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 April 17th, 2015, 06:50 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 09:23 PM.
 April 17th, 2015, 07:29 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,617 Thanks: 2608 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, 07:47 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, 08:15 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, 08:14 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 09:27 PM.

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