# What is a Polynomial? The Elements of Polynomials

## The elements of a polynomial

A polynomial is an algebraic expression made up of two or more terms. Polynomials can be made up of some or all of the following:

**Variables**- these are letters like x, y, and b**Constants**- these are numbers like 3, 5, 11. They are sometimes attached to variables, but can also be found on their own.**Exponents**- exponents are usually attached to variables, but can also be found attached to a constant. Examples of exponents include the 2 in 5² or the 3 in x³.**Addition, subtraction & multiplication**- For example, you can have 2x (multiplication), 2x+5 (multiplication and addition), and x-7 (subtract.)

## Structural Rules to Polynomials

There are a few rules as to what you can't have in a polynomial:

Polynomials __cannot contain division__.

For example, 2y^{2}+7x/4 is not a polynomial as it contains division.

Polynomials __cannot contain negative exponents__.

You cannot have 2y^{-2}+7x-4. Negative exponents are actually a form of division (in order to make the negative exponent positive, you have to do division.) For example, x^{-3} is the same thing as 1/x^{3}.

Polynomials __cannot contain fractional exponents__.

Terms containing fractional exponents (such as 3x+2y^{1/2}-1) are not considered polynomials.

Polynomials __cannot contain radicals__.

For example, 2y^{2} +√3x + 4 is not a polynomial.

## What does 'polynomial' mean?

The "poly" in polynomial means "multiple" and "nomial" refers to terms, so polynomial means "multiple terms."

## How to find the degree of a polynomial

To find the degree, write down the terms of the polynomial in descending order by the exponent. The term whose exponents add up to the highest number is the leading term. The sum of the exponents is the degree of the equation.**Example**: Figure out the degree of 7x^{2}y^{2}+5y^{2}x+4x^{2}.

Start out by adding the exponents in each term.

The exponents in the first term, 7x^{2}y^{2} are 2 (from 7x^{2}) and 2 (from y^{2}) which add up to 4.

The second term (5y^{2}x) has two exponents. They are 2 (from 5y^{2}) and 1 (from x, this is because x is the same as x^{1}.) The exponents in this term add up to 3.

The last term (4x^{2}) only has one exponent, 2, so its degree is just 2.

Since the first term has the highest degree (the 4th degree), it is the leading term. The degree of this polynomial is 4.

## Different types of polynomials

There are different ways polynomials can be categorized. They can be named for the degree of the polynomial as well as by the number of terms it has.

**Monomials - these are polynomials containing only one term ("mono" means one.) 5x, 4, y, and 5y4 are all examples of monomials.****Binomials - these are polynomials that contain only two terms ("bi means two.) 5x+1 and y-7 are examples of binomials.****Trinomials - a trinomial is a polynomial that contains three terms ("tri" mean three.) 2y+5x+1 and y-x+7 are examples of trinomials.**

There are quadnomials (four terms) and so on, but these are usually just simply called polynomials regardless of the number of terms they contain.

A polynomial can also be named for its degree. If a polynomial has the degree of two, it is often called a quadratic. If it has a degree of three, it can be called a cubic. Polynomials with degrees higher than three aren't usually named (or the names are seldom used.)

## Multiplying Polynomials

Now that you understand what makes up a polynomial, it's a good idea to get used to working with them. If you're taking an algebra course, chances are you'll be multiplying polynomials (if you're not already doing so.)

Multiplying polynomials might sound terrifying (I hated it at first), but take a deep breath and slowly work through the guide, "How to Multiply Polynomials, With Examples."

**© 2012 Melanie Shebel**

## Comments

just looking at those equations caused my brain to breakout into a civil war. :)

A great hub. I love maths, but I'm a little rusty on the terminology. So thanks!

Excellent guide. I have a feeling I'll be referring back to it as my kids get a little older! :)

Melbel I will not take your quiz because I already know I will fail hehe Math never was my thing. Oddly enough my daughter (11) is a math genius and I am going to let her read this tomorrow. She will love it :)

Another great math hub Mel. Very useful for those struggling with these concepts and there are many out there including parents struggling to help their kids in grades 6 to 8 with basic algebra.

A very nice treatment of this topic and I think you should also create a YouTube channel and make short videos to go with each of your hubs and before long you will have lots of mathematics students following you. Great work.

Nice basic outlay about polynomials... informative

I have to confess, I got confused and frustrated after the first paragraph. Math and I don't get on.

But from what I could comprehend this seems to be a good hub and I don't doubt you'll be helping loads of people who maybe didn't understand their instructor's explanation.

Voted up and useful.

Excellent explanation of what a polynomial is.

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