几道数学题,不需要翻译只需要过程,着急,感激涕零,不知所云.1.Let S be a set of real numbers that satisfy the following conditions:0 is in S; Whenever x is in S then 2^x+3^x is in S; whenever x^2+x^3 is in S then x is in S.Prove:
来源：学生作业帮助网 编辑：六六作业网 时间：2024/12/23 23:31:07
几道数学题,不需要翻译只需要过程,着急,感激涕零,不知所云.1.Let S be a set of real numbers that satisfy the following conditions:0 is in S; Whenever x is in S then 2^x+3^x is in S; whenever x^2+x^3 is in S then x is in S.Prove:
几道数学题,不需要翻译只需要过程,着急,感激涕零,不知所云.
1.Let S be a set of real numbers that satisfy the following conditions:
0 is in S; Whenever x is in S then 2^x+3^x is in S; whenever x^2+x^3 is in S then x is in S.
Prove: S contains at least two distinct real numbers between 0 and 1.
2.Suppose p(x) is a polynomial with integer coefficients. Show that if p(a)=1for some integer a then p(x) has at most two integer roots(that is, there are at most two integers b and c such that p(b)=0 and p(c)=0.)
算了，我直接先给翻译一下吧。。。
1. S为实数集合并有以下性质
0为S中的一个元素；
若x在S中，那么2^x+3^x也在S中；
若x^2+x^3在S中，那么x也在S中。
证明：S至少含有2个介于0和1之间的不同的实数元素。
2.假设p（x）是整系数多项式。证明若对于整数a，p(a)=1， 则p(x)至多含有两个整数根。（即至多存在两个整数b,c 使p(b)=p(c)=0)
几道数学题,不需要翻译只需要过程,着急,感激涕零,不知所云.1.Let S be a set of real numbers that satisfy the following conditions:0 is in S; Whenever x is in S then 2^x+3^x is in S; whenever x^2+x^3 is in S then x is in S.Prove:
第1题的证明如下：
因为0属于S,所以2^0+3^0=2也是S中的元素.
令x^2+x^3=2可得唯一解x=1,[因为0=x^3+x^2-2=(x-1)(x^2+2x+2)]
这说明1也是S中的元素.
令x^2+x^3=1,
构造函数f(x)=x^2+x^3-1,
因为f(0)=-1,f(1)=1,由连续函数的介值定理可知,f(x)在（0,1）上有且仅有一个零点,不妨设为a.
所以a^3+a^2=1.
令x^2+x^3=a,
显然a不是上述方程x^2+x^3=a的实根（因为a^3+a^2=1>a）,
构造函数g(x)=x^2+x^3-a,
因为g(0)=-a0
所以g(x)在（0,1）上有且仅有一个零点b,且a与b相异,且a,b均属于S.（证毕）