化简:(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1).
来源:学生作业学帮网 编辑:学帮网 时间:2024/06/21 06:15:34
化简:(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1).
原式=(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)/(x-1)
=(x^2-1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)/(x-1)
=(x^4-1)(x^4+1)(x^8+1)(x^16+1)/(x-1)
=(x^8-1)(x^8+1)(x^16+1)/(x-1)
=(x^16-1)(x^16+1)/(x-1)
=(x^32-1)/(x-1)
原式=[(x-1)(x+1)(x²+1)(x^4+1)(x^8+1)(x^16)+1]/(x-1) 【连续使用平方差公式】
=[x^32-1]/(x-1)
将该式当做一个分子为原式、分母为1的分式,分子分母同乘以(x-1)可化简得答案。
(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)
=(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)/(x-1)
=(x^2-1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)/(x-1)
=……
=(x^32-1)/(x-1)
(x-1)/(x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)/(x-1).
=(x^2-1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)
=(x^4-1)(x^4+1)(x^8+1)(x^16+1)
=)(x-81)(x^8+1)(x^16+1)
=(x^16-1)(x^16+1)
=(x^32+1)
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