I love teaching polynomials because I think they offer an opportunity for students to see the rule of 4 constantly (graphical, numerical, algebraic, and verbal). I’m including several links and documents that I use in developing the ideas during the unit.
In Precalculus we start our polynomials unit with a short review of the rules of end-behavior in a lab format. I like this because it gets into an exploration of the “WHY” at the end.
The first part is just sketching what the students notice when they change the leading coefficients for polynomials of different degrees. There are links to sketches of polynomials of different degrees like this one:
Repeated Roots Lab
Here we talk about multiplicity and how repeated roots affect the behavior of the function. I use the language that the function “bounces” off the axes at zeros with even multiplicity, and it “lags” through zeros with an odd multiplicity greater than 1.
In addition to students noticing the behavior of these functions, I ask them to see if they can determine the degree, the leading coefficient, and constant term without expanding the entire polynomial. They use wolframalpha.com to check their answers to that question.
Why the “flat butt” and “sharp arms”?
The next class, I have the students do a little more exploring graphically and numerically. The question is: When a factor is repeated more times, WHY does the graph get “flatter” on the bottom and “sharper” sides?
They explore using desmos, and they look at graphs like this:
I also ask them to make a table of values for the final function, and do the calculations by hand (or at least x=2.8 and x=3.2).
I want them to remember that: repeated multiplication by a number whose magnitude is >1 will give a number whose magnitude is larger (larger than the original), whereas repeated multiplication by a number whose magnitude is <1 will be a number whose magnitude is smaller (smaller than the original).
So the major take-away is that: in the polynomial once I subtract 4, the difference will be repeatedly multiplied by itself, and if it’s taken to the 16th power it will behave much more dramatically than if it’s taken to the 2nd power. 1.2^16 is much larger than 1.2^2, and 0.8^16 is much smaller than 0.8^2. The “flat region” corresponds to the x-values within 1 unit of the zero, and the “sharp arms” correspond to x-values beyond 1 unit of the zero.
Time to Problem Solve:
I am not big into teaching processes if I don’t have to. I’d rather have my students discover or wrestle with the math, so I have the following problem as a warm up on a particular day. The students are using what they know to write an equation for a particular polynomial:
The trick with this one is that most student can get the correct factors for the polynomial, but they need to do a little work to find the leading coefficient.
In this activity, I have the students reinforce what they know about polynomials so far, and they get to use Wolfram Alpha to expand polynomials in factored form to polynomials in standard form.
What else do I do?
I have a lot of white board space, so we spend time practicing finding zeros, expanding polynomials, and drawing sketches of polynomials.We also spend time considering how many zeros and what type–those are fun problems because they require the students to explain verbally their reasoning.
MOSTLY, I care about the students making the connection between the algebraic and the graphical representations.