What is a monomial in algebra?

A monomial in algebra is a single term that is a number, a variable, or a product of numbers and variables with whole-number exponents, such as 7, x, 4x^3, or −2ab^2. It does not include addition, subtraction, or variables in the denominator.

What is a monomial vs a polynomial?

A monomial is one term, like 5x^2 or 3ab. A polynomial is an expression made by adding or subtracting monomials, such as 5x^2 − 3x + 1. Every monomial is a simple kind of polynomial, but not every polynomial is a monomial.

How do you multiply a binomial and a monomial?

To multiply a binomial and a monomial, multiply the monomial by each term in the binomial, then add the results. For example, 3x(2x + 5) = 3x × 2x + 3x × 5 = 6x^2 + 15x.

How do you simplify a fraction with monomials?

To simplify a ratio of two monomials, divide the number parts and use exponent rules on the variable parts. For instance, (12x^3y) ÷ (4xy^2) = 3x^2 ÷ y, because 12 ÷ 4 = 3, x^3 ÷ x = x^2, and y ÷ y^2 = 1 ÷ y = 1/y.

What Is a Monomial? Definition and Examples

Monomial explained for kids

A monomial is like one math "piece" made of numbers and letters multiplied together. For example, 5x, 3ab, and 7 are all monomials. There are no plus or minus signs inside a monomial. As soon as you see a plus or minus, you have more than one term, not a monomial.

Worked examples with monomials

Example 1 (middle school level): Multiply two monomials

Multiply 3x² and 4x.

First, multiply the numbers: 3 × 4 = 12.

Then, multiply the variable parts. x² × x is x³, because when you multiply powers with the same base, you add the exponents: 2 + 1 = 3.

So 3x² × 4x = 12x³, which is still a monomial because it is one term.

Example 2 (high school level): Divide a polynomial by a monomial

Divide 12x³y² + 6x²y by 3xy.

Write the division as a fraction: (12x³y² + 6x²y) ÷ 3xy.

Divide each term in the polynomial by the monomial 3xy:

First term: 12x³y² ÷ 3xy.

Numbers: 12 ÷ 3 = 4.
x variables: x³ ÷ x = x² (subtract exponents: 3 − 1 = 2).
y variables: y² ÷ y = y¹ = y.

So the first term becomes 4x²y.

Second term: 6x²y ÷ 3xy.

Numbers: 6 ÷ 3 = 2.
x variables: x² ÷ x = x¹ = x.
y variables: y ÷ y = 1.

So the second term becomes 2x.

Put the results together: 4x²y + 2x. This is a polynomial made of two monomials.

Going deeper with monomials

Monomials are the building blocks of polynomials, which are expressions made by adding or subtracting monomials. For example, 2x² + 3x − 5 is a polynomial made of three monomials: 2x², 3x, and −5.

As your child moves into more advanced algebra, monomials appear in many skills: multiplying monomials, multiplying a monomial by a binomial or a longer polynomial, and dividing a polynomial by a monomial. These all rely on understanding how to multiply and divide the number part and the variable part of a monomial using the laws of exponents.

Factoring also uses monomials. To factor expressions, students often look for a greatest common monomial factor, which is the largest monomial that divides each term. Pulling this common monomial factor out lets them rewrite an expression as a product, which is important for solving equations and simplifying rational expressions.

Common mistakes with monomials

Mistake: Thinking any expression with variables is a monomial, for example calling x + 3 a monomial.
Correction: A monomial has only one term with no plus or minus signs inside; x + 3 has two terms, so it is a polynomial, not a monomial.
Mistake: Forgetting to add exponents when multiplying monomials with the same base, such as saying x² × x³ = x⁶ because 2 × 3 = 6.
Correction: When multiplying powers with the same base, keep the base and add the exponents: x² × x³ = x²⁺³ = x⁵.
Mistake: Dropping variables when dividing by a monomial, for example simplifying 8x²y ÷ 4x to 2y instead of 2xy.
Correction: Divide numbers and variables separately: 8 ÷ 4 = 2, x² ÷ x = x, and y stays, so the result is 2xy.

How monomials show up in everyday life

Monomials can describe simple formulas your child might see in real life. For example, 5t could represent earning 5 dollars for each hour of babysitting, or 3x² could model how the area of a square grows when each side is x units long. Any time a situation can be written as a single product of numbers and variables, a monomial is being used.

Related algebra terms to explore

Monomials connect to several other key algebra ideas your child will study:

What is a monomial?