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Math Help - Fraction decomposition continued

  1. #1
    Junior Member Greymalkin's Avatar
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    Fraction decomposition continued

    Decompose:
    {x\over {x^4-a^4}}

    answer is:

    {-{{{1\over 2a^2}x}\over {x^2+a^2}}}+{{{1\over 4a^2}\over {x-a}}+{{1\over 4a^2}\over {x+a}}

    I solve for b= {a\over 4a^3}={1\over {4a^2}}, however i can't seem to solve for a or c.(if x=-a then -a^2 and -a makes all factors =0)
    I try using the quadratic method but i end up with equations that are cubic.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Greymalkin View Post
    Decompose:
    {x\over {x^4-a^4}}
    answer is: {-{{{1\over 2a^2}x}\over {x^2+a^2}}}+{{{1\over 4a^2}\over {x-a}}+{{1\over 4a^2}\over {x+a}}
    I solve for b= {a\over 4a^3}={1\over {4a^2}}, however i can't seem to solve for a or c.(if x=-a then -a^2 and -a makes all factors =0) I try using the quadratic method but i end up with equations that are cubic.
    What exactly is the question? I don't understand what you are asking.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Greymalkin View Post
    Decompose:
    {x\over {x^4-a^4}}

    answer is:

    {-{{{1\over 2a^2}x}\over {x^2+a^2}}}+{{{1\over 4a^2}\over {x-a}}+{{1\over 4a^2}\over {x+a}}

    I solve for b= {a\over 4a^3}={1\over {4a^2}}, however i can't seem to solve for a or c.(if x=-a then -a^2 and -a makes all factors =0)
    I try using the quadratic method but i end up with equations that are cubic.
    \displaystyle \begin{align*} \frac{Ax + B}{x^2 + a^2} + \frac{C}{x - a} + \frac{D}{x + a} &\equiv \frac{x}{x^4 - a^4} \\ \frac{(Ax + B)(x - a)(x + a) + C\left(x^2 + a^2\right)(x + a) + D\left(x^2 + a^2\right)(x - a)}{\left(x^2 + a^2\right)(x - a)(x + a)} &\equiv \frac{x}{x^4 - a^4} \\ (Ax + B)(x - a)(x + a) + C\left(x^2 + a^2\right)(x + a) + D\left(x^2 + a^2\right)(x - a) &\equiv x \end{align*}

    Let \displaystyle \begin{align*} x = a \end{align*} to find \displaystyle \begin{align*} 4a^3 C = a \implies C = \frac{1}{4a^2} \end{align*}

    Let \displaystyle \begin{align*} x = -a \end{align*} to find \displaystyle \begin{align*} -4a^3 D = -a \implies D = \frac{1}{4a^2} \end{align*}

    Substituting these values into the equation, we have

    \displaystyle \begin{align*} (Ax + B)(x - a)(x + a) + \frac{1}{4a^2}\left(x^2 + a^2\right)(x + a) + \frac{1}{4a^2}\left(x^2 + a^2\right)(x - a) &\equiv x \end{align*}

    So letting x equal any two numbers you like (say 0 and 1) will give you two equations to solve simultaneously for A and B.
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    Junior Member Greymalkin's Avatar
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    Re: Fraction decomposition continued

    Quote Originally Posted by Plato View Post
    What exactly is the question? I don't understand what you are asking.
    Apologies, had I not broken the cardinal rule of math help for not showing my work you would have understood. Originally my work did not have 4 variables.

    It was along the lines of:

    \frac {A}{x^2+a^2} + \frac {B}{x-a} + \frac {C}{x+a}
    Last edited by Greymalkin; August 14th 2012 at 03:04 PM.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Prove It View Post
    Let \displaystyle \begin{align*} x = -a \end{align*} to find \displaystyle \begin{align*} -4a^3 D = -a \implies D = \frac{1}{4a^2} \end{align*}
    Doesnt -a make D(x^2+a^2)(x-a) = D=D(-a^2+a^2)(-a-a)???
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    Re: Fraction decomposition continued

    Quote Originally Posted by Greymalkin View Post
    Doesnt -a make D(x^2+a^2)(x-a) = D=D(-a^2+a^2)(-a-a)???
    No, if \displaystyle \begin{align*} x = -a \end{align*} then \displaystyle \begin{align*} x^2 = (-a)^2 = a^2 \end{align*}.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Greymalkin View Post
    It was along the lines of:
    \frac {A}{x^2+a^2} + \frac {B}{x-a} + \frac {C}{x+a}
    I don't know why we still teach this topic. I consider is an obsolete topic.
    Look at this.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Plato View Post
    I don't know why we still teach this topic. I consider is an obsolete topic.
    Look at this.
    I consider it highly relevant.
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    Junior Member Greymalkin's Avatar
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    Re: Fraction decomposition continued

    Quote Originally Posted by Plato View Post
    I don't know why we still teach this topic. I consider is an obsolete topic.
    Look at this.

    Yeah I usually use wolfram for finding zeros of high degree polynomials, but only after I forced myself to learn the rational root theorem. I am self teaching myself math so I'm not sure what I'll need what for, so I choose to learn it anyways(Havent started calculus yet, but my goal is to do econometrics). A method that, although time consuming, has proven fruitful in insight. My goal is to have a deep understanding of math, not just to "get by", as I suppose most students tend to do(I used to be one).
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    Re: Fraction decomposition continued

    Quote Originally Posted by Prove It View Post
    No, if \displaystyle \begin{align*} x = -a \end{align*} then \displaystyle \begin{align*} x^2 = (-a)^2 = a^2 \end{align*}.
    How can it be positive if it has not yet been squared though? Isn't that like saying x^2=-x^2, the graphs are reflections of one another. Is it because "a" is a constant and not a variable in cartesian coordinates, so therefore it cannot have a domain/range relationship? think I might have answered my own question but please correct me if I'm wrong.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Prove It View Post
    I consider it highly relevant.
    It is relevant only if one is concerned with teaching basic algebra.
    How is it relevant to a calculus course? In fact, I predict that within five to ten years one will hard pressed to find a calculus textbook that includes a chapter on techniques of integration. I have taught calculus on and off since 1964. I never thought that technology would replace basics techniques. But it has and will continue to do so. Partial fractions are gone as another victim of technology.
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    Re: Fraction decomposition continued

    Quote Originally Posted by Plato View Post
    It is relevant only if one is concerned with teaching basic algebra.
    How is it relevant to a calculus course? In fact, I predict that within five to ten years one will hard pressed to find a calculus textbook that includes a chapter on techniques of integration. I have taught calculus on and off since 1964. I never thought that technology would replace basics techniques. But it has and will continue to do so. Partial fractions are gone as another victim of technology.
    I find it hard to believe that you would believe that it SHOULD be replaced though...
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    Re: Fraction decomposition continued

    Quote Originally Posted by Greymalkin View Post
    How can it be positive if it has not yet been squared though? Isn't that like saying x^2=-x^2, the graphs are reflections of one another. Is it because "a" is a constant and not a variable in cartesian coordinates, so therefore it cannot have a domain/range relationship? think I might have answered my own question but please correct me if I'm wrong.
    What happens when you square a negative value (i.e. multiply a negative value by itself - another negative value)? It becomes positive...
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