in which my mind is blown by polynomial long division

My love of polynomial long division is already documented, but now I’ve found more reason to love. To give some context, I am working on plans for my Intro to the Calculus Unit for my Precalc kiddos. I do it right at the end of the 1st semester followed by an Introduction to Statistics Unit because I want my Sophomores and Juniors to make an informed decision about whether to take AP Calculus or AP Statistics next year.

The Calc unit is fairly straightforward: here are some of the big ideas, hey lets learn some notations, oh noes division by zero?!, take some deep breaths it’s just a limit stop freaking out type of stuff. After doing some work with the delightful Chris Sangwin, I have chosen to play around with the following piece of information (and hopefully some geogebra modeling) to take polynomial long division to a new level in precalculus:

Given a polynomial P with degree n ≥ 2, the remainer when P is divided by (x – a)² is the equation of the tangent line to P at x = a.

Boo Yah. Go try a few examples–it’s rather fun. I am going to set up the kids w/ the long division and then have them graph the original equation in geogebra along with the remaider equation and then have them write down what they notice. I have yet to come up with anything practical as it’s not all that useful for finding extrema, but I need to think about it some more.

Oh, and please feel free to shoot me down if this is wrong. My working knowledge of Taylor Series is rough at best and that’s where this idea comes about.

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5 thoughts on “in which my mind is blown by polynomial long division

  1. Howdy Ashli — this is real similar-like to what I worked on in my functions group at PCMI 2010. I made some Geogebra apps and there are the activities that go along. You want I should share?

    Joe

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  2. Sorry I ran across this so late. Yes, look at PCMI 2010 for a sequence of problems leading to this idea. (Or, CME Project Precalculus, Investigation 3A.) Basically it starts from the Remainder Theorem: f(a) is the remainder when dividing by (x-a).

    So what happens when you divide f(x) by (x-a)(x-b)? You get a linear remainder r(x). But the Remainder Theorem still applies — f(a) gives the same answer as r(a). And f(b) gives the same answer as r(b). If you’re not sure of this, write it out as quotient and remainder:

    f(x) = (x-a)(x-b) * q(x) + r(x)

    Now substitute a… wham! Oh, and b too.

    So here’s the cool upshot: the remainder r(x) is the secant line through the graph of f(x) at (a,f(a)) and (b,f(b)). This is a pretty amazing result with a lot of utility.

    Now start moving a and b really close together, and it still works; or you can fix one point and use a variable for the distance to the other, and it still works. Each of these previews the two limit definitions of derivative, and either can be used to show that if

    f(x) = (x-a)^2 * q(x) + r(x)

    then r(x) is the equation of the tangent line to f(x) at x = a. It’s pretty awesome.

    I’m guessing you already nailed a lot of this 🙂

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  3. Hi, I got a question which i helping my gf with her homework and I have no idea what is this and she having tough time cracking her head for the answer. It will be great if you can help us out with the following question:

    “when a polynomial p(x) with it’s degree >=2 is divided by (x-1) and (x+4), the remainders are -3 and -6. FInd the remainder when p(x) is divided by (x-1)(x+4)”

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