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April 17th, 2015, 06:50 AM   #1
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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.
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April 17th, 2015, 07:29 AM   #2
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I think I would show that the two (complex) roots of the divisor are also roots of the dividend.
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April 17th, 2015, 07:47 AM   #3
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I don't know how to do that. Is there a different way?
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April 17th, 2015, 08:15 AM   #4
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I've actually got it, thank you for your input!
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April 18th, 2015, 08:14 PM   #5
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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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