Degree of a Monomial
Each monomial within a polynomial has a characteristic called degree. This degree is determined by the powers (exponents) of its variables.
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Each monomial within a polynomial has a characteristic called degree. This degree is determined by the powers (exponents) of its variables.
If a monomial has only one variable, like , its degree is the exponent of that variable, which is .
Examples:
If a monomial has more than one variable, its degree is the sum of all the variable exponents.
Examples:
What about a constant (a number without variables), like ? A non-zero constant is considered to have a degree of , because we can write it as (since ).
Here is a summary of monomial degree examples in a table:
| Monomial | Degree | Explanation |
|---|---|---|
| The exponent of is . | ||
| The exponent of is . | ||
| Sum of exponents . | ||
| Non-zero constant. Can be written as . |
The degree of a monomial (with a non-zero coefficient) is the sum of the exponents of all its variables. For a monomial , its degree is .
The degree of a polynomial is determined from the degrees of its non-zero terms.
The degree of a polynomial is the highest degree among all the terms (monomials) that make up the polynomial.
Steps to determine the degree of a polynomial:
Example :
Determine the degree of the following polynomial:
The highest degree among the terms is . Therefore, the degree of this polynomial is .
Example :
Determine the degree of the following polynomial:
The highest degree among the terms is . Therefore, the degree of this polynomial is .
Example :
Determine the degree of the following polynomial:
The highest degree is . Therefore, the degree of this polynomial is .
The degree of a polynomial is the highest degree of its terms.
Is the degree of equal to , since can be written as ?
Generally in mathematics:
The exercises below use the first convention, so the zero polynomial has undefined degree. Non-zero constants have degree . Every other polynomial takes the greatest degree among its non-zero terms.
Determine the degree of
Then explain why the same rule cannot assign degree to the zero polynomial.
Find the total degree of each non-zero term:
The largest term degree is , so .
The zero polynomial has no non-zero term whose degree could be chosen as the largest. Its degree is therefore undefined. Assigning it degree would make some polynomial-degree rules fail.
Published: . Updated: .
| Sum of exponents . |