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Math Help - factoring of polynomials...

  1. #1
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    factoring of polynomials...

    I havn't done too much factoring, so what I'm wondering is if there is way to get rid of the exponentials in the last two equations?

    n^2-n = n(n-1)
    n^3-n = n(n+1)(n-1)
    n^4-n = n(n^3-1)
    n^5-n = n(n^2+1)(n+1)(n-1)
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by 1+1bob View Post
    n^4-n = n(n^3-1)
    Notice that 1 is a root of n^3-1 hence, n^4-n=n(n-1)(an^2+bn+c).
    n^5-n = n(n^2+1)(n+1)(n-1)
    n^2+1=(n-\imath)(n+\imath) but you may want to factor in \mathbb{R} ? In this case you can't go further (or farther ?) than n^2+1.
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  3. #3
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    Well, I think I have to use R in this assignment:

    "Factorize the expression P(n)=n^x - n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k + 1) - P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x."

    Notice that 1 is a root of hence, .
    What do a,b,c do?

    cause n(n - 1)(n^2 + n) = n^4 - n^2 ?
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  4. #4
    Super Member flyingsquirrel's Avatar
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    You should use c to make both -n^2 disappear and n appear : developing n^4-n=n(n-1)(an^2+bn+c) and identifying the coefficients yields a=b=c=1 hence n^4-n=n(n-1)(n^2+n+1).
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  5. #5
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    Quote Originally Posted by 1+1bob View Post
    Well, I think I have to use R in this assignment:

    "Factorize the expression P(n)=n^x - n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k + 1) - P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x."


    What do a,b,c do?

    cause n(n - 1)(n^2 + n) = n^4 - n^2 ?
    This is an interesting way to pose the question. The problem setter wants you to discover a wonderful idea on your own. I hope other people will not spoil it and tell you the idea.

    "Factorize the expression P(n)=n^x - n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x"

    Lets start with 2,
    1) Does 2 always divide n^2 - n?
    Hint: Try some values to be sure before we try to guess the result.n^2 - n = n(n-1). Look at the parity of n and n-1.

    2) Does 3 always divide n^3 - n?
    Hint: n^3 - n = n(n-1)(n+1).
    Try some values to be sure before we try to guess the result.Consider these questions:What if n was divisible by 3, What if n left a remainder of 1 when divided by 3?, what if n left a remainder of -1?

    3)Does 4 always divide n^4 - n?
    Hint:Lets try some values to be sure before we try to guess the result. if n = 2, 4 does not divide 2^4 - 2

    4) Does 5 always divide n^5 - n?
    Hint: Try some values to be sure before we try to guess the result.Just observe that n^5 and n will have the same units place. So the difference is divisible by 10 and hence 5.

    5) Does 6 always divide n^6 - n?
    Hint: Lets try some values to be sure before we try to guess the result. if n = 2, 6 does not divide 2^6 - 2

    Hmm can you see a pattern? Now can you establish it using induction?
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  6. #6
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    Alright, so I did what you said and found that the stuff will be divisible as long as x is a prime number. I didn't need the factorization for this though, so I assume the factorized form will be used in the proof? Anyways, I'm gonna try to write a proof of this now. Thanks for the help all, I guess I will need more pretty soon though.
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  7. #7
    Lord of certain Rings
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    Quote Originally Posted by 1+1bob View Post
    Alright, so I did what you said and found that the stuff will be divisible as long as x is a prime number. I didn't need the factorization for this though, so I assume the factorized form will be used in the proof? Anyways, I'm gonna try to write a proof of this now. Thanks for the help all, I guess I will need more pretty soon though.
    You are welcome
    So remember you have to prove that p|(n+1)^p - (n+1) using the fact that p|n^p - n for a particular prime p.

    Hint: Does p \bigg{|} {p \choose k}
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