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Properties of Roots in Univariate Polynomial of Degree n

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data
    1. Let ##p(x) = a_{0} x^{n} + a_{1} x^{n−1} + ... + a_{n} , a_{0} \neq 0 ##be an univariate polynomial of degree n.
    Let r be its root, i.e. p(r) = 0. Prove that
    ## |r| \leq max(1, \Sigma_{1 \leq i \leq n} | \dfrac{a_{i} }{ a_{0} } | )##
    Is it always true that?
    ## |r| \leq \Sigma_{1 \leq i \leq n} | \dfrac{a_{i} }{ a_{0} } | )##

    2. Relevant equations
    ##ar + br^{2} =0 ##
    ## r(a+br)=0 ##
    ## r=0 & r = -a/b##
    3. The attempt at a solution
    The last equation satisfies what they state but I dont know how to proceed?
    Any help?
     
  2. jcsd
  3. Sep 28, 2013 #2

    epenguin

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    For almost all such polynomial probs there is no loss of generality in making a0 = 1 and then you don't have that denominator in the expression.

    Not sure I know now to proceed either but since they bring in 1 as a case of maxima, it is suggestive when you put r = 1 perhaps.

    I don't follow what you are saying with your relevant equations but when you can't see how to do it in general it is certainly good to try with n=1 and 2.
     
  4. Sep 28, 2013 #3
    I sorta proved it by contradiction.
    Let's say that
    ## p(x)= ax^{2}+bx=0##
    ## 0=x(ax+b), \text{ then the roots are }r_{1}=0 \text{, } r_{2}=\dfrac{-b}{a}##
    ## |r| \geq \dfrac{a+b}{a} \\##
    ## | r_{2}|=| \dfrac{-b}{a}| \geq | \dfrac{ a+b}{a}| \text{, will never be true.}##
     
    Last edited: Sep 28, 2013
  5. Sep 28, 2013 #4

    epenguin

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    That I think is not right, you have misunderstood the formula in the original problem. The Ʃ subscript contains a 1≤ not a 0 ≤. The theorem then is rather trivially true for the first degree case with the ≤ it was required to prove reducing to = . Also your factor x seems an irrelevancy. But there is maybe the germ of an argument.

    As I said you avoid something unnecessary if you make, without loss of generality, a0 = 1. So then the question becomes:
    Let ##p(x) = x^{n} + a_{1} x^{n−1} + ... + a_{n} , ##

    then prove

    |r| ≤ max( 1, | a1 + a2 +... + an| ).

    I think my previous idea leads there but maybe extending what you were trying does too.
     
    Last edited: Sep 28, 2013
  6. Sep 28, 2013 #5

    Dick

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    Here's another hint. Since you have the 1 in the max, it's trivial if |r|<=1. So you only have to do some work for the case |r|>=1. You'll want to use the triangle inequality.
     
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