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We all know how important and ubiquitous polynomials are in mathematics. However, when faced with a (not so much in love with the subject) 14-year-old asking us why they should care about these things, answering that studying polynomials opens the door to more mathematics is probably not the best strategy. Of course, textbooks include some simple geometric and “real life” problems that can be tackled using polynomials, but they are far from exciting to solve.

How do/would you personally motivate your high-school students to study polynomials? More precisely, can you suggest

  1. fun problems (not necessarily from everyday life) that can be solved using polynomials

  2. interesting applications of polynomials (inside or outside of mathematics)

that might convince them that studying these sums of products of numbers and letters is actually worth it?

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    $\begingroup$ An answer from Forbes: forbes.com/sites/forbesfinancecouncil/2018/03/08/… $\endgroup$
    – Raciquel
    Jun 15 at 18:16
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    $\begingroup$ I love James Tanton, and he has lots of ideas about teaching polynomials. I'd recommend googling "james tanton polynomials". $\endgroup$
    – Sue VanHattum ♦
    Jun 15 at 22:09
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    $\begingroup$ @SueVanHattum I haven’t found anything on his website/channel that answers my question tbh, but he sure is an amazing teacher! Thank you for introducing me to him $\endgroup$
    – Federico
    Jun 16 at 11:06
  • $\begingroup$ @Raciquel I edited the question, let me know if it’s still unclear. $\endgroup$
    – Federico
    Jun 16 at 19:16
  • $\begingroup$ There's the problem of non-adaptive group testing, for which solutions may use polynomials. It is in itself a fun puzzle, in addition to being related to COVID testing. There's also Shamir's secret sharing. $\endgroup$
    – Passer By
    Jun 17 at 16:56

7 Answers 7

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  1. Using puzzles to attract attention: "Think of a number, subtract 7, multiply 3, add 30, divide by 3. Then subtract the original number. The result will always be 3. Why does this magic work?" After learning the trick, ask the student to come up with their own puzzles.

  2. Illustrate the idea from intuitive exercises (and without jargon): "I am thinking of a number. I subtracted 5 from it and then squared it. The final answer is 9. Which number was I thinking?" Of course there are two possible answers: 8 or 2. But students often will find only one answer. It starts a discussion about if there are more answers. This nicely sets up discussing degrees and roots.

  3. Tell them the idea is already familiar to them: Essentially every number written in decimal system is a polynomial in '10'. Start discussing various bases like binary and ternary. This also naturally leads to learning the algebra of polynomials. You can add, subtract, multiply, divide polynomials because they are numbers in base 'x'.

  4. Grasp their attention with history: Tell them the story of Italians and their conquests. Very complicated looking monsters like $\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}}$ might actually be a familiar face (4 in this case!). An argument for this involves a cubic.

  5. Set cool goals and motivate them: Results like rational root theorem and ideas about irreducible/minimal polynomials are powerful tools to prove irrationality of certain numbers. I was impressed when I could prove that $\sin 10^{\circ}$ is irrational using polynomials. Maybe you can tell them that, once they learn enough of the theory you will show them how to prove a cool result.

Well, these are the techniques I use to teach 10th grade students. I think they are generally effective.

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    $\begingroup$ The first two examples don’t seem compelling to students uninterested in math. These really work for sone of your students? $\endgroup$
    – KCd
    Jun 15 at 19:38
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    $\begingroup$ I generally have a good mix of students in terms of ability and interest. So I try various things. I teach in a conversational way and see what attracts majority of them to an idea. There are always students who only care about math as a tool or are uninterested. I lay down the methods clearly for them and solve some real world problems. $\endgroup$
    – Spai
    Jun 15 at 20:52
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    $\begingroup$ @SueVanHattum "X such that 0 = X^6 - 45 X^5 + 835 X^4 - 8175 X^3 + 44524 X^2 - 127860 X + 151200", perhaps? If only the students would study polynomials, they could discover what grade they're in. $\endgroup$
    – Ray
    Jun 16 at 5:19
  • $\begingroup$ @Ray 5,6,7,8,9,10 $\endgroup$
    – KingLogic
    Jun 16 at 7:07
  • $\begingroup$ I really like the first two suggestions! And I agree that showing the irrationality of certain numbers using polynomials is really cool. $\endgroup$
    – Federico
    Jun 16 at 11:14
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If the student has ever used a vector-based computer drawing program, they will be familiar with Bézier curves.

Bézier curves are extremely intuitive to understand for humans. They became popular as a digital replacement for wooden splines that were used by car designers to shape the smooth surface of a car's body. People typically very quickly understand how to shape them to their liking when playing around with a graphics program, even without given instructions.

Furthermore, Bézier curves play a fundamental role in computer graphics. Everything visible on your students' phone that is not a photograph is likely being computed from a Bézier polynomial. The text you are looking at right now? A polynomial. Any .svg graphic on a website? A polynomial. The emojis that your students will undoubtedly use extensively? You get the point...

You won't be able to explain how exactly Bézier polynomials work, as that theory will never fit into a high school math curriculum (this video gives some pretty nice visualizations though). But it may be instructive to start with a familiar figure, like a glyph for a letter or an emoji, show how it is composed of curves and then show the polynomial form of the equations for some of those curves. Students are good at pattern matching, so they should be able to identify the common structure of all those equations. Now all you have to do is slap a name on it, and you have motivated polynomials.

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I think this question and the likely answers are heading you down a wrong path, because of unstated assumptions. You would be better off working with the student and helping him achieve. Baby steps and coaching style pedagogy. Direct instruction, mimicking, etc. Greg Ashman has a very good presentation, referencing studies, on how achievement drives motivation more than motivation driving achievement.

In addition many suggestions of puzzles or applications or discovery learning will actually be harder than the standard content...more applicable to top students than to struggling ones.

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    $\begingroup$ What unstated assumptions are you referring to? $\endgroup$
    – Federico
    Jun 16 at 11:16
  • $\begingroup$ @Nij Would you like to explain what led you to this particular politically charged conclusion? $\endgroup$
    – Passer By
    Jun 17 at 16:17
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    $\begingroup$ The reference to any unspecified student as "him" is an exclusive statement, and there is also a significant prejudice suggested that boys will need the help where other students won't. $\endgroup$
    – Nij
    Jun 17 at 21:19
  • $\begingroup$ @Nij in many languages, something called “grammatical gender” exists. For example, in some of them, every single noun is either “masculine” or “feminine” (that doesn’t mean that Italians think that chairs are inherently feminine: grammatical gender is, as its name suggests, just a grammatical device). In such a language, when referring to a generic student, one would use one of the so-called “overextended masculine” and “overextended feminine”. It is plausible that OP’s mother tongue uses the overextended masculine; if that’s the case, OP didn’t mean to refer to boys only. $\endgroup$
    – Federico
    Jun 18 at 16:08
  • $\begingroup$ In the total absence of any reason to expect the author is SEAL, and given the reflection of prejudices well-established in English... whatever. @Federico $\endgroup$
    – Nij
    Jun 19 at 1:11
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I disagree with the claim by others that puzzles are more applicable to top students than struggling ones. Of course, you must choose the right puzzles for the right level, and actually teach them how to solve the puzzles. If you are unable to solve the puzzles yourself, then it's very difficult to teach in a successful manner using them. You may notice that the puzzles I linked to have essentially nothing to do with polynomials. This is not a coincidence; logical reasoning precedes all true mathematical reasoning.

Once your student grasps logical reasoning fully, polynomials will simply show up everywhere and they would have no trouble recognizing and analyzing them. For example, Pascal's triangle literally has a polynomial in each (diagonal) column. You can start there, just playing around with Pascal's triangle and getting students familiar with its interesting properties.

I think the other answer's suggestion of base-n representation is not too good as an example of polynomials, because it actually is not one; a (univariate) polynomial is characterized by having a weighted sum of a variable raised to some natural powers, so to get a true polynomial in base-n representation you would need to have varying n, which is just useless. However, it is alright to say that a polynomial in X is like a number in base X where X is very large.

I think the most common places you would see true polynomials in real life are the parabolic trajectory of a ball (ignoring air-resistance), and the polynomial approximations of a function/curve (related to asymptotic expansion).

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    $\begingroup$ In the last point you could add interpolation. A piecewise linear continuation of a function table is often not sufficient. Bezier splines are frequently used in computer graphics etc. $\endgroup$
    – Lutz Lehmann
    Jun 16 at 7:53
  • $\begingroup$ @LutzLehmann: I did think about splines, but those are not intrinsic applications of polynomials. For example, you might think of cubic splines, but they are not how a real material bends, and of course bogus when two adjacent interpolation points have the same x-coordinate. Anything that is more realistic or coordinate-invariant would no longer be a polynomial. However, you are of course correct that tables are often interpolated via polynomial approximations rather than linear segments. $\endgroup$
    – user21820
    Jun 16 at 8:06
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if you're willing to take a bit of diversion, you could show how polynomials are used for error correcting codes that make the internet possible (reed-Solomon, but do it with real numbers instead of finite fields so their heads don't explode). one really nice thing about this use case is that it also motivate some of the things you want to prove about polynomials (e.g. that n+1 degree polynomials are determined uniquely by n points)

for a slightly harder one, FFTs can be understood as a way of quickly evaluating polynomials, and FFTs are definitely in the top 10 coolest and most useful bits of math of all time.

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    $\begingroup$ I’m afraid we have very different tastes, haha. But I guess a few students might find Reed-Solomon interesting. I don’t think mentioning the FFT as something that lets us evaluate polynomials quickly could be any motivation to study polynomials, however. $\endgroup$
    – Federico
    Jun 16 at 11:23
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Few ideas that come to mind:

  • Let them predict trajectories of vehicles. Data can come from a simulator.
  • Generate splines for scalable fonts targeting fictitious hieroglyphics
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You have two hallways with a different size. What is the maximum length of the segment you can turn from one hallway into the other?

enter image description here

... and now something more realistic: you live in a house with hallways of different sizes. You wife wants to buy a new sofa, but she can't decide to buy the small one or the large one.
The vendors want her to buy the large one, but you are capable of calculating the maximum size of the sofa, proving the vendors wrong.
Your wife will be very proud being married to such an intelligent man.

Do you want your wife to be proud of you? :-)

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    $\begingroup$ That last bit feels like "I don't get this, therefore I'm stupid and inadequate; I should give up now" bait. $\endgroup$
    – wizzwizz4
    Jun 16 at 10:15
  • $\begingroup$ @wizzwizz4: at your age I also didn't get it. Not even a year later I did. :-) $\endgroup$
    – Dominique
    Jun 16 at 10:39

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