What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this? I feel a bit silly because it always bugged me when people asked that in grade school. However, we're both working professionals (I'm a programmer, she's a photographer) and I can't recall ever considering polynomial factoring as a solution to the problem I was solving. Are there real world applications where factoring polynomials leads to solutions? or is it a stepping-stone math that will open my mind to more elaborate solutions that I actually will use? Thanks for taking the time!

$\begingroup$ What kind of homework are a programmer and a photographer doing that has them factoring polynomials?? $\endgroup$

$\begingroup$ @Henning: I've been a self-taught C programmer for the last 14 years. I decided to go back school to get my degree. My wife graduated from Cal Poly in 2004, but she decided to take the math class with me so that we could have some "together" time :) $\endgroup$

$\begingroup$ (too lazy to write an answer) One application is control engineering (think of engineers designing your living room heater thermostat). Often the control circuitry is modeled as differential equations, solution to which give the engineers an idea on how accurate & fast the control system would respond to temperature changes in the room; that is, the system behavior. Equations are typically transformed to polynomial systems using Laplace transforms. Finding roots of these polynomials describe the said system behavior. Checkout this WP. $\endgroup$

17 Answers 17

If you model some phenomenon with a polynomial, it's often of interest to determine when the polynomial evaluates to zero. One of the tools used in deciding when this happens is factoring.

For example, simple trajectory can be modeled with a quadratic function. If you think of time as the input and height as the output, then the positive time for which the polynomial evaluates to zero is precisely the time when the object hits the ground.

$\begingroup$ @Kelmikra Any computer aided solver will be using analytic or numerical methods to determine roots. This may entail factoring. $\endgroup$

$\begingroup$ "easier to use", yes but then you have only solved one particular problem. Having a symbolic answer allows you to answer an infinitude of similar problems using just arithmetic. Large exaggeration there but it is a correct answer :) So knowing how to factor is more important that any particular case and in fact the inability (even in theory) to obtain the factors (in general) is a hindrance in many cases. $\endgroup$

For polynomials with integer coefficients the question is roughly the same as "what are the practical applications of algebraic number theory". The usual answers are coding theory and cryptography where factorization (and related operations such as testing whether a polynomial can be factorized) is part of the basic infrastructure from which systems are built or broken. Coding is necessary for digital communication (including telephone, video and satellites) and cryptography has become a basic feature of everyday computer use and commerce.

For polynomials with real coefficients there is partial fraction expansion used in calculus to compute integrals.

For polynomials with complex numbers as coefficients the factorization is into linear factors so that factoring is practically the same as numerical root finding (and this is in part true for real numbers as well). Problems in engineering where the location of complex roots of a polynomial determines the behavior of the system are common. For example, stability or instability can be decided by whether all the roots are inside the unit circle, or have positive real part, or other location-based criteria. Oscillations might be periodic if roots are $n$'th roots of $1$ for some $n$, or quasiperiodic behavior if roots are on the unit circle but not all at roots of $1$. A system governed by a partial differential equation would show diffusion (like heat) or wave-like behavior based on the factorization of an associated "differential operator", which is essentially a polynomial.

In general, many phenomena are decomposable into components, pieces or subsystems in a way that (when the systems are modeled mathematically) appears as a multiplicative decomposition of polynomials, with one factor per subsystem.

$\begingroup$ "Problems in engineering where the location of complex roots of a polynomial determines the behaviour of the system are common." . A real example is control theory, for example, cruise control or an autopilot. It's nice we can model these before we go test flying :) $\endgroup$

$\begingroup$ Digital filters also demand the testing of polynomial stability. However, in practice, one does not factor the polynomial to check stability, as methods like Routh-Hurwitz or Schur-Cohn merely need to act on the coefficients. $\endgroup$

(This is a very long comment, not a real answer)

When people (including my students) ask me questions like this my internal fuses blow out, I usually reply with a very cynical tone something along the following lines:

This is useless. Everything that you study here is completely useless to you later on in life, if you prefer not to study this you can go to a college, or change profession. This university wants you to enrich you with a broader knowledge, either take it or leave it.

Of course, I am lying. Everything that you study can come into use sometimes, often in unexpected places. It is possible that one day number theory will save your life. In the meantime you can just view your studying as a way of learning to do things abstractly.

Why is that important? Problems are often similar, though one needs to climb one or more level of abstraction to see that.

For example, if I asked you to take out 3 oranges from a pile of 10 oranges. Would this be any different if those were apples? rocks? sheep? bullets? No. It would probably be the same. This level of abstraction is very simple. True.

On the other hand, asking you to find the best route to get from one class to another taking into account the weather, the possible amount of people walking between classes as well, and so on.

This problem may seem very different than asking you to buy food for a week with optimal budget (you don't want to spend all your money on groceries, right?), taking into account the weather and how you are likely to spend the following week.

In reality they are different problems, and one would likely to employ different parts of the brain to solve a spatial reasoning problem and an arithmetical problem about money.

Mathematically speaking one could represent them both as a complicated weighted-graph; probability and statistics; fuzzy logic; multivariable calculus; and perhaps other fields of mathematics.

This is a form of abstraction that people are not usually able to do "just like that". Furthermore, even if you do find a general solution, applying it to each problem is again not a trivial matter and is often complicated just as the abstraction part.

Finally, we reach to the point of my babbling above. Mathematics is a wonderful and abstract tool. If you study it, your ability to make the connections between seemingly unrelated problems is likely to get better, your ability to solve the abstract problems is likely to get better, and as a result your ability to solve the problem at hand is likely to get better.

You are a programmer, you need to be able to deal with a lot of problems, they could come in many forms and many ways. You need to be able to see the abstract similarity, and as a good programmer be able to write abstract tools to handle the general problems. Not to rewrite ad-hoc code to solve each problem on its own.

$\begingroup$ Kinda sad that the "number theory could save your life" link wasn't to the Josephus problem! $\endgroup$

$\begingroup$ Your internal fuses blow when your students ask this question? That's sad - it's a valid question and points out a problem with the way math is taught in our schools. All too often math teachers launch into: "this is a polynomial and you do this, this, and this to factor it," without giving them any clue as to why that would be a useful thing to do to a polynomial. I give a +1 to your students. $\endgroup$

$\begingroup$ @dvanaria: Not everything should have a real world value. Some things should be just beautiful on their own. What are useful thing to do with polynomials? I don't know, and I don't care. As for my students, I never taught them about polynomials. I did teach them about infinite sets, the axiom of choice, the incompleteness theorem, Tarski-Vaught test, and infinitary logic. Can you tell me what these are good for in real life? $\endgroup$

$\begingroup$ @AsafKaragila Not an application of the specific topics you mention, but didn't Hilbert give an application of basic facts about arithmetic of infinite cardinals (like $\kappa+1=\kappa+\kappa=\kappa$) to hotel management? $\endgroup$

None of the answers so far justify making grade 10 students pointlessly factor polynomials. And for most students, it is indeed a waste of time. Unfortunately, if it were removed from the high school math curriculum, it would be impossible to go on. Now I will tell you why.

Sometimes in life you have to solve a quadratic equation. Not just in school, but in life. It is the basic equation that comes into play when competing factors have to be optimized. You don't always write an equation for these things, but that is what is happening. The classic example is the apple orchard, where you get fewer apples per tree the more you crowd the orchard. The optimum solution is given by solving a quadratic equation.

In real orchards with real apple trees, it is true that the actual equation may not be the simplified quadratic equation of the iconic high school math problem. But the principle of optimization is the same, and it is the quadratic equation which most clearly and in the most simple way illustrates this principle.

Perhaps the most important lesson of high school math is that the physical world can be modelled mathematically, and that mathematical equations have solutions. It is possible to simply write out a formula which solves any quadratic equation but this would be wrong. It obscures the basic idea of what it means to solve an equation mathematically. You cannot begin to explain the general solution of a quadratic equation unless you start with the method of factoring. As pointless as it seems when you are doing it, that is where it leads to and that is why you can't teach math without it.

You need polynomial factoring (or what's the same, root finding) for higher mathematics. For example, when you are looking for the eigenvalues of a matrix, they appear as the roots of a polynomial, the "characteristic equation".

I suspect that none of this will be of any use to someone unless they continue their mathematical education at least to the junior classes like linear algebra (which deals with matrices) and differential equations (where polynomials also appear). And I would also bet that the majority of people who take these classes never end up using them in "real life".

$\begingroup$ I should add that I used factoring now and then in my engineering career. I think a lot of this could have been avoided by other means, but if you've got the tool you use it. $\endgroup$

$\begingroup$ In actual practical work, however, one does not obtain eigenvalues by factoring the characteristic polynomial. Instead, it is numerically saner to apply repeated similarity transformations to a given matrix, up to the point where the diagonal elements give sufficient information for obtaining approximate eigenvalues. $\endgroup$

Factoring is often a key skill for solving problems in which you need to find a value for x. What can x equal in real life? Well, about anything. Being able to solve for x is the foundation of algebra, which itself is the foundation for doing trigonometry and calculus and higher math. Want some examples?

Well, suppose you would like to own a business one day. Say you own a painting company and have several employees. You get a rush job to paint a large hotel conference room. Knowing from experience how fast your employees work, you know that Joe can do a room this size in twelve hours, Max can do the job in nine, and Jane can do the job in ten and a half. How long should it take them, then, to do the whole job if you let them work together? To figure this out you need to be able to factor. Only if you know the answer to this question will you be able to tell if they are working hard for you or taking advantage of your mathematical ignorance.

Suppose you want to sell a product, such as mixed nuts. What is the ideal size of box or can for the quantity of product you want to sell? After all, cardboard and metal costs money, and storage of overly large containers wastes cash. To find the ideal size, you will need to be able to factor.

Those nuts you want to mix; prices on nuts change all the time. One day the cost of peanuts may be up, or on another day walnuts may be down. How should you tweak the mix to hold the price you charge constant as the various nut prices change? To figure that out, you will need to know about factoring.

Suppose you want to make computer games which play dice, or launch birds; or perhaps you want to write a program to keep data secure or do scientific research which tracks how owl populations change in relation to weather patterns. To figure all of these things out, you will need to know about factoring.

Factoring is your gateway to doing big things in life. If you want be a chemist or astronomer or ecologist or physicist, or programer or be your own boss and have a competitive edge, or do anything beyond working the 9-to-5; if you want to be a leader in your field and do big things, you will need skills in mathematics which are built upon college algebra, and for that you need to learn about factoring. That is why these skills are important. Those who have math skills earn more money. That is why you should learn about Algebra. That is why you should learn about factoring.

On reflection I realize that a couple of my examples (I will not share which ones) can be figured without factoring because they are linear in nature rather than involving a changing curve. In grade school you learn about math which deals with lines, and you can do quite a bit in life with just those skills. But not everything in life is linear, and to go beyond life-on-a-line, to go beyond grade-school understandings and grade-school skills, to be more, do more and understand more, you need math which can handle change and curves.

There is a secret you can use to stand out in the adult job world, and you don't even have to be particularly gifted at math to take advantage of it. Most people give up learning math the moment they can, and never go beyond whatever math they were required to take in grade school or college. Worse yet, they almost never practice what they did learn. Their failure is your opportunity to get ahead.

The secret is this: Don't stop. Keep learning, keep studying, keep practicing math even when you are out of school and you will leave those who do give it up in the dust. Math isn't something you can cram for, like a quiz or history test. Math isn't something you can fake. Math is like learning to play a musical instrument. It takes practice. Do what you find to be fun, yes, but keep challenging yourself, keep learning and keep pushing ahead.

Even if you never need to know how long a toy rocket should wait to deploy its chute so as to have the longest trip to the ground, the day will come when one of your kids will want your help with factoring; and wouldn't you like to be able to show them how? Math is worthwhile for its own sake; a game which can be played when all you have is a pencil and paper and half an hour to kill. Don't waste those minutes staring at a wall. Math can be fun!

Here is a parting challenge. Which examples in my original post can be calculated using just the math of straight lines (using grade-school math and simple algebra), and which of them require factoring to solve because they are problems about curves intersecting a y = 0 x-axis line? Do you know enough about factoring to say? Wouldn't you like to?