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March 30th, 2014, 10:28 PM  #1 
Newbie Joined: Mar 2014 From: Matrix Posts: 1 Thanks: 0  Prove divisibility of polynomial
Prove that: is divisable by I've no idea how to even start.. have spent a couple of hours already Thanks in advance 
March 31st, 2014, 12:51 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory 
Hint : Prove by induction (or otherwise) that is divisible by for all positive integer .

March 31st, 2014, 02:01 AM  #3 
Senior Member Joined: May 2013 From: España Posts: 151 Thanks: 4 
Hello. And, a bit of brute force, would be worth?: Regards 
March 31st, 2014, 03:11 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
If x is either root of x² + x + 1 = 0, x³ = x(x²) = x²  x = 1. Hence x^2012 = x², and so the factor theorem implies that x² + x + 1 divides x^2012 + x + 1. 
March 31st, 2014, 05:04 AM  #5 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory 
You're the man, skipper, as usual. (clap)


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