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April 17th, 2015, 07:50 AM  #1 
Newbie Joined: Apr 2015 From: United States Posts: 8 Thanks: 0  Polynomial divisible by Polynomial
Prove that the polynomial $(n^2+2n+1)^3+(n^2+8n+16)^3+(9n^2+42n+49)^3+(9n^2+ 48n+64)^3$is divisible by $2n^2+10n+13$.
Last edited by skipjack; April 18th, 2015 at 10:23 PM. 
April 17th, 2015, 08:29 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2551 Math Focus: Mainly analysis and algebra 
I think I would show that the two (complex) roots of the divisor are also roots of the dividend.

April 17th, 2015, 08:47 AM  #3 
Newbie Joined: Apr 2015 From: United States Posts: 8 Thanks: 0 
I don't know how to do that. Is there a different way?

April 17th, 2015, 09:15 AM  #4 
Newbie Joined: Apr 2015 From: United States Posts: 8 Thanks: 0 
I've actually got it, thank you for your input!

April 18th, 2015, 09:14 PM  #5 
Senior Member Joined: Sep 2010 Posts: 221 Thanks: 20 
It's rather simple. Each pair of the polynomials to the power 3 has the same divisor $(n^2+2n+1)+(9n^2+48n+64)=(n^2+8n+16)+(9n^2+42n+49 )=10n^2+50n+65=5(2n^2+10n+13)$ Last edited by skipjack; April 18th, 2015 at 10:27 PM. 

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