Can a Quintic Polynomial be Factored Using a Nonlinear System?

[JungleJesus]

For the sake of doing it, I'm trying to factor a quintic polynomial over the reals using a cool technique I found a few days ago.

It involves stenciling out the general form of the expression you want and then solving a nonlinear system in which there are more variables than there are equations. Here is one way to set it up:

x[SUP]5[/SUP] -x + 1 = (ax[SUP]2[/SUP] + bx + c)(dx[SUP]3[/SUP] + ex[SUP]2[/SUP] + fx + g)

When you multiply out the right side and equate the coefficients on both sides, you get a nonlinear system in 6 equations and 7 variables.

ad = 1
ae + bd = 0
af + be + cd = 0
ag + bf + ce = 0
bg + cf = -1
cg = 1

I know that there probably isn't a factorization over the rationals, which is why I am only concerned with obtaining a particular factorization over the reals. I could always repeat the process and factor over the complex field as well.

I don't know if this method will even work. It makes sense that it should since polynomials can always factor into linear and irreducible quadratic terms over the reals.

Can anyone shed some light on this? Is it nothing more than a really difficult system to solve or is it even possible in the first place? How else could I factor this polynomial exactly?

Please go light on the abstract algebra phrasing. I haven't taken any advanced algebra courses.

 

[AlephZero]

You can get to 6 equations in 6 variables without any loss of generality, by taking either a =1 or d = 1.

Look up Bairstow's method, which is a numerical method for finding a quadratic factor of a polynomial. In your notation, it takes a = 1 and uses Newton's method to find b and c. Once you have a, b, and c, finding the other coefficients is easy.

It is well known (Galois theory) that there is no algebraic solution to this problem for an arbitrary polynomial. So don't waste your time trying to get a closed solution to your equations using school algebra!

 

[JungleJesus]

Is there any way to express the coefficients as infinite sums, products, or continued fractions? What about transcendental coefficients?

I also heard of mechanism called a Bring radical. Is this another option? I really only want to express the coefficients as analytic values. I'd like to avoid using approximations.

 

[disregardthat]

Is there any way to express the coefficients as infinite sums, products, or continued fractions? What about transcendental coefficients?

Infinite sums, products and continued fractions are all examples of limiting sequences, where the elements of the sequence is defined in some particular way. Do you consider recursively defined sequences approximations as opposed to sequences defined another way? Newtons method generates a recursively defined sequence which has a limit in the same way as other sequences and therefore infinite sums, products and continued fractions, and is just as "analytic" as them. You could even extract a recursively defined sequence from some constructive proofs of the fundamental theorem of algebra.

There are no restrictions on the coefficients.

 

[JungleJesus]

To answer Jarle's question, I was looking for something a little more formulaic than the limit of a recursive sequence. The reason is that recursive sequences involve considerably more overhead than infinite sums, products, and the like.

So here's where I'm having trouble. It is my understanding that the fundamental theorem of algebra states that any polynomial can be factored into linear factors over the complex algebraic field as long as its coefficients are in the complex algebraic field. It is also my understanding that as long as the coefficients of a polynomial are real, then any imaginary roots must occur in conjugate pairs, meaning that any polynomial with algebraic real coefficients may be factored into linear and irreducible quadratic polynomials with algebraic real coefficients.

How is it that the values of a, b, c, d, e, f, and g are not expressible in a closed form. Doesn't an algebraic number have to be expressed in closed form by its own definition (otherwise it would be transcendental)? I understand that there is no "general" solution to the zeros of arbitrary polynomials. I also understand that factoring is logically equivalent to finding the zeros of a polynomial. However, I don't see why a "particular" polynomial could not be factored by a "particular" method.

Is my understanding of these concepts correct?

 

[Office_Shredder]

How is it that the values of a, b, c, d, e, f, and g are not expressible in a closed form. Doesn't an algebraic number have to be expressed in closed form by its own definition (otherwise it would be transcendental)? I understand that there is no "general" solution to the zeros of arbitrary polynomials. I also understand that factoring is logically equivalent to finding the zeros of a polynomial. However, I don't see why a "particular" polynomial could not be factored by a "particular" method.

Is my understanding of these concepts correct?

There is a HUGE difference between "knowing there is a solution" and "knowing the solution". For example, if I asked you about solving xe^x-1=0, we know that if x=0 the left hand side is negative, and if x=1 the left hand side is positive, so we find quite easily that there is a solution between 0 and 1. But finding out what that solution is requires defining a new function just for that purpose - or just approximating it with numerical methods

What is your objective of finding the solution here? It sounds like maybe you have a specific application in mind. You can use the Bring radical to solve a quintic polynomial by first reducing it to a certain specific form, but if you want to do computations on a computer that's not better than any other approximation, and if you want to perform some sort of analysis you're probably not going to get much insight into the equation by this method

 

[JungleJesus]

I would like to take a closer look at the relationship between the order of a polynomial and the complexity of its factors. So far, all I have is a few references to Galois theory, which is way over my head right now. I would also like to look more closely at the behavior of rational integrals in the complex plane, which is relevant to my studies right now.

I would also like to find or develop a factorization technique which is broadly useful and which doesn't require too much inter-disciplinary knowledge (ie: something almost anyone could use). This could eventually morph into a computer algorithm for a computer algebra system.

 

[disregardthat]

How is it that the values of a, b, c, d, e, f, and g are not expressible in a closed form. Doesn't an algebraic number have to be expressed in closed form by its own definition (otherwise it would be transcendental)? I understand that there is no "general" solution to the zeros of arbitrary polynomials. I also understand that factoring is logically equivalent to finding the zeros of a polynomial. However, I don't see why a "particular" polynomial could not be factored by a "particular" method.

Is my understanding of these concepts correct?

There is a well-known theorem that the general solutions to polynomials of degree five or more cannot be expressed in terms of the arithmetic rules of multiplication, sum, reciprocation and rational exponentiation. This theorem relies on a study of polynomials in group theory and ring theory known as galois theory (as I see you have mentioned). The book A First Course in Abstract Algebra by John B. Fraleigh is an excellent resource to learn the sufficient theory up to this theorem if you are interested in the details.

Any particular polynomial can be factored by a particular method; a method described in some proofs of the fundamental theorem of algebra. It generates a recursively defined sequence which converges towards a root of the polynomial. You won't in general get it in the form as described above. Polynomials are also so well-behaved that you could easily implement a function in form of a program which gives you all solutions to any polynomial through the use of Newtons method. The latter would probably be more cost-efficient.

A response to Office Shredder: I wouldn't say it's such a huge difference between the two, the most common elementary proof of the intermediate theorem in my opinion explicitly defines a solution, though in form of a recursively defined sequence.

 

[JungleJesus]

So a closed form algebraic solution does exist for particular polynomials, it simply must be found. Is this what I am to understand?

 

[disregardthat]

So a closed form algebraic solution does exist for particular polynomials, it simply must be found. Is this what I am to understand?

How can you find them if they do not exist?

 

[JungleJesus]

Sorry, let me rephrase that. Can an "algebraic" number exist and not be expressible in closed form? If not, then are the coefficients of the factorization mentioned above necessarily algebraic?

I suppose this is what I really need to know because it should determine the forms that I need to look for.

By the way, thank you all for responding to my questions. I know I'm probably way under qualified to be talking about this.

 

[disregardthat]

Sorry, let me rephrase that. Can an "algebraic" number exist and not be expressible in closed form? If not, then are the coefficients of the factorization mentioned above necessarily algebraic?

Yes, if you by "closed form" mean expressible as a composite of summation, multiplication, reciprocation and rational exponentiation of the coefficients, such as $$\sqrt{2+4\sqrt[3]{\frac{3}{2}}}$$. Some polynomials of degree five with rational coefficients have solutions which are not expressible in "closed form", but by definition algebraic (over Q). Galois theory explore what polynomials actually have roots in "closed form".

 

[JungleJesus]

Very cool. Thanks Jarle, you've helped me quite a bit. I just purchased an abstract algebra book which covers Galois Theory. This subject fascinates me and I must learn more.

I heard some time ago that the roots of arbitrary polynomials could be expressed in terms of certain transcendental functions, like the hypergeometric function. Do you know anything of this?

One more thing: is Galois Theory the only tool for characterizing algebraic numbers? How can we differentiate between a "closed form" algebraic number and other algebraic numbers and what properties do these types of numbers share and how do they differ from one another? Will they behave differently in certain situations?