# 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-concept Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/polynomial/polynomial-concept/en.mdx Understand polynomials from the ground up. Learn what monomials are, how they combine to form polynomials, and identify valid polynomial expressions. --- ## Getting to Know Monomials Before defining polynomials, start with their building blocks: **monomials**. Consider the following algebraic expressions: Component: MathContainer Children: ```math \sqrt[3]{p} \quad \text{(not a monomial)} ``` ```math 2x^2y \quad \text{(monomial)} ``` ```math -8 \quad \text{(monomial)} ``` ```math \frac{2}{m} \quad \text{(not a monomial)} ``` ```math 1.24k^4 \quad \text{(monomial)} ``` ```math 5a^{-6} \quad \text{(not a monomial)} ``` From the expressions above, we can group them into two: 1. **Group $$1$$ (Monomials):** $$2x^2y$$, $$-8$$, $$1.24k^4$$ 2. **Group $$2$$ (Not Monomials):** $$\sqrt[3]{p}$$, $$\frac{2}{m}$$, $$5a^{-6}$$ Visible text: 1. **Group (Monomials):** , , 2. **Group (Not Monomials):** , , The algebraic expressions in Group $$1$$ are what we call **monomials**. Visible text: The algebraic expressions in Group are what we call **monomials**. ### What is a Monomial? **A monomial** is a number, a variable raised to a non-negative integer power ($$0, 1, 2, 3, \dots$$), or the product of a number and one or more variables raised to non-negative integer powers. Visible text: **A monomial** is a number, a variable raised to a non-negative integer power (), or the product of a number and one or more variables raised to non-negative integer powers. Let's break down why Group $$1$$ consists of monomials and Group $$2$$ does not: Visible text: Let's break down why Group consists of monomials and Group does not: - **Group $$1$$ (Monomials):** - $$2x^2y$$: The product of a number ($$2$$) and variables ($$x$$, $$y$$) with non-negative integer powers ($$2$$ and $$1$$). - $$-8$$: A constant (just a number). Or this is the same as $$-8x^0$$. - $$1.24k^4$$: The product of a number ($$1.24$$) and a variable ($$k$$) with a non-negative integer power ($$4$$). - **Group $$2$$ (Not Monomials):** - $$\sqrt[3]{p} = p^{1/3}$$: The power of the variable $$p$$ is not a non-negative integer ($$1/3$$). - $$\frac{2}{m} = 2m^{-1}$$: The power of the variable $$m$$ is not a non-negative integer ($$-1$$). - $$5a^{-6} = 5a^{-6}$$: The power of the variable $$a$$ is not a non-negative integer ($$-6$$). Visible text: - **Group (Monomials):** - : The product of a number () and variables (, ) with non-negative integer powers ( and ). - : A constant (just a number). Or this is the same as . - : The product of a number () and a variable () with a non-negative integer power (). - **Group (Not Monomials):** - : The power of the variable is not a non-negative integer (). - : The power of the variable is not a non-negative integer (). - : The power of the variable is not a non-negative integer (). So, the key characteristic of a monomial is that **the exponents of the variables must be non-negative integers**. The number multiplying the variable (like $$2$$ in $$2x^2y$$) is called the **coefficient**. Visible text: So, the key characteristic of a monomial is that **the exponents of the variables must be non-negative integers**. The number multiplying the variable (like in ) is called the **coefficient**. ## Definition of Polynomial After understanding monomials, we can now define a **polynomial**. **A polynomial** is an algebraic expression that is a monomial or the sum (and subtraction) of two or more monomials. Consider the following examples: Component: MathContainer Children: ```math 4x^3y - 3x^2 ``` ```math x + 2\sqrt{x} ``` ```math 2x^3 - 5x^{-2} + 1 ``` Let's identify which are polynomials and which are not: 1. $$4x^3y - 3x^2$$ - The term $$4x^3y$$ is a monomial. - The term $$-3x^2$$ is a monomial. - Conclusion: **Polynomial** (subtraction of two monomials). 2. $$x + 2\sqrt{x}$$ - The term $$x$$ is a monomial. - The term $$2\sqrt{x} = 2x^{1/2}$$ is not a monomial (exponent is not a non-negative integer). - Conclusion: **Not a Polynomial**. 3. $$2x^3 - 5x^{-2} + 1$$ - The term $$2x^3$$ is a monomial. - The term $$-5x^{-2}$$ is not a monomial (exponent is not a non-negative integer). - The term $$1$$ is a monomial (constant). - Conclusion: **Not a Polynomial**. Visible text: 1. - The term is a monomial. - The term is a monomial. - Conclusion: **Polynomial** (subtraction of two monomials). 2. - The term is a monomial. - The term is not a monomial (exponent is not a non-negative integer). - Conclusion: **Not a Polynomial**. 3. - The term is a monomial. - The term is not a monomial (exponent is not a non-negative integer). - The term is a monomial (constant). - Conclusion: **Not a Polynomial**. ### Addition and Subtraction in Polynomials You might ask, "The definition of a polynomial involves the _sum_ of monomials, but example $$1$$ has _subtraction_ ($$4x^3y - 3x^2$$). How does that work?" Visible text: You might ask, "The definition of a polynomial involves the _sum_ of monomials, but example has _subtraction_ (). How does that work?" Recall that subtraction can be viewed as adding the negative. So, $$4x^3y - 3x^2$$ is the same as $$4x^3y + (-3x^2)$$. Visible text: Recall that subtraction can be viewed as adding the negative. So, is the same as . Since both $$4x^3y$$ and $$-3x^2$$ are monomials, their sum is still a polynomial. This is why subtraction between monomials also results in a polynomial. Visible text: Since both and are monomials, their sum is still a polynomial. This is why subtraction between monomials also results in a polynomial. So, an algebraic expression is called a polynomial if all its terms are monomials, meaning every variable has a non-negative integer exponent.