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 polynomial problem 3 (Posted on 2016-10-08)
Let P(x) be a nonzero polynomial such that

(x-1)P(x+1) = (x+2)P(x)

for every real x, and P(2)^2 = P(3).

Then find P(2016).

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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 D4? (spoiler) | Comment 1 of 2
p(x+1) = (x+2)*p(x)/(x-1)

when x=2

p(3) = 4*p(2) = p(2)^2

p(2) = 4

p(3) = 16

Programming the transition from p(x) to p(x+1):

5   open "poly3.txt" for output as #2
10   X=3:P=16
20   for Newx=4 to 2016
30      P=P*(X+2)//(X-1)
40      X=Newx:print X,P
45      print #2,X,P
50   next
60   print X,P
70   close #2

Adding third differences manually, shows a 3rd degree polynomial:

` x  P(x)       differences 4   40  5   80      40 6   140     60    20 7   224     84    24    4 8   336    112    28    4 9   480    144    32    4 10   660   180    36    4 11   880  12   1144  13   1456  14   1820  15   2240  16   2720  17   3264  18   3876  19   4560  20   5320 ... 2007   5389528224  2008   5397588336  2009   5405656480  2010   5413732660  2011   5421816880  2012   5429909144  2013   5438009456  2014   5446117820  2015   5454234240  2016   5462358720 `

 Posted by Charlie on 2016-10-08 09:32:36

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