Given that f(x) is a polynomial of degree EIGHT such that

f(m) = 1/m for m = 1,2,3,4,5,...,9

Find f(10) 

Answer: 2/10

Solution:

Consider another polynomial, g, defined as g(x) = x f(x) - 1

g(1)=0, g(2)=0, …, g(9)=0

So g(x) = A(x-1)(x-2)…(x-9)

g(0) = 0 f(0) - 1 = -1

also, g(0) = A(-1)(-2)…(-9), so A = 1/9!

g(10) = A(9)(8)…(1) = (1/9!)(9!) = 1

g(10) = 1 = 10 f(10) - 1, so f(10) = 2/10