Polynomials, it turns out, are useful for more than just input-output assignments!

Perspectives on Polynomials

(it’s a witch!)

Courtney Gibbons
Hamilton College

It was a dark and stormy night… Okay, it was probably more like 3:30 in the afternoon on a crisp fall day back when I was teaching Calc 1 for the first time as “Professor Gibbons,” and I was looking through my colleagues’ past syllabi to see what problems they liked to assign. One of the problems sent me off on a tangent (pun intended!) because it evocatively named the rational function $y = \frac{1}{1+x^2}$ “The Witch of Agnesi” and, in this problem and others in subsequent chapters, proceeded to use differential calculus to tease out its secrets.

Graph of “The Witch of Agnesi”.

In fact, the Witch is one of a family of plane curves, $x^2 y + 4c^2y – 8c^3 = 0$, parametrized by $c$ (just take $c = \frac{1}{2}$).

I don’t want to spoil anyone’s calculus homework, but Evelyn Lamb has written a nice blog about the Witch over at Roots of Unity. (Okay, one spoiler: the Italian for “curve” is “versiera” while “witch” is “avversiera” – so the name of the curve is a mistranslation at best, or one of those groan-inducing mathematician puns at worst.)

My encounter with the Witch left me thinking about the many ways we discuss polynomials (and rational functions) with our college students, and I wanted to share some of those perspectives in this column.

As functions

A math student’s first encounter with polynomials often comes hot on the heels of the definition of a function. In this context, we meet our friends the single-variable polynomials $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ and spend a lot of time with the functions that plot lines ($y = mx + b$) and parabolas ($y = ax^2 + bx + c$). Implicit in this introduction is that the coefficients belong to the real numbers.

There are lots of ways to write down a line, and thinking back to high school algebra, you might remember that you and your math class best buddy might look at the line plotted below and come up with different point-slope equations.

A line passing through $(-4,0)$ and $(0,2)$.

You write down $y – 5 = \frac{1}{2}(x – 6)$ while your pal writes down $y – 3 = \frac{1}{2}(x-2)$. These should be the same because we say two functions are equal if they make the exact same input-output assignments (and in the case of functions we can plot in the $xy$-plane, this means they sketch out the same graph). Now, if you and your high school math pal wanted to check that your lines are the same in a different way, you’d each wrangle your line into slope-intercept form and make sure they match (in this case, you’d both end up with the single-variable polynomial expression $\frac{1}{2}x + 2$).

A Witchy Little Twist

Okay, but. That last part? That relies on the definition of polynomial equality (two polynomials are equal if they are equal coefficient by coefficient) matching the definition of function equality, and that doesn’t always work!

Consider the field $F$ with two elements, $0$ and $1$, with addition and multiplication defined modulo $2$. That is, $0 + 0 = 0 = 1+1$ and $0+1 = 1 = 1+0$, while $0\cdot 1 = 1\cdot 0 = 0\cdot 0 = 0$ and $1\cdot 1 = 1$.  We’re going to take our polynomial coefficients from this field.

As functions, $f(x) = x^2 + 1$ and $g(x) = x+1$ take elements of $F$ and assign them to elements of $F$ the same way. Indeed, $f(0) = g(0) = 1$ and $f(1) = g(1) = 0$. So, as functions, $f(x)$ and $g(x)$ are equal! But as polynomials, they are not, because when we do a coefficient comparison, we see that $f$ has a nonzero coefficient for $x^2$ while $g$ does not. (In this weird little example, we actually have that $f = g^2$.)

It’s a fair question to ask why we would bother with a special definition of polynomial equality if it’s going to pull sneaky tricks like this. Polynomials, it turns out, are useful for more than just input-output assignments!

When polynomials sneak into the denominator

I will admit to taking a certain joy in teaching partial fraction decomposition in calculus. There are pedagogical arguments in favor and against including it in the syllabus, but my enthusiasm is epicurean: I like that students are seeing an example of a basis for a pretty weird vector space before they have taken linear algebra!

If your integrand contains the rational function $y = \frac{p(x)}{(x+2)^2(x^2 +1)}$, you may remember that you can decompose it by first performing polynomial long division to write $\frac{p(x)}{(x+2)^2(x^2+1)} = q(x) + \frac{r(x)}{(x+2)^2(x^2+1)}$ and then break the “reduced” rational function down into a sum of the form $$\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{Cx}{x^2+1} + \frac{D}{x^2+1}.$$ This technique works because $\frac{1}{x+2}$, $\frac{1}{(x+2)^2}$, $\frac{x}{x^2+1}$, and $\frac{1}{x^2+1}$ form a basis for the vector space of rational functions with denominator $(x+2)^2(x^2 +1)$. One technique for finding the constants $A$, $B$, $C$, and $D$ is to use that rational functions are, well, functions! If two expressions really are the same functions, they’ll have the same input-output assignments. So, trying some handy values for $x$ (who doesn’t love $x = 0$?) gives us leverage to solve for the missing constants.

Polynomials as storage devices

More generally, many abstract mathematical objects have features that you might want to collect (or count), and you might want to stash your collection (or count) somewhere. For example, the characteristic polynomial $p(x) = |A – Ix|$ of a matrix $A$ stashes the eigenvalues of $A$ in its linear factors! The entries in a Young tableau are stashed as the coefficients of a Schur polynomial (read all about it)! It’s a bonus when these polynomials turn out to be invariants, and even better when you can specialize to all sorts of other polynomial invariants, as you can with the Tutte polynomial of a graph (or link, or matroid, or…)!

A plethora of polynomials, and a few words about Maria Agnesi

An 1836 engraved portrait of Maria Gaetana Agnesi.

In marshalling my resources for this blog post, I spent a little time reviewing the AMS Notices “What is…?” collection. There are some recent entries that piqued my interest and might pique yours, too: a column about multiple orthogonal polynomials and another about Sobolev orthogonal polynomials.

And, finally, Florian Cajori writes this entry in his A History of Mathematics (available for free through Project Gutenberg):