.在有理数范围内分解因式x^{12}+x^9+x^6+x^3+1.？
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x^{12}+x^9+x^6+x^3+1
=(x^3-1)(x^12+x^9+x^6+x^3+1)/(x^3-1)
=(x^15-1)/(x^3-1)
=(x^5-1)(x^10+x^5+1)/(x^3-1)
=(x-1)(x^4+x^3+x^2+x+1)(x^10+x^5+1)/(x^3-1)
=(x^4+x^3+x^2+x+1)(x^11-x^10+x^6-x^5+x-1)/(x^3-1)
其中x^11-x^10+x^6-x^5+x-1
=(x^11-x^5)-(x^10-x)+(x^6-1)
=(x^6-1)(x^5+1)-x(x^9-1)
=(x^3-1)(x^3+1)(x^5+1)-x(x^3-1)(x^6+x^3+1)
=(x^3-1)[(x^8+x^5+x^3+1)-(x^7+x^4+x)]
=(x^3-1)(x^8-x^7+x^5-x^4+x^3-x+1)
∴原式=(x^4+x^3+x^2+x+1)(x^8-x^7+x^5-x^4+x^3-x+1),6,
=(x^3-1)(x^12+x^9+x^6+x^3+1)/(x^3-1)
=(x^15-1)/(x^3-1)
=(x^5-1)(x^10+x^5+1)/(x^3-1)
=(x-1)(x^4+x^3+x^2+x+1)(x^10+x^5+1)/(x^3-1)
=(x^4+x^3+x^2+x+1)(x^11-x^10+x^6-x^5+x-1)/(x^3-1)
其中x^11-x^10+x^6-x^5+x-1
=(x^11-x^5)-(x^10-x)+(x^6-1)
=(x^6-1)(x^5+1)-x(x^9-1)
=(x^3-1)(x^3+1)(x^5+1)-x(x^3-1)(x^6+x^3+1)
=(x^3-1)[(x^8+x^5+x^3+1)-(x^7+x^4+x)]
=(x^3-1)(x^8-x^7+x^5-x^4+x^3-x+1)
∴原式=(x^4+x^3+x^2+x+1)(x^8-x^7+x^5-x^4+x^3-x+1),6,
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