英语翻译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.Suppo

英语翻译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.Suppo
英语翻译
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.)
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)

英语翻译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.Suppo
1、设 S 是实数构成的集合,满足以下条件：
（1）0∈S ；
（2）如果 x∈S ,那么 2^x+3^x ∈S ；
（3）如果 x^2+x^3∈S ,那么 x∈S .
证明：S 至少包含 0、1 之间的两个不同实数.
由（1）得 0∈S ,
由（2）得 2^0+3^0=2∈S ,
令 x^2+x^3=2 ,解得 x=1∈S ,
令 x^2+x^3=1 ,解得 x ≈ 0.75∈S ,
令 x^2+x^3=0.75 ,解得 x ≈ 0.67∈S ,
因此结论成立.（本题实际上是要证明 x^2+x^3=1 有实根 x0 满足 0