# Math Help - polynomial

1. ## polynomial

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

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

Find f(10)

2. Originally Posted by perash
Given that f(x) is a polynomial of degree EIGHT such that

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

Find f(10)
let $g(t)=tf(t) - 1.$ so $g$ is a polynomial of degree 9 with 9 roots t = 1, 2, ... , 9. thus $tf(t)-1=k(t-1)(t-2) \ ... \ (t-9), \ \ \ \ (1)$

for some constant $k.$ now in (1) put t = 0 to to get: $k=\frac{1}{9!}.$ therefore (1) becomes: $tf(t)=\frac{1}{9!}(t-1)(t-2) \ ... \ (t-9)+1. \ \ \ \ \ \ (2)$

finally in (2) put t = 10 to get: $10f(10)=\frac{1}{9!} \times 9! + 1 = 2,$ which gives us: $f(10)=\frac{1}{5}. \ \ \ \square$

Remark: it's clear that the problem can be easily generalized in different ways and the above method will still work quite nicely!