\(x^7+x^2+1\)
\(=\left(x^7-x\right)+\left(x^2+x+1\right)\)
\(=x\left(x^6-1\right)+\left(x^2+x+1\right)\)
\(=x\left(x^3+1\right)\left(x^3-1\right)+\left(x^2+x+1\right)\)
\(=x\left(x^3+1\right)\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left[x\left(x-1\right)\left(x^3+1\right)+1\right]\)
\(=\left(x^2+x+1\right)\left(x^5-x^4+x^2-x+1\right)\)
\(=\left(x^7-x\right)+\left(x^2+x+1\right)\)
\(=x\left[\left(x^3\right)^2-1^2\right]+\left(x^2+x+1\right)\)
\(=x\left(x^3-1\right)\left(x^3+1\right)+\left(x^2+x+1\right)\)
\(=x\left(x^3+1\right)\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left[x\left(x-1\right)\left(x^3+1\right)+1\right]\)
\(=\left(x^2+x+1\right)\left[x^2\left(x^3+1\right)-x\left(x^3+1\right)+1\right]\)
\(=\left(x^2+x+1\right)\left(x^5+x^2-x^4-x+1\right)\)
