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
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What Are Polynomials?
Let’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.
Variables in polynomials can never be raised to a fractional or negative power.
You can think of building a polynomial as the process of adding or subtracting these building blocks. For example, we can add up five “x2” building blocks (which we can picture as x-block-long by x-block-tall squares), two “x” building blocks (which we can picture as x-block-long rectangles), and the numerical constant 1 (which we can picture as a single block).
This creates the polynomial expression: x2 + x2 + x2 + x2 + x2 + x + x + 1.
Combining Like Terms in Polynomials
That’s a pretty big polynomial we’ve built. It has 8 terms! But you may have noticed that many of these terms are identical. As we’ve talked about before, the word “identical” here means that many of these terms are what we call like terms. In other words, we can think of them as representing the same shaped and sized objects—which means we can imagine stacking them.
In particular, in the polynomial we’ve been building, there are 5 square-shaped “x2” terms and 2 line-shaped “x” terms. Since these are each the same “size” and “shape,” we can stack the “x2” terms on top of each other to make a cube that’s 5 squares high, and we can also stack the pair of “x” terms on top of each other. Which means we can simplify our 8 term polynomial by combining like terms to build the equivalent 3 term polynomial: 5x2 + 2x + 1.
To put all of this a bit more algebraically, “like terms” in polynomials are those terms that are raised to the same power of the variable. In the case of our polynomial, all terms with a power of 2 on the variable were put into one group, all terms with a power of 1 on the variable were put into another group, and the lone term without a variable was put into its own lonely group.
The Anatomy of a Polynomial
Now that we’ve seen an example of a polynomial, it’s time to take a proper look at their anatomy and the language used to describe them. The polynomial we built—5x2 + 2x + 1—contains three terms. The first two are made from constant numerical coefficients and a variable raised to different powers, and the final term is known as a constant term since it doesn’t contain a variable.
Terms can be separated by either addition or subtraction. And, of course, each of the terms can be much more complex then the simple examples seen here. Since the exponents of variables in polynomials can only be positive integers, any expression with a negative number, fraction, or irrational number as an exponent must not be a polynomial.
Why Are Polynomials Important?
So that’s what polynomials are, but why have we bothered to build them? And why do we have all of these rules about no negative or fractional exponents? The answer is simple: It turns out that many important relationships between quantities in the world—and therefore many important math and science problems—can be expressed by equations involving polynomials. So polynomials just turn out to be really useful in the real world.
While the puzzle we talked about at the outset wasn’t exactly the most important problem in the world, it turns out that it too can be understood in terms of an equation involving a polynomial. How exactly? I’m going to let you think about that for a bit, and I’ll have the answer for you next time.
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with numbers and math, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Polynomial image from Shutterstock.
About the Author
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.