# 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 are factoring polynomials and we both had the same thought at the same time: when are we going to use this?

I feel a little silly asking that question because it always bugged me when people would ask that in grade school. However, we're both grown, working professionals (I'm a programmer, she's a photographer), and I can't think of a single time where I found myself thinking that polynomial factoring would be a good solution to the problem I was trying to solve.

So, my question is this: are there real world applications to learning how to factor polynomials, 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!

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What kind of homework are a programmer and a photographer doing that has them factoring polynomials?? – Henning Makholm Nov 20 '11 at 4:42
Define "real world" :-) – lhf Nov 20 '11 at 15:07
@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 :) – Mr. JavaScript Nov 20 '11 at 17:39
Great opening line. – zyx Nov 20 '11 at 21:04
(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. – user2468 Jan 5 '12 at 20:48

## 16 Answers

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.

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Would it not be much easier to use an automatic equation solver? – Kelmikra Oct 8 '15 at 3:44

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.

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"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 :) – Damien Nov 20 '11 at 11:26
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. – J. M. Nov 20 '11 at 14:58

(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.

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Yeah, this was way too long for a comment. – Asaf Karagila Nov 20 '11 at 7:45
Kinda sad that the "number theory could save your life" link wasn't to the Josephus problem! – The Chaz 2.0 Nov 20 '11 at 8:50
@TheChaz: Feel free to add a link in the comments. – Asaf Karagila Nov 20 '11 at 9:14
+1 seems like an answer to me, and a pretty good one at that – KennyPeanuts Nov 20 '11 at 12:51
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. – dvanaria Apr 15 '14 at 5:10

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.

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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".

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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. – Carl Brannen Nov 20 '11 at 6:23
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. – J. M. Nov 20 '11 at 9:53

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.

Addendum:

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?

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+1 /standing applause – Mr. JavaScript Oct 4 '12 at 2:56
Actually, history and literature if taught at a proper level are not things you can cram either. If on the other hand mathematics is taught as a bunch of formulas and "fit A,B,C into D,E,F" sort of approach then it can be easily crammed. Studying and understanding something, anything, requires insightful views and thinking, neither can be gained by cramming. – Asaf Karagila Oct 5 '12 at 20:59
To be fair, I don't see how this answers the question directly. This is more of a rant as my answer below is. It is a good rant, it is a fair rant about why mathematics is good for you. It is not an answer why factoring polynomials is useful, though. – Asaf Karagila Oct 5 '12 at 21:00
Guilty as charged. But I wanted to offer more than just a couple of quick examples which kids will not be able to relate to. I want to inspire. I hope both kids and adults will find reasons to want to know things mathematical. If they do, than I am happy. To Karagila, I meant no slight to history or literature. Your point is granted. – Science_1 Oct 5 '12 at 21:14

It may be that you never have to factor a polynomial. But that's beside the point.

You're factoring polynomials to hone your symbolic manipulation skills and give you a more intuitive feel, deeper understanding, and quickness to reasoning about and manipulating polynomials, or any algebraic equations.

By analogy: If you are playing football, the coach may tell you to go lift weights in the gym. You could complain: "There are no barbells in football. At no point during a football game will I ever need to do a bench press. Why am I doing this?" But to ask that is to misunderstand; repetitive practice to strengthen your fundamentals does, ultimately, make you better at a variety of tasks, even when they do not superficially resemble the abstracted exercise.

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I agree with your reasoning, but unfortunately I would have to argue that this is totally lost on students that are busy trying to get it "right" instead of getting "it". – Mr. JavaScript Dec 4 '12 at 22:34

As a high-school math teacher, I've struggled with finding engaging applications for factoring quadratics over the integers, for the following reasons:

1. Most real-life situations involve variables that cannot be assumed to be rational, making factoring an inappropriate strategy.

2. The genuine applications of polynomial factoring (for example, integration by partial fractions, cryptography, manipulating complex power series) are far too advanced for the grades in which factoring is taught.

The best solution I've come up with so far is to introduce the unit by introducing students to the Ulam Spiral and telling them we're going to develop some tools to understand the insanity of it. This has no real-world utility at all (that I know of) but is fascinating enough that most of the students buy in.

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I use the football analogy with the kids too. How often do you end up running down the football field and encounter a series of tires that you have to run through?

Yes, I would like some "real life" problems to demonstrate the need for factoring, but the brain flexibility idea and practice of setting up the problems to help recognize similar situations tend to motivate the kids through the tasks. I even finally convince about half of them that factoring is FUN (honestly). Why do we do sudokus or crosswords?

I mention the optimizing idea, and looking for zeros and the need for it in the sciences or calculus. There's also the "because it is there" argument.

I hope the math class brought about some intimate moments...

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Polynomials define simple curves in the language of mathematics so that they may be easily analyzed and modified. Simple curves can be combined to closely approximate more complicated curves. Planets, weather, etc. move in curves. Mechanical forces, chemical and biological processes, etc. are not constant but change over space and time. These changes and other changes like fluctuations in the economy can be approximated by curves. Also, televisions, computers, phones, music players, etc. all receive signals that are sine waves (curves). Does an electrical engineer factor polynomials on a daily basis? No. A novelist doesn't analyze the structure of each sentence, but at some point the novelist needed to learn sentence structure to write books. In the same way, polynomials are the building blocks of all these sciences.

Changes can be approximated by curves, since polynomials define curves in the language of mathematics, we can approximate these changes (from fluctuations in the economy to weather changes) Everything changes over time. Changes can be approximated to curves. Polynomials define simple curves in the language of mathematics so that they may be easily analyzed and modified. Simple curves can be combined to closely approximate more complicated curves. Therefore we can approximately calculate changes by analyzing the curve it is approximated to.

Eg:Halley's Comet takes about 76 years to travel abound our sun. Edmund Halley saw the comet in 1682 and correctly predicted its return in 1759. Although he did not live long enough to see his prediction come true, the comet is named in his honor.

Read more: http://www.physicsforums.com

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Wow! Thanks for that explanation! – Mr. JavaScript Sep 1 '13 at 0:38

Two things I notice about the answers in this site is that polynomials need not be reduced to the quadratic case and that real life is obviously intended for the sort of exploits that a school leaver or high school graduate would be interested in or find themselves doing as that is when they typically learn about polynomials.

So given that real life is a restricted version of what mathematicians, engineers and scientist and economists might consider, the answer to this question might better be stated without the mention of control theory for thermostats or self driven cars. I also would not mention the use in authentication of transmitted data over the internet. Certainly the specifications of pumps and electric motors that describe their operation characteristics is out of the question for a school leaver or high school graduate because these things are not of any interest or concern to them. The design or construction of a roller coaster is well beyond their focus as is the planting of orchards...come on lets face it, school leavers and high school students who have learnt polynomials have virtually no use for them in the short term. Quadratics though is a simple case of a polynomial that just might allow them to optimise something...but what ? Alcohol consumption ?? or perhaps facebook likes ??

Its quite true that polynomials is of no use whatsoever to high school leavers for 'real life' exploits of a teenager. They are perfectly right to state this obvious fact. Even basic Algebra has limited use to them as teenagers.

Yes if forms the very basis of their further study towards becoming a professional at their chosen career, but what possible use can they get from them at school age. Perhaps its the same as learning the times tables, if they even do that anymore. They may rue the endless recitals of the various tables, but only later will they see their usage in multiple purchases or dividing up the stocks of lollies or whatever. Perhaps polynomials are like that. We might not know how useful they are until we are actually called to use them and this might be on their first job as a professional engineer or scientist or economist or FX trader or mathematician or whatever.

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While I could think of problems related To area of land with a house on it, or a path cutting through it, the motivation for the question is my issue.

However poorly or well math is taught in schools the topic through high school and early college tend to be precursors to latter courses. It would be better if students under stood that. Not everything will be total practical. No one would have enough math experience on university entrance if we avoid teaching fundamentals the may not have rock solid day to day practical value.

Much what is taught could be considered casting a net to catch the people will go on the further in fields that require more math. For every one else the effort was about learn how to difficult things which you may have to in another field of study. This could be true about the practicalness of iambic pentameter in English.

Yes, the teaching programs could be more interesting but problems for any technique with feel contrived and impractical to most.

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If one has a 2x2 two-person zero-sum matrix game M (where neither row dominates the other, nor column dominates the other) where Row can play row I or II and Column can play column 1 or 2, what is the optimal mixed strategy for each player? If Row plays Row I with probability p and Row II with probability 1-p and Column plays Column 1 with probability q and Column 2 with probability 1-q then one can compute the expected value from Row's point of view. This expected value is a polynomial in p and q (and constants from the matrix M). One can factor this polynomial into the form C(p-s)(q-t) + V. V is the value of the game (from Row's point of view) and s and t are the optimal mixed strategies and C is a constant. The beautiful result is that if Row plays optimally (p = s) then whatever Column does not matter (similarly reasoning for Column about what Row does), since the term (p-s)(q-t) will be 0. So if V is positive Row can get a gain of V on average with each play of the game in the long run. (When V is negative the game is "biased" towards Column; when V is 0 the game is fair.) if Column wants to keep losses to a minimum (V positive case) the optimal play is q = t. This is not a "standard" high school factoring problem but it is a very nifty way to see a not obvious lovely result by factoring a polynomial.

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You probably will not use this in real life, as many before me have stated. However, a real world example is a basic trajectory problem.

Maybe if you and your wife build a rocket one day and take some calculations, you can create a program (using a polynomial in the form of $f(t) = at^2 + vt + h$, $a$= acceleration, $v$=initial velocity, $t$=time in seconds, setting it equal to zero, and factoring it) to find how long it took for your rocket to hit the ground, and you can probably even model it as they do in games such as Angry Birds.

Also, if you do programming for a research company, understanding any kind of advanced math can really help you help engineers solve their problems and/or create new equations and to get to advanced math, as one stated before, you have to get the basics, and factoring polynomials is definitely a basic algebraic skill.

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A tiny bit of factoring is immediately useful in some way. For example, can you name the numbers whose square is equal to itself? Set x squared equal to x, pull the x from the right side to the left, and factor, giving x(x – 1) = 0, and then read off the answers: 0 and 1. You don’t want to be humiliated by having some 10-year-old point this out to you at a party, or, worse yet, at the office on visiting day. Sparing yourself that humiliation qualifies as a “real-world application” does it not? Such small-potatoes instances of factoring come up all the time. To take another example, the formula for the number of diagonals of a convex n-gon is n(n – 3)/2. If you encounter the unfactored form of this formula, would you be at a loss? I think not. I think you are taking small-potatoes factoring for granted, but small-potatoes factoring is still factoring.

Remember that the purpose of learning mathematics is not to help you with your shopping, but to make you more useful to your employer. Coming to grips with factoring beyond the small-potatoes scale gains you ground in contending with one of the cardinal semantic conundrums everyone faces everywhere – in our personal lives and in our professional lives – namely, the problem of synonyms. It doesn’t take much along this line to, for example, save your employer money. I can give you a concrete example from my own experience. I was once in a computer programming shop (like you), where we were asked to verify that all modules contained a certain piece of logic, and if it didn’t, to change it so that it did. One of the other programmers, in looking at one of the modules, felt intuitively that the logic was there, but in a different form. He asked me to take a look at it, and I assured him that his hunch was right: it was indeed equivalent. Now, this particular case involved De Morgan’s Law. The spec was phrased in terms of one side of the double arrow, but what my colleague encountered was what was on the other side of the double arrow. He did not have explicit knowledge of De Morgan’s Law, but was astute enough to suspect the equivalence that it states. The point is that since this module did not need to be changed, and so did not need to be (re-)tested. Testing is a time-consuming and therefore expensive, but necessary, process, and so the employer was spared that expense. This example was about De Morgan’s Law, but it could easily have been about two equivalent algebraic expressions, whose equivalence was most easily demonstrable by means of factorization.

Also, presumably, calculus is not far away in your mathematics educational progress. However, calculus is heavily dependent on taking limits, but taking limits is heavily dependent on factoring. In practice, taking limits often boils down to this: factor both the numerator and denominator. Then, do all cancellations possible. Then, just plug in the value in question if that will not produce a denominator that is zero, in which case what you get is the limit being sought. (And if a denominator of zero would be produced, then divide everything by the highest power of the independent variable that occurs, and consider what happens as the limit is approached…)

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Polynomials and rational functions of polynomials (aka transfer functions) are a cornerstone of linear system theory - a theory used to approximate dynamic systems as linear models. They are used extensively by engineers and physicists to model dynamic systems of all sorts where one can relate input and output behavior. The factoring of the polynomials mathematically reveals the roots, but physically reveals the characteristic modes of the system under study. Modes are frequencies at which system behavior undergoes change in magnitude and phase.

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