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 Post subject: PolynomialPosted: Thu, 5 Aug 2010 08:22:52 UTC
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A polynomial f is given. All we know about it is that all its coefficients are non-negative integers, f(1) = 6 and F(7)=3438. What is the value of f(3)?

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 Post subject: Re: PolynomialPosted: Thu, 5 Aug 2010 09:27:50 UTC
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CrowsZero wrote:
A polynomial f is given. All we know about it is that all its coefficients are non-negative integers, f(1) = 6 and F(7)=3438. What is the value of f(3)?

Think you mean f(7)=3438, not F(7)=3438.

Just bound, and fit.

Spoiler:
From f(7)=3438 you know f is at most a quartic. Also, since f(1)=6<7, you know you only need to covert to base 7 and read off the answer. Now , so , hence f(3)=166.

_________________

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 Posted: Tue, 10 Aug 2010 02:14:05 UTC
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Joined: Tue, 10 Aug 2010 02:11:46 UTC
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I'm sorry I don't understand the reply given. Could you be more explicit? Thanks.

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 Post subject: Posted: Tue, 10 Aug 2010 03:01:08 UTC
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where all coefficients are non-negative integers.

The smallest possible quintic would be , but that makes f(7) = 16807, which is too high, so f(x) must be of order 4 or lower.

f(1)=6, so a+b+c+d+e=6, so no coefficient exceeds 6.

f(7)=3438, so . Because all coefficients are in {0,1,2,3,4,5,6}, solving this is equivalent to finding 3438 in base 7.

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 Post subject: Posted: Tue, 10 Aug 2010 03:13:53 UTC
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Joined: Tue, 10 Aug 2010 02:11:46 UTC
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Thanks, Aswoods. Rafick

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