A monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.
A monomial is one term: a coefficient times variables with whole-number exponents. Add the exponents to get its degree.
Formal definition and notation
A monomial is an expression of the form ax1k1x2k2 ⋯ xmkm, where the coefficienta is a real number and each exponent ki is a nonnegative integer. Its degree is k1 + k2 + ⋯ + km; a nonzero constant has degree 0. A sum of monomials is a polynomial, so a monomial is a polynomial with exactly one term.
Symbol
Read it as
Meaning
5x3
five x cubed
coefficient 5, variable x, exponent 3, degree 3
ab
a b
letters side by side are multiplied; the hidden coefficient is 1 and the degree is 1 + 1 = 2
−y
negative y
the hidden coefficient is −1
Key facts
A plain number such as 12 is a monomial of degree 0. The number 0 is also a monomial, but it is given no degree (or degree −∞ in some books).
Dividing by a number is allowed: x4 = 14x is a monomial. Dividing by a variable is not: 4x = 4x−1 has a negative exponent.
The product of two monomials is always a monomial: (2x3)(−5x2y) = −10x5y.
The sum of two monomials is a monomial only when they are like terms; otherwise it is a binomial.
Name by number of terms: monomial (1), binomial (2), trinomial (3), polynomial (any number).
Rules for working with monomials
Rules for working with monomials
axm · bxn = abxm + n
a, b
the coefficients: multiply them
m, n
exponents on the same base: add them
To multiply monomials, multiply the coefficients and add the exponents of matching variables. To add monomials, they must be like terms: 3x2 + 5x2 = 8x2, but 3x2 + 5x stays as two terms.
Monomial examples, step by step
Example 1: Find the coefficient and degree
Warm-up
Give the coefficient and degree of −4a3b2c.
The coefficient is the number in front: −4.
List the exponents: a has 3, b has 2, c has 1 (an unwritten exponent is 1).
Add them: 3 + 2 + 1 = 6.
Answer: The coefficient is −4 and the degree is 6.
Answer:(3x2y)(−6x4y3) = −18x6y4, a monomial of degree 10.
Example 3: Which expressions are monomials?
Test-style
Which of these are monomials? 23m5, 72, 9n2, x2 − 1
23m5: a fraction coefficient is fine, and the exponent 5 is a whole number. Monomial.
72 = 49: a constant. Monomial (degree 0).
9n2 = 9n−2: negative exponent. Not a monomial.
x2 − 1: two terms joined by subtraction. Not a monomial (it is a binomial).
Answer: Only 23m5 and 72 are monomials.
Common mistakes
✗ Mistake3x2 + 4x3 = 7x5.
✓ Instead3x2 + 4x3 cannot be combined; it stays as two terms.
You can only add monomials that are like terms (same variables with the same exponents). Adding exponents is the rule for multiplying, not adding.
✗ MistakeThe degree of 5x2y3 is 3, the biggest exponent.
✓ InsteadThe degree is 2 + 3 = 5.
For one monomial, the degree is the sum of all the variable exponents. The highest exponent is used to compare the terms of a polynomial, not for a single term.
✗ Mistake6x is a monomial because it is one term.
✓ Instead6x = 6x−1 is not a monomial.
Monomial exponents must be whole numbers (0, 1, 2, …). A variable in the denominator means a negative exponent.
✗ MistakeThe degree of the monomial 9 is 1.
✓ InsteadThe degree of 9 is 0.
A constant has no variable, which you can think of as 9x0. Its degree is 0.
Every like term is a monomial, but “like” describes a pair: 2xy and −7xy are like terms because their variable parts match.
Quick check: 3 questions
Question 1 of 3
Which expression is a monomial?
Show answer and explanation
Answer: C.−12x4y is one term with a number coefficient and whole-number exponents. 3x + y has two terms, 5x has a variable in the denominator and √x is x to the one-half power.
Question 2 of 3
What is the degree of 8a2b5?
Show answer and explanation
Answer: D. Add the exponents: 2 + 5 = 7. 8 is the coefficient, 5 is only the largest exponent and 10 multiplies the exponents.
Question 3 of 3
Simplify (−2m3)(4m2).
Show answer and explanation
Answer: A. Multiply coefficients, (−2)(4) = −8, and add exponents, 3 + 2 = 5. m6 multiplies the exponents, 2m5 adds the coefficients, and m9 is a slip.
Monomial: frequently asked questions
Is a constant like 7 a monomial?
Yes. A constant is a monomial with no variables, so its degree is 0 (think of 7 as 7x0). The number 0 is also counted as a monomial, but most textbooks say it has no degree, because 0 = 0x5 = 0x9 and so on, so no single exponent fits.
Can a monomial have a fraction or a negative coefficient?
Yes. The coefficient can be any real number: −6x, 34y2 and 0.5ab are all monomials. The restriction is on the variables, not the number: every exponent on a variable must be a whole number, so no variables in denominators and no roots of variables.
How do you find the degree of a monomial?
Add the exponents of all its variables. In 6x3y2z, the exponents are 3, 2 and 1, so the degree is 6. Remember that a variable with no written exponent has exponent 1, and that the coefficient never affects the degree.
VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Real NumberA real number is any number that can be placed on the number line. 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Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. 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