Monomial — Definition, Examples & Degree

TL;DR
A monomial is an algebraic expression made of a single term — a constant coefficient multiplied by one or more variables, each raised to a non-negative integer exponent. This article covers the definition, the parts of a monomial, how to find the degree, three worked examples, and the line that separates a monomial from a binomial, trinomial, and general polynomial.

The Smallest Building Block of Algebra

Every polynomial — from x² + 3x + 5 to a thirty-term mess — is built from monomials, joined by plus and minus signs. The monomial is to algebra what the atom is to chemistry: the smallest meaningful piece.

What a Monomial Is

A monomial is an algebraic expression that consists of a single term. A term, in turn, is a product of:

• a constant (the coefficient), and
• zero or more variables, each raised to a non-negative integer exponent.

Examples — 555, 3x³, −7y², 12ab, 4x²y², πr² are all monomials.

What is not a monomial:

• 3x + 2 — two terms (a binomial, not a monomial).
• x = x¹/² — fractional exponent (not a non-negative integer).
• 1/x = x⁻¹ — negative exponent.
• 2x²^x — variable in the exponent (an exponential expression, not a monomial).

Quick facts.

• Single term: no addition or subtraction inside the expression.
• Coefficient: any real number (including 0, 1, fractions, π).
• Exponents: must be non-negative integers (0, 1, 2, 3, …).
• Variables: any number of them, including zero (a constant is a monomial).
• Degree: sum of all variable exponents.

The Three Parts of a Monomial

Every monomial has three identifiable parts.

1. The coefficient.

The numerical factor in front. In 7x², the coefficient is 7. In −3xyz, the coefficient is −3. In x², the coefficient is the unwritten 1.

2. The variables.

The letters. In 7x², the variables are x and y.

3. The exponents.

The power each variable is raised to. In 7x², the exponents are 2 (on x) and 1 (on y, unwritten).

How to Find the Degree

The degree of a monomial is the sum of the exponents of all its variables.
deg(monomial) = ∑(variable exponents).

• Degree of 555 — no variables, exponent sum is 0. Degree 0.
• Degree of 3x³ — one variable with exponent 1. Degree 1.
• Degree of 4x² — one variable with exponent 2. Degree 2.
• Degree of 7x²y — exponents 2 and 1. Degree 2 + 1 = 3.

A constant (with no variables) has degree 0 — sometimes called a zero-degree monomial.

Worked Examples of Monomial

Quick.

Identify the coefficient, variables, and degree of −6a²b.

• Coefficient: −6. Variables: a and b. Exponents: 2 (on a) and 1 (on b).
• deg = 2 + 1 = 3.
  Final answer: coefficient −6, variables a and b, degree 3.

Standard (Wrong Path First — Tripping Points to Avoid).

Find the degree of 3x⁴yz².

• The wrong path: Degree 7.
• The flaw: Negative exponent violation.
• Final answer: 3x⁴yz² is not a monomial.

Stretch.

Multiply 4x²y⋅(−3xy³)⋅2x⁴y².

• Multiply coefficients: −24.
• Combine variables: -24x⁷y⁶. Degree: 7 + 6 = 13.
• Final answer: −24x⁷y⁶, degree 13.

Where Monomials Show Up in the Real World

• Area and volume formulas. A=πr² (degree 2 in r). V=43πr³ (degree 3).
• Newton's law of gravitation. F=Gm₁m₂/r² (more strictly a rational expression).
• Polynomial regression. y=a+bx+cx²+dx³ is a sum of monomials.
• Computer science. Time complexity is a monomial in input size.

Common Confusions With Monomial

1. Treating x as a variable to the power 1.

The correct way: x = x¹/² is not a monomial.

2. Confusing the coefficient with the degree.

The correct way: The degree comes from the exponents, not the coefficient.

3. Missing the hidden exponent of 1.

The correct way: Every variable carries an exponent.

4. Confusing monomial with polynomial.

The correct way: 3x+5 is a binomial. A monomial has exactly one term.

Conclusion

• A monomial is a single-term expression — coefficient times variables raised to non-negative integer exponents.
• The degree of a monomial is the sum of its variable exponents.
• Common mistake: negative or fractional exponents are not monomials.

Practice These Three Before Moving On

1. Find the degree of −5x³y²z⁴.
2. Multiply 2a²b⋅(−4ab³) and report the result as a single monomial.
3. Is x⋅y² a monomial? Explain why or why not.