The degree of a monomial is the sum of all the exponents on its variables. For a single-variable term like 4x³, the degree is simply 3. For a term with multiple variables like 6x²yz³, you add the exponents together: 2 + 1 + 3 = 6.
How It Works With One Variable
A monomial is a single term: a number, a variable, or a number multiplied by one or more variables. When only one variable is involved, the degree equals the exponent on that variable. So 5x³ has a degree of 3, and 2y⁷ has a degree of 7.
One detail that trips people up: a variable written without a visible exponent actually has an exponent of 1. The term 9x really means 9x¹, so its degree is 1.
How It Works With Multiple Variables
When a monomial contains more than one variable, the degree is the sum of all the exponents. Take the term x²y as an example. The exponent on x is 2 and the exponent on y is 1, so the degree is 2 + 1 = 3.
Here are a few more examples to make the pattern clear:
- 10a²b²: degree = 2 + 2 = 4
- −6x²yz³: degree = 2 + 1 + 3 = 6
- 9xy⁷: degree = 1 + 7 = 8
Notice that the number in front of the variables (the coefficient) never affects the degree. Whether the coefficient is 10 or −6 or 900, you ignore it and focus only on the exponents.
The Degree of a Constant
A plain number with no variable, like 5 or −12, is still a monomial. Its degree is 0. You can think of it this way: 5 is the same as 5x⁰, because any number raised to the zero power equals 1. The exponent is 0, so the degree is 0.
There is one exception. The number 0 itself is a special case. Because it has no nonzero terms, mathematicians generally leave the degree of 0 undefined. Some textbooks define it as negative infinity for convenience, but if you encounter this on homework, “undefined” is the standard answer.
What Counts as a Monomial
For an expression to qualify as a monomial, its exponents must be whole numbers (0, 1, 2, 3, and so on). A term like x⁻² or x^(1/2) does not meet this requirement. If you see a negative exponent, rewrite it as a fraction first. For instance, x⁻² becomes 1/x², which is not a monomial because the variable is in the denominator. You would not assign it a monomial degree.
Why the Degree Matters for Polynomials
A polynomial is just a sum of monomials, and the degree of each individual term determines the degree of the entire polynomial. The polynomial’s degree equals the highest degree among all its terms. In 3x⁴ + 2x² + 7, the term degrees are 4, 2, and 0, so the polynomial has degree 4.
This is also why polynomials are written in standard form with terms listed from highest degree to lowest. That ordering puts the most important term, the one that controls the polynomial’s long-run behavior, first.