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Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and while its easy to remember and its possible for a person to use this method by hand, there are many better techniques available (which do not rely on factoring a polynomial).

So I was wondering, why on earth is it actually important to have techniques available to solve polynomial equations? (to be specific, I mean solving over $\mathbb{C}$)

I actually used to be fairly interested in how to do it, and I know a lot of the different methods that people use. I was just thinking about it though, and I'm actually not sure what sort of applications there are for those techniques.

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    Solving polynomial equations is indeed an addiction. Good to see that you have gotten over it. To actually help others, though, general statements on a website are not enough, it must be personal support, perhaps in small gatherings where people discuss their struggles with addiction.2012-02-13
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    Are you talking about exact solutions in radicals, or approximate numerical solutions?2012-02-13
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    @Will Nice! Do you offer courses?2012-02-13
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    @AlexB., thanks, I do my best. It is easier in writing, in front of a classroom I used to just get angry.2012-02-13
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    If I understood correctly. Polynomial systems, for one, have many industrial applications. [This book](http://www.amazon.ca/Linear-Systems-Thomas-Kailath/dp/0135369614) is a good reference.2012-02-13
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    Polynomial systems also arise in Computer Aided Geometric Design.2012-02-13

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