求(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)的值

求(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)的值
求(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)的值

求(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)的值
原式=(x-1)(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)/(x-1)
=(x²-1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)/(x-1)
反复用平方差
=(x^128-1)/(x-1)

先把最前面乘以（x-1），然后就能做下去了，最后的结果别忘了再除以（x-1）

(x+1)(x^2+1)(x^4+1)(x^8+1)……(x^64+1)
=(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)……(x^64+1)/(x-1)
=(x^2-1)(x^2+1)(x^4+1)(x^8+1)……(x^64+1)/(x-1)
反复用平方差
=(x^128-1)/(x-1)

令t=(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)则
t(x-1)=(x-1)(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)=(x^2-1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)
=---=x^128-1
=>t=（x^128-1）/（x-1）
即(x+1)(...

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令t=(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)则
t(x-1)=(x-1)(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)=(x^2-1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)
=---=x^128-1
=>t=（x^128-1）/（x-1）
即(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)=（x^128-1）/（x-1）（x不等于1）
当x=1时，(x+1)(x²+1)（x^4+1）(x^8+1)...(x^64+1)=2^6=64

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