# Polynomials

### Topics

There are two different definitions of **polynomial equation** that appear in books, on web sites, and in bathroom stalls, but the two definitions actually mean the same thing. Here, we'll prove it. In the future though, please stop reading bathroom stalls. We'd like you to hold onto what's left of your innocence.

**Definition 1:** A polynomial equation is any equation that can be written in the form (polynomial) = 0.

### Sample Problem

*x*^{7} + 8*x* – 43 = 0.

**Definition 2:** A polynomial equation is any equation that sets one polynomial equal to another:

(one polynomial) = (other polynomial).

### Sample Problem

*x*^{2} + 3*x* = 8*x* + 17.

Any equation of the form (polynomial) = 0 is basically the same as setting one polynomial equal to another, since 0 is a polynomial. Anything that counts as a polynomial equation according to the first definition is also a polynomial equation according to the second definition. In other words, the first definition is definitely defined by the definitive definition of the...huh. We should have stuck with the first way we said it.

On the other hand, any equation that sets one polynomial equal to another can be written in the form (polynomial) = 0.

For example, the polynomial equation 7*x* = 5*x* can be written as 2*x* = 0.

Any polynomial equation under the second definition is also a polynomial equation under the first definition.

The short story is that you can pick whichever definition makes more sense to you and go with that. For now, we'll use Definition 1, which is more convenient for the next section. However, just because we're using it for the time being doesn't mean it's fundamentally any better; we love all our definitions equally.