# Thread: Polynomial regression

1. ## 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 $p(d) = d!-1$

2. ## Re: Polynomial regression

Originally Posted by romsek
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 $k=0,1,2,...,d-1)$.

3. ## Re: Polynomial regression

Originally Posted by Walagaster
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 $d=2008$

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

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

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

$p(2008)=2008!-1$