Extension Problems for Patterns in Polynomials
Oved Shisha (1932-1998), whose quote opens this project, was a prominent mathematician in the second half of the twentieth century. His primary area of research was approximation theory, which, among other things, is concerned with properties of polynomials.
To give you an idea of what he meant when he said that "Polynomials are temperamental creatures...," we are going to ask you to try squeeze a polynomial into a "box" and see what happens. You'll need either a graphing calculator or some other software that lets you draw graphs quickly. We will start with cubic polynomials, using 
as an example. For now, our concern will be in the domain 
. The graph of 
should be familiar:
Notice that this graph is neatly contained within the box 
and 
, which is two units high. Our objective is to find a cubic polynomial that fits into a shorter box. This isn't a very challenging problem unless we add one more restriction: Any polynomial that we use must be monic; that is, the leading coefficient must be 1. Therefore, the allowable cubic polynomials are 
for any real numbers a, b, and c.
You can do much better than the box of height 2. For example, in the graph below we see that 
does better than 
, which is displayed as a dashed curve. We can squeeze 
into a smaller box.
Here are some questions for this extension:
1. What monic cubic polynomial(s) needs the smallest box for graphing between -1 and 1? Can you prove it?
2. Change the word "cubic" in Question 1 to "constant," "linear," and "quadratic" and answer the question again for each.
3. Find a monic quartic (degree 4) polynomial that fits into the smallest box for graphing between -1 and 1.
4. Do you detect any patterns in the answers you have found to the first three questions? Do these observations give you a hint at what the answer might be for a monic polynomial of degree n, 
?
5. Based on Oved Shisha's statement about polynomials, what do you suppose happens to the graph of the polynomials that fit into the smallest boxes when these polynomials go outside the box?
