# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Division of Polynomials",
  description: "Learn polynomial division algorithm with clear examples. Understand quotient, remainder concepts, and verify results using fractional & multiplication forms.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/04/2025",
  subject: "Polynomial",
};

## Basic Concept of Polynomial Division

Polynomial division is similar to the division of integers we are familiar with. When we divide one number by another, we get a quotient and a remainder.

For example, when dividing 7 by 4:

- <InlineMath math="\frac{7}{4}" /> can be written as <InlineMath math="1 \frac{3}{4}" /> (1
  remainder 3)
- This can also be written as <InlineMath math="7 = 4 \cdot 1 + 3" />

Here:

- 7 is the **dividend**
- 4 is the **divisor**
- 1 is the **quotient**
- 3 is the **remainder**

The same concept applies to polynomials.

## Polynomial Division Algorithm

The division algorithm states the relationship between the dividend polynomial, the divisor, the quotient, and the remainder.

If <InlineMath math="P(x)" /> (the dividend) and <InlineMath math="Q(x)" /> (the divisor) are two polynomials, with <InlineMath math="Q(x) \neq 0" />, then there exist unique polynomials <InlineMath math="H(x)" /> (the quotient) and <InlineMath math="S(x)" /> (the remainder) such that:

<BlockMath math="P(x) = Q(x) \cdot H(x) + S(x)" />

or it can be written as:

<BlockMath math="\frac{P(x)}{Q(x)} = H(x) + \frac{S(x)}{Q(x)}" />

with the condition that the degree of <InlineMath math="S(x)" /> is less than the degree of <InlineMath math="Q(x)" />, or <InlineMath math="S(x) = 0" /> (zero remainder).

**Terminology:**

- <InlineMath math="P(x)" />: Dividend Polynomial
- <InlineMath math="Q(x)" />: Divisor Polynomial
- <InlineMath math="H(x)" />: Quotient Polynomial
- <InlineMath math="S(x)" />: Remainder Polynomial

**Illustrative Example:**

The division of <InlineMath math="x^3 + 4x^2 + 5x + 8" /> by <InlineMath math="x + 3" /> yields:

- Quotient <InlineMath math="H(x) = x^2 + x + 2" />
- Remainder <InlineMath math="S(x) = 2" />

This can be written in two forms according to the algorithm:

1.  Fractional form:

    <BlockMath math="\frac{x^3 + 4x^2 + 5x + 8}{x + 3} = x^2 + x + 2 + \frac{2}{x + 3}" />

2.  Multiplication form:

    <BlockMath math="x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2" />

Note that the degree of the remainder (<InlineMath math="S(x)=2" />, degree 0) is less than the degree of the divisor (<InlineMath math="x+3" />, degree 1).

### Verifying the Division Algorithm

We can prove the correctness of the second form above by multiplying the quotient by the divisor and then adding the remainder.

Prove that <InlineMath math="x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2" />.

Let's expand the right side:

<div className="flex flex-col gap-4">
  <BlockMath math="(x + 3)(x^2 + x + 2) + 2" />
  <BlockMath math="= [x(x^2 + x + 2) + 3(x^2 + x + 2)] + 2 \quad \text{(Distribute } (x+3))" />
  <BlockMath math="= [x^3 + x^2 + 2x + 3x^2 + 3x + 6] + 2 \quad \text{(Distribute } x \text{ and } 3)" />
  <BlockMath math="= x^3 + (x^2 + 3x^2) + (2x + 3x) + 6 + 2 \quad \text{(Group like terms)}" />
  <BlockMath math="= x^3 + 4x^2 + 5x + 8 \quad \text{(Final result)}" />
</div>

Since the right side equals the left side, the equation is proven true.
