<s>[INST] What is the value of $f(0)+f(8)$? Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?
 [/INST] The given values are the values of the quadratic polynomial $X^2+1$. Therefore, we have $f(X) - X^2 - 1$ a polynomial of degree $7$ with the seven zeros $1,2,3,4,5,6,7$, so
$$f(X) = c\cdot\prod_{k=1}^7 (X-k) + X^2+1.$$
with an unknown $c$ ($c\neq 0$ if the degree is exactly $7$). Then we have
$$f(8) = c\cdot 7! + 65$$
and
$$f(0) = c\cdot(-1)^7 7! + 1,$$
whence $$f(0) + f(8) = 65 + 1 + c(7! - 7!) = 66.$$
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