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Given the equation, $$\sum _{i=0}^{n}a_{i}x^{i}=\sum _{j=1}^{m}b_{j}x^{-j}$$ solve for x. I'd typically solve this by multiplying across by $x^{m}$, then shifting everything to one side and finding the roots: $$x^{m} \cdot \sum _{i=0}^{n}a_{i}x^{i}=x^{m} \cdot \sum _{j=1}^{m}b_{j}x^{-j}$$ $$\sum _{k=0}^{m+n}a_{k}x^{k}=0$$ Keeping in mind that $x\neq 0$, the roots are the solutions to the problem (neglect the fact that polynomials of degree > 5 have no closed-form expression for their roots). Question: is there some approach that capitalizes on the structure of the equation to find the values of x?

In one example, two solutions are obvious: $$x^2+x=\frac{1}{x}+\frac{1}{x^2} \rightarrow x=\pm1$$ but that is by inspection. In another example: $$x+2=-\frac{1}{x}$$ $$x \cdot \left(x+2+\frac{1}{x} \right)=0$$ $$ x^{2}+2x+1=0$$ $$ x=-1$$ but that is the typical way (convert problem into root-finding problem). I wonder if this kind of problem has a general approach. I've considered performing substitutions, and borrowing concepts from complex analysis but I'm very rusty and don't recall any that would facilitate "finding x."

Edit: Given that this problem is essentially a re-written polynomial, the answer may already be here (How to solve an nth degree polynomial equation)

Eric
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    I would be very surprised if a general solution were to exist. If there was a way, couldn't we convert arbitrary polynomials into the form of this question and then solve them? We know from the answer to the question you linked that this is imposible. – Polygon Jun 29 '20 at 02:53
  • Just got through having a discussion with a mathematician friend of mine. I agree. It hurts though :( – Eric Jun 29 '20 at 03:19

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Based on the answer to the linked question - it seems that there is no general approach to this problem if the resulting polynomial is of degree > 4.

Eric
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This is one of the famous consequences of the deep investigation into roots of polynomials that is now called "Galois theory". But to be precise it needs to be stated carefully.

For any polynomial P(z) of degree equal to 1, 2, 3, or 4, there is a general formula for the roots of any polynomial of that degree, which involves the coefficients of P and finitely many arbitrary constants using any of the four arithmetic operations as well as possibly taking integer roots. It's been proven that such a general formula cannot exist for all polynomials of any degree that is at least 5.

This discovery is usually called the "Abel-Ruffini Theorem". The first proof believed to be complete was published by Abel in 1824.

Dan Asimov
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