# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/polynomial/polynomial-degree Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/polynomial/polynomial-degree/en.mdx Learn finding polynomial degrees by identifying the highest power term. Learn monomial degrees, multi-variable cases, and worked examples. --- ## 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 $$n$$. Visible text: If a monomial has only one variable, like , its degree is the exponent of that variable, which is . **Examples:** - The monomial $$4x^5$$ has a degree of $$5$$. - The monomial $$0.12x$$ (or $$0.12x^1$$) has a degree of $$1$$. Visible text: - The monomial has a degree of . - The monomial (or ) has a degree of . ### 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 $$3 + 1 + 3 = 7$$. Visible text: - The monomial has a degree of . - The monomial (remember ) has a degree of . ### 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 $$x^0 = 1$$). Visible text: 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 | | :-------------------------------------- | :----- | :---------------------------------------------------------------- | | $$4x^5$$ | $$5$$ | The exponent of $$x$$ is $$5$$. | | $$\frac{3}{4}x^2y^7$$ | $$9$$ | Sum of exponents $$2+7=9$$. | | $$0.12x$$ | $$1$$ | The exponent of $$x$$ is $$1$$. | | $$2.17x^3yz^3$$ | $$7$$ | Sum of exponents $$3+1+3=7$$. | | $$10$$ | $$0$$ | Non-zero constant. Can be written as $$10x^0$$. | Visible text: | Monomial | Degree | Explanation | | :-------------------------------------- | :----- | :---------------------------------------------------------------- | | | | The exponent of is . | | | | Sum of exponents . | | | | The exponent of is . | | | | Sum of exponents . | | | | Non-zero constant. Can be written as . | ### 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 $$n$$. Visible text: 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 . ## 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. Visible text: 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** $$1$$: Visible text: **Example** : Determine the degree of the following polynomial: ```math 8x^3 - 36x^2 + 54x - 27 ``` - 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 $$0$$. Visible text: - The term has degree . - The term has degree . - The term (or ) has degree . - The term (constant) has degree . The highest degree among the terms is $$3$$. Therefore, the degree of this polynomial is $$3$$. Visible text: The highest degree among the terms is . Therefore, the degree of this polynomial is . **Example** $$2$$: Visible text: **Example** : Determine the degree of the following polynomial: ```math 5x^4y^2 + xy^2 - 2x^5y^6 ``` - 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 $$5 + 6 = 11$$. Visible text: - The term has degree . - The term (or ) has degree . - The term has degree . The highest degree among the terms is $$11$$. Therefore, the degree of this polynomial is $$11$$. Visible text: The highest degree among the terms is . Therefore, the degree of this polynomial is . **Example** $$3$$: Visible text: **Example** : Determine the degree of the following polynomial: ```math 0.13x^3 + 1.56x^2 - 2.24x + 1.72 ``` - 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 $$0$$. Visible text: - The term has degree . - The term has degree . - The term has degree . - The term has degree . The highest degree is $$3$$. Therefore, the degree of this polynomial is $$3$$. Visible text: The highest degree is . Therefore, the degree of this polynomial is . ### 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 $$0x^0$$? Visible text: Is the degree of equal to , since can be written as ? 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. Visible text: - **Non-zero** constants (like ) have a degree of . - The **zero polynomial** (the number itself) is often considered to **have no degree** or sometimes is said to have a degree of **negative infinity** (). 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. Visible text: However, for the high school level, understanding that non-zero constants have degree and the degree of a polynomial is the highest degree of its terms is sufficient.