A polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"

Polynomials with one variable make nice smooth curves:

algebra girl

A polynomial can have:

constants (like 3, −20, or ½)
variables (like x and y)
exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc are allowed

that can be combined using addition, subtraction, multiplication and division ...

... except ...

... not division by a variable (so something like 2/x is right out)


A polynomial can have constants, variables and exponents,
but never division by a variable.

Also they can have one or more terms, but not an infinite number of terms.

Polynomial or Not?


These are polynomials:

(Yes, "5" is a polynomial, one term is allowed, and it can be just a constant!)


These are not polynomials


But these are allowed:

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

monomial, binomial, trinomial

How do you remember the names? Think cycles!

monocycle bicycle tricycle

There is also quadrinomial (4 terms) and quintinomial (5 terms),
but those names are not often used.


Polynomials can have no variable at all

Example: 21 is a polynomial. It has just one term, which is a constant.

Or one variable

Example: x4 − 2x2 + x   has three terms, but only one variable (x)

Or two or more variables

Example: xy4 − 5x2z   has two terms, and three variables (x, y and z)

What is Special About Polynomials?

Because of the strict definition, polynomials are easy to work with.

For example we know that:

So you can do lots of additions and multiplications, and still have a polynomial as the result.

Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

Example: x4−2x2+x


See how nice and
smooth the curve is?

You can also divide polynomials (but the result may not be a polynomial).


The degree of a polynomial with only one variable is the largest exponent of that variable.


4x3 − x + 2 The Degree is 3 (the largest exponent of x)

For more complicated cases, read Degree (of an Expression).

Standard Form

The Standard Form for writing a polynomial is to put the terms with the highest degree first.

Example: Put this in Standard Form: 3x2 − 7 + 4x3 + x6

The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

x6 + 4x3 + 3x2 − 7

You don't have to use Standard Form, but it helps.


342, 343, 1095, 1096, 3182, 3183, 3184, 3185, 1097, 4002