Key takeaway: A monomial is an algebraic expression made of just one term: a constant, a variable, or several of these multiplied, with exponents that are whole numbers.
Grade 9 – Algebra 1A-SSE.A.1aExpressions and Equations
Definition of Monomial
A monomial is an expression with exactly one term. The numbers \(-6\) and \(0.4\), a lone variable such as \(y\), and products like \(3xy^{2}\) or \(-9a^{3}b\) are all monomials. The prefix “mono” means one.
What disqualifies an expression? A plus or minus sign between pieces, since \(2x + 1\) has two terms. A variable in a denominator, such as \(\frac{5}{x}\), a negative exponent, a fractional exponent, or a variable under a radical sign. The numerical factor is the coefficient, and the sum of the exponents on the variables is the degree of the monomial.
Formula and notation
A monomial in standard form looks like
\[k\,x^{a}y^{b}\cdots\]
with a coefficient \(k\) and whole-number exponents \(a, b\), and so on. The degree is \(a + b + \cdots\). For \(3xy^{2}\), the coefficient is 3 and the degree is \(1 + 2 = 3\).
Key facts
- A nonzero constant such as 12 has degree 0.
- The degree adds the exponents of every variable in the term.
- Multiplying such terms multiplies coefficients and adds exponents of like variables.
- Two of them with identical variable parts are like terms.
Monomial example
Decide which of \(-6\), \(3xy^{2}\), \(\frac{5}{x}\), \(2x + 1\) are monomials, and find the degree of each.
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\(-6\) is a single constant term, so it qualifies with degree 0.
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\(3xy^{2}\) is one product; the exponents are 1 and 2, so its degree is \(1 + 2 = 3\).
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\(\frac{5}{x}\) has a variable in the denominator, and \(2x + 1\) has two terms, so neither qualifies.
\(-6\) (degree 0) and \(3xy^{2}\) (degree 3) are the monomials.
Common mistake
Watch out
Remy gives the degree of \(3xy^{2}\) as 2, because the biggest exponent he sees is 2.
Why it is wrong
For a term with several variables, add all the exponents. A variable written without an exponent has exponent 1, so \(3xy^{2}\) has degree \(1 + 2 = 3\). Taking only the largest exponent works for one variable but not for two or more.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Expressions and Equations