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Math101learn.math101.caMonomial
A monomial is a single polynomial term: a constant times variables raised to nonnegative integer exponents.
Monomials are polynomial building blocks. Their coefficient and exponent rules power multiplication, division, factoring, dimensional analysis, and scientific notation.
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
- Simplify products and powers and combine numerical factors.
- Check that the expression has one top-level term.
- Inspect every variable exponent; each must be a nonnegative integer.
- Ensure no variable remains in a denominator or radical.
- If valid, state coefficient and degree.
Worked example
Representations and interpretation
A monomial can be encoded by a coefficient and exponent vector, such as $-8x^5y^5\leftrightarrow(-8;5,5)$. Multiplication adds exponent vectors and multiplies coefficients.
Reasoning about variations
$x^0=1$ for nonzero $x$, so $5x^0$ simplifies to constant monomial $5$. By contrast, $x^{-1}=1/x$ leaves the polynomial family and is not a monomial under this definition.
Common mistakes
How to check your work
- Rewrite variable denominators as negative exponents to expose invalid cases.
- Count top-level terms after full simplification.
- Reconstruct the term from its coefficient and exponent vector.
Practice
- Is $-6a^3b^2$ a monomial, and what is its degree?
- Is $4/x$ a monomial?
- Simplify $(3x^2)(-2x^5)$.
Answers and brief solutions
Show answers
- Yes; degree $5$ It is one term and $3+2=5$.
- No $4/x=4x^{-1}$ has a negative exponent.
- $-6x^7$ Multiply coefficients and add exponents.
Synthesis and transfer
A rectangular volume written as one coefficient times powers of its dimensions is a monomial; exponent addition records repeated dimensions when the factors are multiplied.
A box with dimensions $3x$, $2x^2$, and $5y$ has volume $30x^3y$, one product with nonnegative integer exponents. Coefficients multiply and exponents on the same base add because repeated factors are being counted. The expression $30x^3/y$ is not a monomial in the polynomial sense, since rewriting it gives exponent $-1$ on $y$; neither is $\sqrt{x}$, whose exponent is $1/2$. A zero coefficient yields the zero polynomial, whose degree needs special treatment even though it can appear as a single term. Structural classification should precede applying monomial exponent rules.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Is $-6a^3b^2$ a monomial, and what is its degree?
- It is one term and $3+2=5$.
End of lesson
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