Solve x^19-x^57

$$x ^ { 19 } - x ^ { 57 }$$

Factor

$$\left(x-1\right)\left(x+1\right)\left(-x^{18}+x^{17}-x^{16}+x^{15}-x^{14}+x^{13}-x^{12}+x^{11}-x^{10}+x^{9}-x^{8}+x^{7}-x^{6}+x^{5}-x^{4}+x^{3}-x^{2}+x-1\right)\left(x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)x^{19}$$

Solution Steps

Factor out x^{19}.

$$x^{19}\left(1-x^{38}\right)$$

Consider 1-x^{38}. Rewrite 1-x^{38} as 1^{2}-\left(-x^{19}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(1+x^{19}\right)\left(1-x^{19}\right)$$

Reorder the terms.

$$\left(x^{19}+1\right)\left(-x^{19}+1\right)$$

Consider x^{19}+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.

$$\left(x+1\right)\left(x^{18}-x^{17}+x^{16}-x^{15}+x^{14}-x^{13}+x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)$$

Consider -x^{19}+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.

$$\left(x-1\right)\left(-x^{18}-x^{17}-x^{16}-x^{15}-x^{14}-x^{13}-x^{12}-x^{11}-x^{10}-x^{9}-x^{8}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}-x^{2}-x-1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -x^{18}-x^{17}-x^{16}-x^{15}-x^{14}-x^{13}-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^{18}-x^{17}+x^{16}-x^{15}+x^{14}-x^{13}+x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1.

$$x^{19}\left(x+1\right)\left(x^{18}-x^{17}+x^{16}-x^{15}+x^{14}-x^{13}+x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)\left(x-1\right)\left(-x^{18}-x^{17}-x^{16}-x^{15}-x^{14}-x^{13}-x^{12}-x^{11}-x^{10}-x^{9}-x^{8}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}-x^{2}-x-1\right)$$

Evaluate

$$x^{19}-x^{57}$$