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March 30th, 2014, 10:28 PM   #1
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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
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March 31st, 2014, 12:51 AM   #2
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Hint : Prove by induction (or otherwise) that is divisible by for all positive integer .
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March 31st, 2014, 02:01 AM   #3
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Hello.

And, a bit of brute force, would be worth?:
















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March 31st, 2014, 03:11 AM   #4
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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.
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March 31st, 2014, 05:04 AM   #5
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You're the man, skipper, as usual. (clap)
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