Factor 1+a^10-2a^5

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

Rearrange terms.

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

Rewrite $1$ as $1^{2}$ .

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

Rewrite $a^{10}$ as ${(a^{5})}^{2}$ .

$1^{2} - 2 a^{5} + {(a^{5})}^{2}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a^{5} = 2 \cdot 1 \cdot a^{5}$

Rewrite the polynomial.

$1^{2} - 2 \cdot 1 \cdot a^{5} + {(a^{5})}^{2}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = a^{5}$ .

${(1 - a^{5})}^{2}$