分解因式：（1-7t-7t²-3t³）（1-2t-2t²-t³）-（t+1）⁶
2013-11-27 · 知道合伙人教育行家
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解：
（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-(t+1)^6
=（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-[(t+1)^3]^2
=（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-[t^3+3t^2+3t+1]^2
这里再化简有点复杂
不妨设t+t^2=m,t^3=n，
原式
=(1-7m-3n)(1-2m-n)-(3m+n+1)^2
=(3n²誉兄+7mn-n+6mn+14m²-2m-3n-7m+1)-(n²+6mn+9m²+2n+6m+1)
=3n²+13mn+14m²-4n-9m+1- n²-6mn-9m²-2n-6m-1
=2n²+7mn+5m²-6n-15m
=(2n²+7mn+5m²)-(6n+15m)
=(2n+5m)(n+m)-3(2n+5m)
=(2n+5m)(n+m-3)
=(2t³+5t²+5t)(t³+t²+t-3)
=t(2t²激稿+5t+5)[(t³明虚孝-1)+(t²-1)+(t-1)]
=t(2t²+5t+5)[(t-1)(t²+t+1)+(t-1)(t+1)+(t-1)]
= t(2t²+5t+5)(t-1)(t²+2t+3)
（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-(t+1)^6
=（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-[(t+1)^3]^2
=（1-7t-7t^2-3t^3)(1-2t-2t^2-t^3)-[t^3+3t^2+3t+1]^2
这里再化简有点复杂
不妨设t+t^2=m,t^3=n，
原式
=(1-7m-3n)(1-2m-n)-(3m+n+1)^2
=(3n²誉兄+7mn-n+6mn+14m²-2m-3n-7m+1)-(n²+6mn+9m²+2n+6m+1)
=3n²+13mn+14m²-4n-9m+1- n²-6mn-9m²-2n-6m-1
=2n²+7mn+5m²-6n-15m
=(2n²+7mn+5m²)-(6n+15m)
=(2n+5m)(n+m)-3(2n+5m)
=(2n+5m)(n+m-3)
=(2t³+5t²+5t)(t³+t²+t-3)
=t(2t²激稿+5t+5)[(t³明虚孝-1)+(t²-1)+(t-1)]
=t(2t²+5t+5)[(t-1)(t²+t+1)+(t-1)(t+1)+(t-1)]
= t(2t²+5t+5)(t-1)(t²+2t+3)
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