}+1\right)$$

Consider x^{11}+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.

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

Consider -x^{99}+1. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power -x^{99} and n divides the constant factor 1. One such factor is x^{33}-1. Factor the polynomial by dividing it by this factor.

$$\left(x^{33}-1\right)\left(-x^{66}-x^{33}-1\right)$$

Consider x^{33}-1. Rewrite x^{33}-1 as \left(x^{11}\right)^{3}-1^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

$$\left(x^{11}-1\right)\left(x^{22}+x^{11}+1\right)$$

Consider x^{11}-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.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -x^{66}-x^{33}-1,x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1,x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1,x^{22}-x^{11}+1,x^{22}+x^{11}+1,x^{66}-x^{33}+1.

$$x^{2}\left(x+1\right)\left(x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)\left(x^{22}-x^{11}+1\right)\left(x^{66}-x^{33}+1\right)\left(x-1\right)\left(x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\left(x^{22}+x^{11}+1\right)\left(-x^{66}-x^{33}-1\right)$$

Evaluate

$$x^{2}\left(1-x^{22}\right)\left(\left(x^{22}+1\right)^{2}-x^{22}\right)\left(\left(x^{66}+1\right)^{2}-x^{66}\right)$$