Polynomial Degree

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. For one variable the degree is the exponent itself, and for several variables it is the sum of all the exponents in the term.

Degree of a Single Variable Monomial

If a monomial has only one variable, like $ax^n$, its degree is the exponent of that variable, which is $n$.

Examples:

• The monomial $4x^5$ has a degree of $5$.
• The monomial $0.12x$ (or $0.12x^1$) has a degree of $1$.

Degree of a Multiple Variable Monomial

If a monomial has more than one variable, its degree is the sum of all the variable exponents.

Examples:

• The monomial $\frac{3}{4}x^2y^7$ has a degree of $2 + 7 = 9$.
• The monomial $2.17x^3yz^3$ (remember $y = y^1$) has a degree of $3 + 1 + 3 = 7$.

Degree of a Constant

What about a constant (a number without variables), like $5$? A non-zero constant is considered to have a degree of $0$, because we can write it as $5x^0$ (since $x^0 = 1$).

Here is a summary of monomial degree examples in a table:

Definition of Monomial Degree

The degree of a monomial (with a non-zero coefficient) is the sum of the exponents of all its variables. For a monomial $ax^n$, its degree is $n$.

Determining the Degree of a Polynomial

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:

1. Identify all the terms (monomials) in the polynomial.
2. Determine the degree of each term.
3. Choose the highest degree among all the terms. That is the degree of the polynomial.

Example 1:

Determine the degree of the following polynomial:

• The term $8x^3$ has degree $3$.
• The term $-36x^2$ has degree $2$.
• The term $54x$ (or $54x^1$) has degree $1$.
• The term $-27$ (constant) has degree $0$.

The highest degree among the terms is $3$. Therefore, the degree of this polynomial is $3$.

Example 2:

Determine the degree of the following polynomial:

• The term $5x^4y^2$ has degree $4 + 2 = 6$.
• The term $xy^2$ (or $x^1y^2$) has degree $1 + 2 = 3$.
• The term $-2x^5y^6$ has degree $5 + 6 = 11$.

The highest degree among the terms is $11$. Therefore, the degree of this polynomial is $11$.

Example 3:

Determine the degree of the following polynomial:

• The term $0.13x^3$ has degree $3$.
• The term $1.56x^2$ has degree $2$.
• The term $-2.24x$ has degree $1$.
• The term $1.72$ has degree $0$.

The highest degree is $3$. Therefore, the degree of this polynomial is $3$.

Definition of Polynomial Degree

The degree of a polynomial is the highest degree of its terms. In $4x^3 - 2x + 7$ the three terms have degrees $3$, $1$, and $0$, so the highest one fixes the degree of the whole polynomial.

Degree of the Zero Polynomial

Is the degree of $0$ equal to $0$, since $0$ can be written as $0x^0$?

Generally in mathematics:

• Non-zero constants (like $5, -27, 1.72$) have a degree of $0$.
• The zero polynomial has no non-zero term. One common convention leaves its degree undefined. Another assigns the formal value negative infinity ($-\infty$), which keeps rules such as $\deg(PQ)=\deg P+\deg Q$ valid when $P$, $Q$, or both are the zero polynomial.

The exercises below use the first convention, so the zero polynomial has undefined degree. Non-zero constants have degree $0$. Every other polynomial takes the greatest degree among its non-zero terms.

Exercise

Determine the degree of

Then explain why the same rule cannot assign degree $0$ to the zero polynomial.

Worked Solution

Find the total degree of each non-zero term:

• $4x^3y^2$ has degree $3+2=5$.
• $-7x^2y^5$ has degree $2+5=7$.
• $6$ has degree $0$.

The largest term degree is $7$, so $\deg P=7$.

The zero polynomial has no non-zero term whose degree could be chosen as the largest. Its degree is therefore undefined. Assigning it degree $0$ would make some polynomial-degree rules fail.