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March 21st, 2019, 09:42 PM  #1 
Newbie Joined: Mar 2019 From: Romania Posts: 3 Thanks: 0  Polynomials divisibilty problem
If P(x) and Q(x) are two polynomials such that P(x)  P(Q(x)), what are the restrictions for Q such that the statement P (x) = 0 => Q(x) = x to be true (I was thinking the restriction has to be Q to be strictly monotone on Im(Q), but I’m not quite sure)? If the left to right implication is true, furthermore, can someone please help me prove whether the inverse implication ( If P(x) and Q(x) are two polynomials such that P(x)  P(Q(x)), then Q (x) = x => P (x) = 0 ) holds or is actually false? 
March 22nd, 2019, 06:00 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
If $Q(x)=x$ then $P(Q(x))=P(x)$, which is obviously divisible by $P(x)$ regardless of anything else.
Last edited by v8archie; March 22nd, 2019 at 06:03 AM. 
March 22nd, 2019, 06:31 AM  #3 
Newbie Joined: Mar 2019 From: Romania Posts: 3 Thanks: 0 
I agree, however I think you didn’t exactly understood my question. We are given two polynomials, P(X) and Q(X), we know that P(x)  P (Q(x)), but we need to find the restriction for Q(x) such that P(x)=0 to imply that Q(x)=x ( normally, if P(x)P(Q(x)) and P(x)=0, then Q(x) must be one of the roots of P(x), but not necessarily x. We need to find the restriction for Q such that the only possibility to actually be Q(x)=x). If you allow me to rephrase it a bit differently: Let P(X),Q(X) belong to R[X] such that P(X)  P(Q(X)) and let’s consider the sets: A = { x in R  P(x) = 0} , B = {x in R  Q(x) = x}. I’m trying to find a necessary condition for Q such that A = B. 

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