# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Polynomial Function",
  description: "Learn polynomial functions: understand P(x) notation, identify leading terms, coefficients, and degrees. Master function components with clear examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/04/2025",
  subject: "Polynomial",
};

## Understanding Polynomial Functions

Essentially, a polynomial function is a rule that maps an input value (variable) to an output value using a polynomial expression.

## General Form of a Polynomial Function

A polynomial function in the variable <InlineMath math="x" /> is generally written in the form:

<BlockMath math="P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0" />

Let's break down the important components of this general form:

- **<InlineMath math="P(x)" />**:

  Function notation, read "P of x", indicating the function's value depends on the value of <InlineMath math="x" />.

- **<InlineMath math="x" />**:

  The variable of the polynomial function.

- **<InlineMath math="n" />**:

  The highest power of the variable <InlineMath math="x" />. This value <InlineMath math="n" /> must be a **non-negative integer** (0, 1, 2, 3, ...). This non-negative integer <InlineMath math="n" /> also determines the **degree** of the polynomial function.

- **<InlineMath math="a_n, a_{n-1}, \dots, a_1, a_0" />**:

  The coefficients of the polynomial function. These coefficients are **real numbers**.

- **<InlineMath math="a_n x^n" />**:

  The term with the highest power. This term is called the **leading term**.

- **<InlineMath math="a_n" />**:

  The coefficient of the leading term. This is called the **leading coefficient**. It's important to note that the leading coefficient <InlineMath math="a_n" /> **cannot be zero** (<InlineMath math="a_n \neq 0" />) for the function to truly have degree <InlineMath math="n" />.

- **<InlineMath math="a_0" />**:

  The term without the variable <InlineMath math="x" /> (or can be considered <InlineMath math="a_0 x^0" />). This term is called the **constant term** or **constant**.

## Example of a Polynomial Function

Suppose we have the function: <InlineMath math="f(x) = 5x^3 - 2x^2 + 7x - 1" />

- This is a polynomial function in the variable <InlineMath math="x" />.
- Its degree is 3 (the highest power of <InlineMath math="x" />).
- Its leading term is <InlineMath math="5x^3" />.
- Its leading coefficient is 5 (<InlineMath math="a_3 = 5" />).
- Other coefficients are <InlineMath math="a_2 = -2" />, <InlineMath math="a_1 = 7" />.
- Its constant term is -1 (<InlineMath math="a_0 = -1" />).

Thus, a function can be called a polynomial function if it follows this general form, with the main conditions being that the variable exponents must be non-negative integers and the leading coefficient is not zero.
