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Thread: Constructing a polynomial -relatively simple

  1. #1
    Junior Member hercules's Avatar
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    Constructing a polynomial -relatively simple

    Construct a single polynomial f(x) in Z5[x] (the set of all polynomials in Z mod 5) such that every element of Z5 = {[0],[1],[2],[3],[4]} is a root of f(x). (Z is the set of integers.)

    Please tell how you can quickly construct a polynomial with such properties
    Thank you.
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    Quote Originally Posted by hercules View Post
    Construct a single polynomial f(x) in Z5[x] (the set of all polynomials in Z mod 5) such that every element of Z5 = {[0],[1],[2],[3],[4]} is a root of f(x). (Z is the set of integers.)

    Please tell how you can quickly construct a polynomial with such properties
    Thank you.
    $\displaystyle f(x) = x(x-1)(x-2)(x-3)(x-4)$
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    Junior Member hercules's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    $\displaystyle f(x) = x(x-1)(x-2)(x-3)(x-4)$

    lol hacker, give me another one please

    Quote Originally Posted by ThePerfectHacker View Post
    $\displaystyle f(x) = x(x-1)(x-2)(x-3)(x-4)$

    I never thought looking at that smilie was this annoying. And thanks for the answer. I was trying to figure if possible what other polynomials met the condition and an easy way for that....wasting my brain.
    Last edited by ThePerfectHacker; May 5th 2008 at 07:32 PM.
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    Quote Originally Posted by hercules View Post
    lol hacker, give me another one please
    f(x) = 0
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    Junior Member hercules's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    f(x) = 0

    hey a polynomial needs more than one term....i can't believe your giving the answer but completely dodging the answers i'm hoping for-the reason i made this thread. But good ones. ....No more smilies please....nightmares.
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    Quote Originally Posted by hercules View Post
    hey a polynomial needs more than one term....i can't believe your giving the answer but completely dodging the answers i'm hoping for-the reason i made this thread. But good ones. ....No more smilies please....nightmares.
    The zero polynomial is still a polynomial.

    For any polynomial $\displaystyle f(x)$ the polynomial $\displaystyle g(x) = x(x-1)(x-2)(x-3)(x-4)f(x)$ will have the desired properties.
    Furthermore, any other such polynomial which contains zeros of $\displaystyle 0,1,2,3,4$ must have this form.
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    Junior Member hercules's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    The zero polynomial is still a polynomial.

    For any polynomial $\displaystyle f(x)$ the polynomial $\displaystyle g(x) = x(x-1)(x-2)(x-3)(x-4)f(x)$ will have the desired properties.
    Furthermore, any other such polynomial which contains zeros of $\displaystyle 0,1,2,3,4$ must have this form.

    Thank you ...needed to clear my misconceptions.
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