When studying algebra, and learning how to perform more complicated rearrangements and calculations, you will frequently see math terms such as “monomials” and “polynomials.” They may sound like some kind of advanced mathematical concepts, but in reality, they are not. In fact, their definitions are really quite simple, and understanding these will help you when working with problems that include them. Understanding the words is important here, because their definitions essentially explain the concepts.
Consider first the term “monomial.” As you probably know already, the mono- part of the word refers to one (or single), just like the word monologue refers to one person speaking, or monotone refers to a sound that has a single pitch. The –nomial part of the word come from the Latin word for name, and refers to terms or expressions (think of them as math names). So, if you put the two parts together, you can see that a monomial is an expression that contains only a single term. Here are a few examples of monomials, and yes they really can be as simple as they look:
All of these are single terms, therefore these are all monomials. Any standalone whole number is a monomial, as are variables. You can also combine these to have many parts multiplied or divided, as seen in the examples, but you cannot add or subtract within a term. You also can’t have negative or fractional exponents. Think about a monomial as being anything where all the numbers and letters can be crammed together without having to break it up with an operation. A few things to keep in mind when you work with monomials: a monomial times a monomial is a monomial (eg. x times y is xy), and a constant time a monomial is a monomial (eg. 3 times x is 3x).
So, now that I have defined monomials as being math expressions consisting of only a single term, do you think we have a way of describing expressions that have more than one term? Ha ha! Of course!
Polynomial is the word used to identify a mathematical expression that is composed of multiple terms. Any expression with one term is a monomial, whereas any expression with more than one term is a polynomial. You can break this word down into its components as well, and realize that poly- refers to multiple or many, so now you can easily remember what this word means. Here are some examples of polynomials, and yes these can get very complicated:
- x + 3
- 2x – 10
- x2 + 5x – 100
- 81x6 – 43x5 + 23x4 – 12x3 + 2x2 + 6x –1000
All of these expressions are polynomials, and they are all made up of several monomials. Each individual term in these polynomial expressions are monomials on their own. So, as I said to consider monomials as being terms where all the parts are multiplied and crammed together into a single term, as soon as you introduce an operation such as addition, you are creating a polynomial. As you can see on that last example, I ordered all of terms from highest x exponent to lowest x exponent. The highest exponent refers to the degree of the polynomial, which I will discuss in a future post.
While polynomial in a general terms that encompasses all expressions having more than a single term, there are a few other terms that you may see used to describe some specific cases. Binomial refers to polynomials that have exactly two terms, sincebi– refers to two. Similarly, using the prefix tri- gives you the word trinomial for dealing with expressions that have exactly three terms.
So, now that you are familiar with these common math definitions, you can see that they aren’t introducing any additional advanced techniques to your math problems. They are merely words used to describe what you are already used to working with!
In my next post, I will go into a bit more detail on performing calculations with polynomials. For example, how do you divide a polynomial by a monomial, or how do you FOIL two binomials? Follow along, and you’ll learn how to do polynomials problems with ease!