# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/polynomial/polynomial-concept
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/polynomial/polynomial-concept/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Polynomial Concept",
  description: "Understand polynomials from the ground up. Learn what monomials are, how they combine to form polynomials, and identify valid polynomial expressions.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/04/2025",
  subject: "Polynomial",
};

## Getting to Know Monomials

Before we delve into the definition of polynomials, let's first get acquainted with their building blocks: **monomials**. Consider the following algebraic expressions:

<div className="flex flex-col gap-4">
  <BlockMath math="\sqrt[3]{p} \quad \text{(not a monomial)}" />
  <BlockMath math="2x^2y \quad \text{(monomial)}" />
  <BlockMath math="-8 \quad \text{(monomial)}" />
  <BlockMath math="\frac{2}{m} \quad \text{(not a monomial)}" />
  <BlockMath math="1.24k^4 \quad \text{(monomial)}" />
  <BlockMath math="5a^{-6} \quad \text{(not a monomial)}" />
</div>

From the expressions above, we can group them into two:

1.  **Group 1 (Monomials):** <InlineMath math="2x^2y" />, <InlineMath math="-8" />, <InlineMath math="1.24k^4" />
2.  **Group 2 (Not Monomials):** <InlineMath math="\sqrt[3]{p}" />, <InlineMath math="\frac{2}{m}" />, <InlineMath math="5a^{-6}" />

The algebraic expressions in Group 1 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, ...), 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:

- **Group 1 (Monomials):**

  - <InlineMath math="2x^2y" />:

    The product of a number (2) and variables (<InlineMath math="x" />, <InlineMath math="y" />) with non-negative integer powers (2 and 1).

  - <InlineMath math="-8" />:

    A constant (just a number). Or this is the same as <InlineMath math="-8x^0" />.

  - <InlineMath math="1.24k^4" />:

    The product of a number (1.24) and a variable (<InlineMath math="k" />) with a non-negative integer power (4).

- **Group 2 (Not Monomials):**

  - <InlineMath math="\sqrt[3]{p} = p^{1/3}" />:

    The power of the variable <InlineMath math="p" /> is not a non-negative integer (<InlineMath math="1/3" />).

  - <InlineMath math="\frac{2}{m} = 2m^{-1}" />:

    The power of the variable <InlineMath math="m" /> is not a non-negative integer (-1).

  - <InlineMath math="5a^{-6} = 5a^{-6}" />:

    The power of the variable <InlineMath math="a" /> is not a non-negative integer (-6).

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 <InlineMath math="2" /> in <InlineMath math="2x^2y" />) 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:

<div className="flex flex-col gap-4">
  <BlockMath math="4x^3y - 3x^2" />
  <BlockMath math="x + 2\sqrt{x}" />
  <BlockMath math="2x^3 - 5x^{-2} + 1" />
</div>

Let's identify which are polynomials and which are not:

1.  <InlineMath math="4x^3y - 3x^2" />

    - The term <InlineMath math="4x^3y" /> is a monomial.
    - The term <InlineMath math="-3x^2" /> is a monomial.
    - Conclusion: **Polynomial** (subtraction of two monomials).

2.  <InlineMath math="x + 2\sqrt{x}" />

    - The term <InlineMath math="x" /> is a monomial.
    - The term <InlineMath math="2\sqrt{x} = 2x^{1/2}" /> is not a monomial (exponent is not a non-negative integer).
    - Conclusion: **Not a Polynomial**.

3.  <InlineMath math="2x^3 - 5x^{-2} + 1" />

    - The term <InlineMath math="2x^3" /> is a monomial.
    - The term <InlineMath math="-5x^{-2}" /> is not a monomial (exponent is not a non-negative integer).
    - The term <InlineMath math="1" /> 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_ (<InlineMath math="4x^3y - 3x^2" />). How does that work?"

Recall that subtraction can be viewed as adding the negative. So, <InlineMath math="4x^3y - 3x^2" /> is the same as <InlineMath math="4x^3y + (-3x^2)" />.

Since both <InlineMath math="4x^3y" /> and <InlineMath math="-3x^2" /> are monomials, their sum is still a polynomial. This is why subtraction between monomials also results in a polynomial.

In essence, an algebraic expression is called a polynomial if all its terms are monomials (variables have non-negative integer exponents).
