Polynomials
Definitions
A monomial is an algebraic expression that contains exactly one term.
The term may be a constant, or the product of a constant and one or more
variables. The exponent of any variable must be a nonnegative integer (that
is, a whole number).
The following are monomials:
A monomial in one variable, x, can be written in the form ax
r, where a is
any real number and r is a nonnegative integer.
The following are not monomials:
 |
The denominator contains a variable with a positive exponent.
So the term cannot be written in the form axr where r is a
nonnegative integer. |
 |
There is a squared variable under a cube root symbol.
So the term cannot be written in the form axr where r is a
nonnegative integer. |
A polynomial is the sum of one or more monomials.
Here are some examples:
| 2x3 - 5x + 2 |
x + 2 |
5 |
3xy2 - 7x + 5y - 1 |
A polynomial with one, two, or three terms has a special name.
| Name
|
Number
of terms |
Examples |
| monomial
binomial
trinomial |
1
2
3 |
x, 5y, 3xy3, 5
x + 1, 2x2 - 3, 5xy3 + 4x3y2
x2 + 2x + 1, 3x2y3 - xy + 5 |
Note:
A polynomial with 4 terms is called a
“four term polynomial.â€
A polynomial with 5 terms is called a
“five term polynomial,†and so on.
Example 1
Determine if each expression is a polynomial.
Solution
a. The expression is a polynomial.
It has one term, so it is a monomial.
The term has the form awr, where a = -4 and r = 3.
b. The expression is not a polynomial.
The term
cannot be written in the form axr where r is a
nonnegative integer.
c. The expression is not a polynomial.
The term
cannot be written in the form axr where r is a
nonnegative integer.
d. The expression is a polynomial.
It has two terms, so it is a binomial.
Each term can be written in the form axr:
x + 2
1x1 + 2x0