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.