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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. Tags divisibilty, polynomials, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post miket Elementary Math 0 May 10th, 2014 05:12 AM earth Algebra 3 September 4th, 2010 01:13 PM Boka Abstract Algebra 1 June 4th, 2010 04:57 AM ElMarsh Linear Algebra 3 October 15th, 2009 04:14 PM momo Number Theory 4 July 23rd, 2008 05:58 PM

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