polinom

$\hspace{-10}P(1)=827\Leftrightarrow P(1)-1\times827=0$\\\\ $P(1)=1654\Leftrightarrow P(1)-2\times827=0$\\\\ $P(1)=2481\Leftrightarrow P(1)-3\times827=0$\\\\ So We can write as\\\\ $P(x)-x\times 827=0$, Where $x=1,2,3$ are the roots of a Polynomial.\\\\ but $P(x)$ is a $4^{th}$ degree polynomial. So It has $4$ Roots.\\\\ Let $x=\delta$ be $4^{th}$ of Given equation.\\\\ $P(x)-x\times 827=(x-1)(x-2)(x-3)(x-\delta)=0$\\\\ $P(x)=(x-1)(x-2)(x-3)(x-\delta)+x\times 827$\\\\ Put $x=9,$\;We Get\\\\ $P(9)=8\times7\times6\times(9-\delta)+9\times 827$\\\\ Similarly Put $x=-5,$\; We Get\\\\ $P(-5)=6\times 7\times8\times(5+\delta)-5\times827$\\\\ $P(9)+p(-5)=6\times7\times8\times14+4\times827$\\\\ $\boxed{\boxed{\frac{P(9)+p(-5)}{4}=6\times7\times2\times14+827=2003}}$$

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