Sharing: Polynomial Functions

We’re studying polynomial functions in my freshman Advanced Algebra course.  I’ve been struggling with determining what it is they really need to know about polynomials. I’m not convinced that long division of polynomials is an essential enduring skill (feel free to try to convince me that I’m wrong). We now have CAS and W|A to do this. I do want them to be able to successively divide by linear factors to reduce a polynomial function to a product of linear factors and irreducible quadratics. Long dividing a 7th degree by a 5th degree? Eh.

So, I had to decide, what else do I want them to know? I have been talking to the pre-calc and calculus teachers to find out with what they struggle when they get to their courses. These are important conversations to have. We still have some work to do, but we’re getting there. In light of these conversations, I made this:

Among other things, I’m asking them to: identify the zeros, y-intercept, vertical scale factor and write the equation — determine if the function is decreasing faster between A & B or between H & I — determine the equation of the secant through C & G and through D & F — then for the big preview to calc piece: draw a tangent at E and determine which of those two secant’s slope better approximates the slope of the tangent. (link to .pdf or .docx).

Any suggestions for other questions I should be asking are welcome!

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7 Responses to Sharing: Polynomial Functions

  1. Riley says:

    I thought long division of polynomials was pointless too until I realized the power they give for finding roots, and for leading into the factor theorem. Now I freaking love it. But: don’t use long division, use polydokus!

  2. jd2718 says:

    I like giving a few points and asking them to write a polynomial that fits them – easiest if you supply the x-intercepts + one point off the axis. Might be too easy for your kids.

    Pat Bellew had some interesting stuff recently on geometrically locating imaginary roots on a polynomial graph. Shoot him an e-mail or dig back a couple of months in his archives…

    Long division of polynomials is analogous to long division of numbers – essential? no. But it forces practice of lots of little polynomial work, and kids could always use more of that. (divide monomial by monomial, multiply polynomial by monomial, subtract polynomials, repeat).


  3. Scott Carter says:

    As a mathematics professor at a state university, I often find that I have to reteach students to perform these operations. We use them often to sketch graphs, and to gain a deeper understanding at and beyond the level of calculus. And yes the calculator knows how to sketch graphs, but most college level math profs want students to perform these operations without a calculator. All of the arithmetic and algebra skills that are being glossed over by avoiding such topics are useful in many more broad mathematical contexts.

    By analogy, you can pretty much get anywhere you want to go by car, but walking is often more practical.

    It is really a good idea for high school teachers to talk to their peers at local colleges and universities to find out what we expect students to know when they arrive.

  4. benblumsmith says:

    For those of them that are headed toward advanced math, especially algebra, at a future point, polynomial division is totally awesome and central. (Not saying I favor the standard long-division algorithm over Riley’s polydoku, just that being able to divide polynomials fluently is a big and exciting deal.) I didn’t understand this myself until I studied graduate-level algebra, and it may not be worth it to you to take into account the probably small fraction of your students that may be headed for graduate level matematical study, but I think it’s worth knowing that this topic has a very big and rich future. Dividing something degree 5 into something degree 7 seems possibly overly cumbersome/annoying but if I am one day teaching them higher-level algebra, I might find it useful if they can divide big polynomials by quadratics or cubics, and I more importantly I’d definitely want them familiar with the mechanics of the process so they can think about its theoretical implications. (This is the big reason why teaching it is still valuable in the age of CAS.) For example, getting on a gut level that you can always arrange to have a remainder lower in degree than the divisor will allow them to understand why the polynomials over any field form a Euclidean domain and therefore also a unique factorization domain. Or, my favorite, the construction of quotient rings depends on understanding how division works. For example, the algebraist’s formal construction of the complex numbers from the reals:
    Take the set of real polynomials, and regard them as equivalent if they have the same remainder when divided by x^2+1. These classes of equivalent polynomials are the complex numbers. The imaginary unit i is the set of polynomials whose remainder is x. Square any polynomial like that (e.g. x^2 + x +1; squared is x^4 + 2x^3 + 3x^2 + 2x + 1), and you will get something whose remainder is -1 (try it). This was really confusing when I first learned it; now I think it’s totally awesome. My change in perspective came when I thought about the process of dividing x^2+1 into any polynomial and taking the remainder, and realized that this is basically taking every x^2 + 1 found in the polynomial and annihilating it (making it 0), leaving only the remainder. Therefore it can be thought of informally as setting x^2 + 1 to 0; thus x is “becoming” i. I never would have understood this if I didn’t have a pretty intimate working knowledge of polynomial division.

    I’m not making a case about what you should do in class. I just used to have the same feeling that polynomial division might be kind of outmoded and now I think it’s rad, so I wanted to share.

  5. Avery says:

    Agree with Ben that polynomial division can be quite interesting (although I’m not sure I completely followed…wouldn’t dividing by x^2+1 “make it 1”, not 0?) It’s also quite useful with generating functions, which are super cool and within the grasp of high school students. My big issue, though, is motivating this in algebra 2. I never really like the “you’ll use this 2, 3, 4 years from now” argument. Why not just wait to introduce it in said algebra (the fancy college variety) class?

  6. cecil kirksey says:

    Just a couple of clarifications please. Your writeup suggests that the graph is for a third degree polynomial since there are three real zeros (I am assuming that the end points on the graph are really arrows that have been omitted.). But that is not certain based on the graph, correct?

    Also, the direction to draw a tangent line at a given point on the curve is very misleading. There is one and only one tangent line and it has a slope given by the derivative of the function evaluated at the point. I assume you know this but perhaps phrased the exercise imprecisely: Draw a line approximating what a tangent line would be.

  7. benblumsmith says:

    @ Avery – I’m mad late as always but in case you subscribed to this comment feed, here is (hopefully) a clarification:

    The algebraic construction of the complex numbers from the reals involves taking the set of all polynomials over the reals and sorting them into big equivalence classes based on the criterion “Two polynomials are equivalent if they have the same remainder when divided by x^2+1.” The reason this is like turning every x^2+1 into 0 (rather than 1) is because you’re only looking at the remainder, not the quotient. The only part of the division you care about is the remainder. So it’s like making every whole x^2+1 that fits in your polynomial = 0 and leaving only the remainder.

    It’s exactly analogous to forming the integers mod n. For example mod 5. You take the integers, and you sort them into big equivalence classes based on their remainder when divided by 5. Thus, y’know, 24, 39, 109, etc. are all equivalent to 4, while -4, 31, 106, etc. are all equivalent to 1. This is kind of like making every whole set of 5 = 0. (With 109 for example, 21 5’s become 0 and you are left with 4.)

    (This wasn’t your question, but to spell out the rest of the construction:
    With integers, normal addition and multiplication induce well-defined addition and multiplication on the set of equivalence classes mod 5 because if ab=c, then c’s remainder mod 5 is totally determined by a’s and b’s remainders mod 5; i.e. it doesn’t matter which representatives of each equivalence class you pick; the equivalence classes of a and b uniquely determine the equivalence class of c. Same for a+b=c. And the exact same thing works if a, b and c are polynomials over the reals, and the equivalence classes are determined by the remainder mod x^2+1. So the equivalence classes of polynomials have a well-defined addition and multiplication. These equivalence classes can be regarded as the complex numbers, with x’s equivalence class being i. If taking the integers mod 5 is like setting 5=0, then taking the real polynomials mod x^2+1 is like setting x^2+1=0, which is why x’s equivalence class behaves like i.)

    The motivation issue is a whole other beast. I’d never want to tell a kid, “I’m making you learn this so that if you decide to be a math graduate student in 6 years you will understand how to construct a quotient ring.” I do want teachers to know that this stuff is theoretically important. I feel like whenever I am teaching a topic whose mathematical future I’m aware of, I naturally convey a sense of excitement and awesomeness about the topic at hand, and unity and coherence about mathematics as a whole. So I want teachers to know polynomial division is not a disconnected and outmoded skill but a theoretically central and important tool of higher math, and I want us all to have a taste of why. Mainly for the sake of how it will affect their tone when they teach about it. Looking for a level-appropriate way to motivate it in Alg. II is a whole other question. (But I bet an answerable one. I think I basically always think that if an idea or skill is awesome and important in the long run, and accessible at a certain level, it can probably be compellingly motivated at that level. No promise that it won’t take a lot of time though.)

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