Solve 1-2a^5+a^10

$$1 - 2 a ^ { 5 } + a ^ { 10 }$$

Factor

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

Solution Steps

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{10} and m divides the constant factor 1. One such factor is a^{5}-1. Factor the polynomial by dividing it by this factor.

$$\left(a^{5}-1\right)\left(a^{5}-1\right)$$

Consider a^{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 a-1.

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

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

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

Evaluate

$$\left(a^{5}-1\right)^{2}$$