I am in high school and currently studying polynomials. This includes factorisation, finding zeroes and graphing. However I have realised that all the polynomials given to us usually have discrete zeroes ( such as $1,-1,2,-2$ and so on ). This makes them quite easy to find using the factor remainder theorem. However if a polynomial has roots which are not discrete ($\sqrt2,2.243$) it can be quite difficult, maybe even impossible to find their roots using the theorem. I have heard of two other methods of approximating the roots: The Newton Method and The Method Of Bisection. If I am correct, these two methods do not return exact values and take some time to implement on large polynomials. Does there exist a more efficient method which can return exact values and in turn allow for the factorisation, by hand, of a difficult polynomial? Any input is much appreciated. :)
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Check this: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem – Panglossian Oporopolist Mar 12 '17 at 03:31
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@Kugelblitz So from what I have understood, if a polynomial has arbitrary coefficients and it is of degree 5 or higher, it cannot be factored. Is this correct? – billy606 Mar 12 '17 at 03:40
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1You already pointed out something in the right direction. Moreover, you can check the Laguerre Method. – Felix Marin Mar 12 '17 at 05:03
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@FelixMarin Very interesting... I applyed that method on a simple polynomial which i know the roots of and it obtained the exact value after one iteration. This method would work like magic with a bit of C programming. – billy606 Mar 12 '17 at 05:46
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I've given an example here of an exact method to find quadratic factors of quartic polynomials, this can be generalized to find nth degree factors of mth degree polynomials, however, you then end up with having to find integer solutions of an $\binom{m}{n}$th degree polynomial. So, this method generalized the rational root theorem, but it becomes too unwieldy for large degree polynomials. – Count Iblis Mar 12 '17 at 20:34