Mathwords: Monomial

Q: How do you find the degree of a monomial?

Add up all the exponents on the variables. For example, $6x^4y$ has degree $4 + 1 = 5$. A nonzero constant like $-3$ has degree 0.

Key Formula

ax^m y^n \cdots

Where:

$a$ = The coefficient — any real number (e.g., 5, −3, or 1)
$x, y, \ldots$ = Variables
$m, n, \ldots$ = Non-negative integer exponents (0, 1, 2, 3, …)

Worked Example

Problem: Identify which of the following are monomials:

7x^2y

,

3x + 2

,

\dfrac{5}{x}

,

-9

,

x^{1/2}

.

Step 1: Check

7x^2y

. It is a single term with coefficient 7, and both exponents (2 on

x

, 1 on

y

) are non-negative integers.

7x^2y \quad \checkmark \text{ — monomial}

Step 2: Check

3x + 2

. This expression has two terms separated by addition, so it is a binomial, not a monomial.

3x + 2 \quad \times \text{ — not a monomial}

Step 3: Check

\frac{5}{x}

. Rewrite it as

5x^{-1}

. The exponent

-1

is negative, so this is not a monomial.

\frac{5}{x} = 5x^{-1} \quad \times \text{ — not a monomial}

Step 4: Check

-9

. A constant counts as a monomial (think of it as

-9x^0

).

-9 \quad \checkmark \text{ — monomial}

Step 5: Check

x^{1/2}

. The exponent

\frac{1}{2}

is not an integer, so this is not a monomial.

x^{1/2} \quad \times \text{ — not a monomial}

Answer:

7x^2y

and

-9

are monomials. The other three are not.

Another Example

Problem: Find the degree of the monomial

-4x^3y^2z

.

Step 1: Identify the exponent on each variable:

x

has exponent 3,

y

has exponent 2, and

z

has exponent 1.

-4x^3y^2z^1

Step 2: The degree of a monomial is the sum of all the variable exponents.

3 + 2 + 1 = 6

Answer: The degree of

-4x^3y^2z

is 6.

Frequently Asked Questions

Is a single number like 7 a monomial?

Yes. A plain number (constant) is a monomial of degree 0. You can think of 7 as

7x^0

, which fits the definition: one term with a non-negative integer exponent.

How do you find the degree of a monomial?

Add up all the exponents on the variables. For example,

6x^4y

has degree

4 + 1 = 5

. A nonzero constant like

-3

has degree 0.

Monomial vs. Binomial / Trinomial

A monomial has exactly one term (e.g.,

5x^2

). A binomial has exactly two terms (e.g.,

3x + 1

), and a trinomial has exactly three terms (e.g.,

x^2 + 2x + 1

). All three are specific types of polynomials, classified by their number of terms.

Why It Matters

Monomials are the building blocks of all polynomials — every polynomial is a sum of monomials. Understanding monomials makes it easier to add, subtract, and multiply polynomials term by term. They also appear in exponent rules, factoring, and the distributive property, so recognizing them quickly is a core algebra skill.

Common Mistakes

Mistake: Thinking that

\frac{5}{x}

or

x^{-2}

are monomials because they look like single terms.

Correction: Monomial exponents must be non-negative integers (0, 1, 2, …). A variable in a denominator produces a negative exponent, which disqualifies the expression.

Mistake: Confusing the coefficient with the degree.

Correction: The coefficient is the numerical factor in front (e.g., 4 in

4x^3

). The degree is the sum of the variable exponents (3 in

4x^3

). They measure different things.

Related Terms

Polynomial — General expression; sum of one or more monomials
Binomial — Polynomial with exactly two terms
Trinomial — Polynomial with exactly three terms
Term — Each monomial within a polynomial
Coefficient — The numerical factor of a monomial
Degree of a Polynomial — Highest degree among its monomial terms
Exponent — Power to which a variable is raised