43.5 Monomial Degree

Monomial Degree is the value describing the total exponent weight carried by the variable factors of a single monomial term, calculated as the exponent itself for a single variable or as the sum of all variable exponents when more than one variable is present.

Single-Variable Monomial Degree

Definition

For a monomial containing only one variable, the degree of that monomial is simply the exponent attached to that variable.

Example

The monomial $5x^4$ has a degree of $4$.

Multivariable Exponent-Sum Degree

Definition

For a monomial containing more than one variable, the degree of that monomial is the sum of the exponents attached to every variable factor present in the term.

Example

The monomial $3x^2{y}^3$ has a degree of $2+3$, which equals $5$.

Nonzero Constant Monomial Degree

Definition

A nonzero constant monomial, containing no variable factors at all, is assigned a degree of zero, consistent with the understanding that a constant can be represented as a variable raised to the zero power.

Example

The monomial $8$ has a degree of $0$.

Zero-Coefficient Monomial Degree Exclusion

Special Case

A monomial with a coefficient of exactly zero, evaluating to zero regardless of its variable factors, is excluded from having a meaningfully defined degree, since this term contributes no value and is typically omitted from a polynomial entirely.

Reasoning

Assigning a specific degree to the zero monomial would require choosing among the exponents of variables that were never truly contributing to a nonzero value, so this case is treated as a special exception outside the standard degree definitions.

Monomial Degree Calculation

Procedure

To calculate the degree of any monomial, every variable factor present in the term is identified along with its exponent, and these exponents are added together, treating a single-variable monomial as a special case of this same sum with only one term.

Example

For $7x^1{y}^2{z}^0$, the degree is calculated as $1+2+0$, which equals $3$.

Monomial Degree Comparison

Procedure

Two monomials can be compared by their degree, with the monomial carrying the larger summed exponent value considered to have the higher degree, regardless of how many distinct variables each one contains.

Example

The monomial $2x^3$, with a degree of $3$, is compared against $4xy$, with a degree of $2$; the first monomial has the higher degree, even though the second monomial involves two different variables.