Polynomial Degree

Understanding the 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.

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 Multi-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

Once we know how to determine the degree of each monomial (term), finding the degree of a polynomial becomes easier.

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.

What About the Degree of Zero?

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 (the number 0 itself) is often considered to have no degree or sometimes is said to have a degree of negative infinity ($-\infty$). The reason is a bit complex, but essentially it helps keep properties of degrees (like the degree of the product of two polynomials) consistent.

However, for the high school level, understanding that non-zero constants have degree 0 and the degree of a polynomial is the highest degree of its terms is sufficient.