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, 2y2+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/x3.
Polynomials cannot contain fractional exponents.
Terms containing fractional exponents (such as 3x+2y1/2-1) are not considered polynomials.
Polynomials cannot contain radicals.
For example, 2y2 +√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 7x2y2+5y2x+4x2.
Start out by adding the exponents in each term.
The exponents in the first term, 7x2y2 are 2 (from 7x2) and 2 (from y2) which add up to 4.
The second term (5y2x) has two exponents. They are 2 (from 5y2) and 1 (from x, this is because x is the same as x1.) The exponents in this term add up to 3.
The last term (4x2) 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.)
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."