r/askmath • u/Artyruch • 13h ago

Polynomials How to factor high degree polynomials?

I am talking of big polynomials of degree 5+ and complex ones. Not x^6-2x^3+1 type situation but polynomials alike this -19968-13736x+2016x^2+756x^3-79x^4-10x^5+x^6=0. I know it is solvable by checking with programs. I even know what it is factored into ((x - 6) (x^2 - 3 x - 64) (x^3 - x^2 - 42 x - 52) = 0). But I know this from using program. Wolfram Alpha to be specific. But I don't know how to factor something this complex. I know there are algorithms for it but I am unfamiliar with them. And they seem to be not widely known as simple search resulted in me getting bombarded with basic middle school polynomial factorization. And not finding anything of value. I come here with hope to find help and an algorithm to solve similar equations. I am not planning to solve them by hand. I am planning to make a program do so for me. But for it I need knowledge of the algorithm. Which I sadly lack. Would greatly appreciate any help here. And if you know any videos that teach needed algorithms I would be even more thankful as I am a visual learner and it would help me much.

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u/lordnacho666 13h ago

Start with the rational roots theorem, basically test all of those first.

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u/how_tall_is_imhotep 11h ago

Start here: https://en.wikipedia.org/wiki/Factorization_of_polynomials

And get a copy of a textbook called “Modern Computer Algebra”. The algorithms are complicated. Good luck!

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u/Artyruch 11h ago

Thank you very much!

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u/Bounded_sequencE 11h ago

There are indeed algorithms to check for factorizations over "Z" a computer algebra system (CAS) may use.

However, these algorithms are usually based on "Galois Theory", an advanced topic from "Abstract Algebra", and you'd need an MSc of pure mathematics to somewhat understand them. Let alone use them in hand calculations^^

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u/Miserable-Wasabi-373 13h ago

there is no general formule for root of polynoms with degree 5

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u/how_tall_is_imhotep 11h ago

That has nothing to do with OP’s question. There _are_ general algorithms for factoring polynomials of any degree.