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Math Help - Use identity to prove result about polynomials

  1. #1
    r45
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    Use identity to prove result about polynomials

    Hi there,

    I am stuck on the following problem:

    If f(x) is a polynomial and a is a number, prove that there exists a polynomial g(x) such that f(x) = (x - a)g(x) + f(a)

    Hint: use the identity x^n -a^n = (x - a)(x^{n-1} + ax^{n-2} + ... + a^{n-2}x + a^{n-1})

    I can't really see how to apply the hint. Apart from the obvious inclusion of (x - a), I'm out of ideas.

    Many thanks!
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  2. #2
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    A further hint

    Let f(x)=b_nx^n+b_{n-1}x^{n-1}+\ldots+b_2x^2+b_1x+b_0. Now look at f(x)-f(a).

    --Kevin C.
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  3. #3
    r45
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    Aha, gotcha! Very nice indeed. Thankyou!
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