Monomial

Intuition and core definition

A monomial is a single polynomial term: a constant times variables raised to nonnegative integer exponents. Examples include $7$, $-3x^2y$, and $a^5$. Expressions with variable denominators, negative or fractional variable exponents, or addition between nonzero terms are not monomials.

Notation, language, and conditions

$c x_1^{a_1}\cdots x_n^{a_n}$ is a monomial when $c$ is constant and each exponent $a_i$ is a nonnegative integer. Its total degree is $a_1+\cdots+a_n$ when $c\ne0$. A nonzero constant has degree $0$.

Why this idea matters

A monomial is one polynomial term with nonnegative integer variable exponents, a restriction that separates it from rational and radical expressions.

A dependable method

1. Simplify products and powers and combine numerical factors.
2. Check that the expression has one top-level term.
3. Inspect every variable exponent; each must be a nonnegative integer.
4. Ensure no variable remains in a denominator or radical.
5. If valid, state coefficient and degree.

Worked example

Common mistakes