x^8+x+1=(x^8-x^2)+(x^2+x+1)=x^2(x^6-1)+(x^2+x+1)=x^2(x^3-1)(x^3+1)+(x^2+x+1)=x^2(x-1)(x^2+x+1)(x^3+1)+(x^2+x+1)=(x^2+x+1)(x^6-x^5+x^3-x^2)+(x^2+x+1)=(x^2+x+1)(x^6-x^5+x^3-x^2+1)