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Math Help - polynomial

  1. #1
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    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)
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  2. #2
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    Quote Originally Posted by perash View Post
    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!
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