A monomial is an expression in algebra that contains one term, like 3xy. Monomials include numbers, variables or multiple numbers and/or variables that are multiplied together. Any number all by itself is a monomial, like 5 or 2,700. A monomial can also be a variable, like "b" or "y." It can also be a combination of these, like 98b or xy. Explore the rules of monomials, monomial degrees and monomial examples.
Monomial Definition
One of the first math terms you learn in algebra is monomials. When it comes to understanding the definition of a monomial, it’s pretty easy. Mono means “one.” So, monomial functions are those expressions that only have the one term. While a monomial can be a single number, variable or combination of a number and variables, it can’t be a negative exponent. Therefore, monomials have two rules.
The Rules of Monomials
Math always includes a few rules and monomials aren’t any different. There are two rules to remember about monomials. In these examples, the * symbol stands for multiplication.
1. A monomial multiplied by a monomial is also a monomial.
2 * 2 = 4 (a monomial)
2 * x = 2x
2 * 6 = 12
2 * y = 2y
2. A monomial multiplied by a constant (number) is also a monomial.
-13 * 7z = -91z (13 is the constant and 7z the monomial)
(⅛) * 8mn = -mn (⅛ is the constant and 8mn the monomial)
(⅕) * 5p = p (⅕ is the constant and 5p is the monomial)
Examples: Numbers That Are Monomials
Now, it's time to really look at a few examples of monomials. Monomials are positive numbers. It doesn't matter how big they are, they are still a monomial. See a few examples of monomial numbers in action.
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- 100
- 500
- 1,000
- 5,000
- 10,000
- 3,598,772
- 4,000,000
- 14,000,000
- 14,100,300
- 20,000,000
Examples: Variables That Are Monomials
It might be hard to think of a variable as a monomial, especially when you get to a group like abc. But remember one of the rules of monomials, a monomial multiplied by a monomial is still, you guessed it, a monomial. Therefore, variables multiplied by each other are also monomials. Look at a few examples.
- x
- y
- xy
- abc
- mx
- n
- b
- w
- l
- s
- bxy
- a
- ax
Examples: Combinations of Numbers and Variables That Are Monomials
Numbers and variables aren't going to stand alone when it comes to monomials. They can work together. Therefore, 645a is still a monomial. Explore a few other examples of combinations of monomials.
- 1x
- 2y²
- 32x³y
- 653abc
- 2g7g9g
Degrees of a Monomial Function
You might have noticed in the combinations that some monomials have an exponent. For example, 2y² has an exponent of 2. It also has a degree of 2. In a monomial, you can add the exponents of the variables together to find the degree of a monomial function. The exponent for a constant is always 0, and the exponent for a variable that doesn't have an exponent listed is always 1. For example:
5ab3c4
5, exponent = 0
a, exponent = 1
b, exponent = 3
c, exponent = 4
The total degree of this equation is 0 + 1 + 3 + 4 = 8
What Is Not a Monomial
A monomial is an algebraic expression with only one term, so 132y÷ 9y or 47z + 2y are not monomials. These are called polynomials since they have a finite number of terms. Explore a few examples of equations that are not monomials.
- 9a2 + 45a - 2
- 8x5 + 21x3 - 9x
- 8a - 15
Understanding Monomials
All algebraic expressions with one term are examples of monomials — so there are countless examples that encompass the entire number system. Ready to keep your math knowledge going? Take a look at polynomials and monomials.