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Some Algebra II

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Let $$P(x)$$ be a polynomial with degree 2008 and leading coefficient 1 such that

$$P(0)=2007, P(1)=2006, P(2)=2005, ... , P(2007)=0$$

What is the value of $$P(2008)$$ ?

I don't really understand how to solve it all, so help would be greatly appreciated.

Thank you very much!

:P

Dec 16, 2018
edited by CoolStuffYT  Dec 16, 2018

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I don't know why yet.

Dec 16, 2018
edited by Rom  Dec 16, 2018
edited by Rom  Dec 16, 2018
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$$\text{consider the polynomial } \\ q(x) = p(x) - (d-1-x)\\ q(x) = 0,~\forall x = k=0,1,\dots ,(d-1)\\ q(x) = \prod \limits_{k=0}^{d-1}~(x-(d-1-k))\\ \text{now let }d=2008 \\ q(x) = \prod \limits_{k=0}^{2007}~(x-2007+k)\\ p(2008) = q(2008) +(2007-2008) = q(2008)-1\\ q(2008) = \prod \limits_{k=0}^{2007}~k+1 = 2008!\\ p(2008)=2008!-1$$

Many thanks to Walagaster over at MHF for the idea of q(x)

Rom  Dec 16, 2018
edited by Rom  Dec 16, 2018
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Rom, I'm not sure what that bold pi looking thing is. Could you explain that?

CoolStuffYT  Dec 25, 2018
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∑ - This is the capital Greek letter "sigma" for "summation".
∏ - This is the capital Greek letter "Pi" for "Product of"

Guest Dec 25, 2018
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That is the Product of Sequences symbol, sometimes known as “Big Pi;” it means to multiply.

It’s analogous to the Summation symbol for addition.

Why did you wait until the eleventh hour to reply? This post is about to lock, and Rom will not be able to answer your question without messaging one of the Moderators to unlock it. That’s a big pain in the ass for all parties.  You are probably not worth the trouble.  If you care enough to ask the question, you should care about the solution enough to ask follow up questions within a reasonable time.

GA

GingerAle  Dec 25, 2018
edited by GingerAle  Dec 25, 2018