x^18+x^15+x^12+x^9+x^6+x^3+1分解因式

x^18+x^15+x^12+x^9+x^6+x^3+1
=(x^3-1)(x^18+x^15+x^12+x^9+x^6+x^3+1)/(x^3-1)
=(x^21-1)/(x^3-1)
=(x^7-1)(x^14+x^7+1)/(x^3-1)
=(x^6+x^5+x^4+x^3+x^2+x+1)(x-1)(x^14+x^7+1)/(x^3-1)
其中(x-1)(x^14+x^7+1)=x^15-x^14+x^8-x^7+x-1
=(x^15-1)-(x^14-x^8)-(x^7-x)
=(x^3-1)(x^12+x^9+x^6+x^3+1)-x^8(x^6-1)-x(x^6-1)
=(x^3-1)[x^12+x^9+x^6+x^3+1-(x^8+x)(x^3+1)]
=(x^3-1)(x^12+x^9+x^6+x^3+ 1-x^11-x^8-x^4-x)
=(x^3-1)(x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1)
∴原式=(x^6+x^5+x^4+x^3+x^2+x+1)(x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1)

等比数列的求和
其中a0=1，q=x^3;
那么
和S=1-x^21/1-x^3

(x^6+x^5+x^4+x^3+x^2+x+1)*(x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1)

=x^18+x^15+x^12+x^9+x^6+x^3+1+x^3-1+5-4
=(x^18+1)+(x^15+1)+(x^12+1)+(x^9+1)+(x^6+1)+(x^3+1)+(x^3-1)-4
=(x^18+1)+(x^15+1)+(x^12+1)+(x^9+1)+(x^6+1)+(x^6-1)-4
=(x^18+1)+(x^15+1)+(x^12+1)+(x^9+1)+(x^9-1)-4
=(x^18+1)+(x^15+1)+(x^12+1)+(x^12-1)-4
=(x^18+1)+(x^15+1)+(x^15-1)-4
=(x^18+1)+(x^18-1)-4
=x^36-5