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