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Math Help - Root Linear Coefficient Theorem

  1. #1
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    Root Linear Coefficient Theorem

    Hi Forum (I'm new)

    Root Linear Coefficient Theorem: "The sum of the reciprocal of the roots of a polynomial equals the negative of the coefficient of the linear term in the Maclaurin series" (Root Linear Coefficient Theorem -- from Wolfram MathWorld)

    I get it equals the negative of the ratio of the linear coefficient to the constant term.
    \frac{1}{r_1}+\frac{1}{r_2}+...+\frac{1}{r_n}=\fra  c{S_{n-1}}{S_{n}}=\frac{(-1)^{n-1}\frac{a_1}{a_n}}{(-1)^{n}\frac{a_0}{a_n}}=-\frac{a_1}{a_0}
    where P(x)=a_nx^n+...+a_1x+a_0=a_n(x-r_1)(x-r_2)...(x-r_n) and I'm using Vieta's Formulas (Vieta's Formulas)

    I'm pretty sure that's right... What went wrong?
    Last edited by sleepingcat; May 8th 2008 at 08:21 PM.
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  2. #2
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    I believe that you require the polynomial itself is monic, i.e., the leading coefficient is 1; otherwise, "the negative of the coeff. of the linear term" may not even be defined. For instance, f(x)=x^2-3x and g(x)=2x^2-6x, both have precisely the same roots, but the coefficients of x are definitely distinct.

    Hope this helps.
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  3. #3
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    Thanks for answering. If the constant term is non-zero, can I assume the sum is -a_1/a_0 then?
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  4. #4
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    Quote Originally Posted by sleepingcat View Post
    Thanks for answering. If the constant term is non-zero, can I assume the sum is -a_1/a_0 then?
    Yes
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