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Thread: Polynomial regression

  1. #1
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    Polynomial regression

    we have a monic polynomial $p(x)$ of degree $d$ such that

    $p(k) = d-k-1,~\forall k = 0,1,\dots, d-1$

    Show $p(d) = d!-1$
    Last edited by romsek; Dec 16th 2018 at 11:34 AM.
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  2. #2
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    Re: Polynomial regression

    Quote Originally Posted by romsek View Post
    we have a monic polynomial $p(x)$ of degree $d$ such that

    $p(k) = d-k-1,~\forall k = 0,1,\dots, d-1$

    Show $p(d) = d!-1$
    Hint: Think about the polynomial $q(x) = p(x) - (d-1-x)$ and its values for $k=0,1,2,...,d-1)$.
    Thanks from romsek
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  3. #3
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    Re: Polynomial regression

    Quote Originally Posted by Walagaster View Post
    Hint: Think about the polynomial $q(x) = p(x) - (d-1-x)$ and its values for $k=0,1,2,...,d-1)$.
    $q(x) = p(x) - (d-1-x)$

    $q(x) = 0,~\forall x = k=0,1,\dots ,(d-1)$

    $q(x) = \prod \limits_{k=0}^{d-1}~(x-(d-1-k))$

    now let $d=2008$

    $q(x) = \prod \limits_{k=0}^{2007}~(x-2007+k)$

    $p(2008) = q(2008) +(2007-2008) = q(2008)-1$

    $q(2008) = \prod \limits_{k=0}^{2007}~k+1 = 2008!$

    $p(2008)=2008!-1$
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