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Math Help - Functions

  1. #1
    Member integral's Avatar
    Joined
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    Arkansas
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    Functions

    If you have
    f(x)=x^{2}+3
    I know f(x) just means the function of x, or just y.

    But take this:

    If
    \frac{f(x)}{x-k}
    then
    r=f(k)
    Where r= remainder
    This makes no since to me. some function divided by x-k yields the function of k.

    How do you come up with a function?

    Edit: normally f(x),f(c),f(g) are all put in the place of possible functions, but this one is not.
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  2. #2
    Newbie
    Joined
    Jan 2010
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    Quote Originally Posted by integral View Post
    If
    \frac{f(x)}{x-k}
    then
    r=f(k)
    Where r= remainder
    This makes no since to me. some function divided by x-k yields the function of k.
    This statement is deduced based on the Polynomial Remainder Theorem. That is, if you have a function f(x), and you divide it with another function x - c, then the remainder of the division is equals to f(c). Proof:

    Let's say you have a function f(x) = (x - c)q(r) + r(x), where q(r) is the quotient (the result of dividing with x - c) and r(x) is the remainder (after dividing with x - c). According to the Polynomial Division Algorithm, the remainder of the division must have a smaller degree than the divisor, in this case, x - c. Since x - c has a degree of 1, r(x) must be a constant polynomial.

    Now, if you substitute x = c into the function, you'll end up with:

    f(c) = (c - c)q(r) + r(c)
    f(c) = (0)q(r) + r(c)
    f(c) = r(c)

    Therefore, the remainder is equals to f(c).
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