Let $p(x)$ be an eighth degree polynomial. Given that:
$p(1) = 1$, $p(2) = 1/2$, $p(3) = 1/3$, .... $p(9) = 1/9$
Find the value of $p(10)$. I tried to approach this by taking $h(x) = p(x) - 1/x$ but then $h(x)$ won't be a polynomial. I tried to manipulate the equation but I failed to find anything helpful. I have found a relation between the coefficients and some values; 9 equations and 9 variables but it would be too cumbersome to solve.
Some guidance would be appreciated. Thanks! (Sorry for the poor title; if anyone can improve it please do so)