## Polynomials: A Matter of Degrees

Last time we examined why polynomials are defined as they are. This time, let’s look at some tricky aspects of the concept of “degree”, mostly involving something being zero.

Last time we examined why polynomials are defined as they are. This time, let’s look at some tricky aspects of the concept of “degree”, mostly involving something being zero.

A question last week (Hi, Zahraa!) led me to dig up some old discussions of how we define a polynomial (or monomial, or term) and, specifically, why the exponents have to be non-negative integers. Why can we only multiply, and not divide by, variables? Since we’ve been looking at polynomials, let’s continue.

Last week’s discussion about zeros of a polynomial, and other conversations, have reminded me of a past discussion of the shape of the graph of a polynomial near its zeros. Let’s take a look, starting with some other questions that nicely lead up to it.

A recent question from a student demonstrates that not everything on the Internet should be taken at face value – and that it’s easy to think you are right when you are not.

(A new question of the week) We often see polynomials in a simplistic way, imagining that any function whose graph resembles a polynomial is a polynomial. Much as an attempt to mimic random data often lacks essential properties of genuine randomness, so what we intend to be a polynomial often is not. As we observe …

(An archive question of the week) There are several ways to restrict the range of values you need to test when you are searching for zeros of a polynomial (using the Rational Zero Test or the Intermediate Value Theorem, for example). One of them can be quite useful for difficult problems, but can be hard …