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Thread: Polynomial Question

  1. #1
    Newbie
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    Polynomial Question

    Hey there,

    I was reading an article, where the author claims that the following polynomial:
    $\displaystyle x^{2}-(1+\frac{1}{\theta }+\frac{1}{\theta\alpha })x+\frac{1}{\theta }=0
    $

    has two roots, first one root is $\displaystyle \lambda $, which is between 0 and 1, and the other can be expressed as $\displaystyle \frac{1}{\theta\lambda }$

    note that this polynomial is obtained after factoring a second order difference equation with forward and lag operators. Now I have no idea how he arrives at that identity, could anyone explain me the trick behind it?

    And while you are at it, could you tell me a similar identity between the roots of the following polynomial:

    $\displaystyle x^{2}-(1+\frac{1}{\beta }+\frac{\gamma\kappa }{\beta })x+\frac{1}{\beta }=0$

    So for example let $\displaystyle \lambda $, be the root between 0 and 1 of this polynomial, what is the second root in terms of $\displaystyle \lambda $?

    Thanks a lot

    Burak
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ichoosetonotchoosetochoos View Post
    Hey there,

    I was reading an article, where the author claims that the following polynomial:
    $\displaystyle x^{2}-(1+\frac{1}{\theta }+\frac{1}{\theta\alpha })x+\frac{1}{\theta }=0
    $

    has two roots, first one root is $\displaystyle \lambda $, which is between 0 and 1, and the other can be expressed as $\displaystyle \frac{1}{\theta\lambda }$

    note that this polynomial is obtained after factoring a second order difference equation with forward and lag operators. Now I have no idea how he arrives at that identity, could anyone explain me the trick behind it?

    And while you are at it, could you tell me a similar identity between the roots of the following polynomial:

    $\displaystyle x^{2}-(1+\frac{1}{\beta }+\frac{\gamma\kappa }{\beta })x+\frac{1}{\beta }=0$

    So for example let $\displaystyle \lambda $, be the root between 0 and 1 of this polynomial, what is the second root in terms of $\displaystyle \lambda $?

    Thanks a lot

    Burak
    The constant term in a monic quadratic (one that has coefficien of $\displaystyle $$ x^2$ equal to $\displaystyle $$ 1$) is the product of the two roots. So for a monic quadratic if the constant term is $\displaystyle $$ c$ and one root is $\displaystyle $$ \lambda$ then the other root is $\displaystyle c/\lambda$

    CB
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