What Are Polynomials?

What are polynomials? How are they built? What can you do with them? And why are they important? Keep on reading to find out!

Jason Marshall, PhD
5-minute read
Episode #285

Polynomial EquationLet’s kick things off today with a puzzle:

First, pick any whole number you like (although you’ll probably want to keep it reasonably sized so it’s easier to do the following in your head). Once you have your whole number, subtract 1 and then square the result. Now, subtract the number you just got from the square of the original number. With me so far? If so, it’s time to subtract your original number from your result, and then finally add 1. What did you end up with? Hopefully the same number you started with … otherwise, you made a mistake!

How can I possibly know that? Obviously, it’s because I was imbued with magical powers as a baby and can now interact with your mind from anywhere on the planet. Or not … not at all, actually. The truth is all of this was just a bit of (not exceptionally clever) mathematical slight of hand. Because the particular rigamarole I put you through was really just a way of building up a mathematical equation containing something called a polynomial.

To understand exactly what this all means and how it works, let’s begin digging a bit deeper into the world of polynomials.

What Are Polynomials?

To begin with, what are polynomials? As we’ll discuss today, polynomials are algebraic expressions made up of one or more terms of a particular type. As we’ve discussed before, algebraic expressions are sort of the phrases of the algebra world—they’re made from a combination of numbers, variables, operators, as well as things like parentheses and brackets that manage the order of operations.

Polynomials are algebraic expressions made up of one ore more terms of a particular type.

For example, “1” is a very simple expression made from a single number. But not all expressions have to be so simple. The expression “5 + 10” uses two numbers and an operator, and the expression “100 • x + 10” is a bit more complicated yet with two numbers, two operators, and a variable.

So polynomials are a particular set of these algebraic expressions that contain one or more terms. But these terms can’t just be any old term you can think of—there are a few rules and restrictions about what exactly polynomials can be built from. In other words, there are rules about how each term is constructed.

How to Build a Polynomial

What do these rules and restrictions look like? And how specifically do we go about building polynomials? The answer is term by term. As we’ve discussed, a term is what you get when you multiply one or more numbers and/or variables together. For example, 3x, x2/3, and 2/x are all terms. But not all of these terms are legal in polynomials since, as we’ve alluded to, the terms in polynomials can’t be any old terms … only particular terms will do.

To find out which, let’s actually build a polynomial out of polynomial building blocks. These “building blocks” are allowed to be any product of a numerical constant with a variable raised to a positive integer power. Which means that variables in polynomials can never be raised to a fractional or negative power.


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.