# Nakafa Framework: LLM
URL: https://nakafa.com/en/subject/high-school/11/mathematics/polynomial/remainder-theorem
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/polynomial/remainder-theorem/en.mdx
Output docs content for large language models.
---
export const metadata = {
title: "Remainder Theorem",
description: "Discover the Remainder Theorem to quickly find polynomial division remainders without long division. Master this shortcut with clear examples and proofs.",
authors: [{ name: "Nabil Akbarazzima Fatih" }],
date: "05/04/2025",
subject: "Polynomial",
};
## Understanding the Remainder Theorem
Have you ever wondered if there's a quick way to find the remainder of a polynomial division without performing long division or the lengthy Horner's method? The answer lies in the **Remainder Theorem**!
The Remainder Theorem provides an interesting relationship between the remainder of a polynomial division and the value of the polynomial itself.
### Statement of the Remainder Theorem
If a polynomial is divided by , then the remainder is .
This means that to find the remainder when is divided by , we simply need to evaluate the polynomial at .
### Why Does the Remainder Theorem Work?
This theorem stems directly from the polynomial division algorithm we already know:
Where:
- is the dividend polynomial.
- is the divisor polynomial (degree 1).
- is the quotient.
- is the remainder (a constant, since the divisor is degree 1).
Now, let's substitute into the division algorithm equation:
This proves that the value of the polynomial at is equal to the remainder when is divided by .
## Calculating with the Remainder Theorem
The Remainder Theorem is very useful for quickly determining the remainder of a division or for evaluating a polynomial at a specific point.
Find the remainder if is divided by .
### Using Horner's Method
The divisor is , or , so .
Coefficients of (completing the x term): .
From Horner's method, we get:
- Quotient:
- Remainder:
### Using the Remainder Theorem
According to the Remainder Theorem, the remainder when is divided by is .
Let's calculate :
The result is the same! Using the Remainder Theorem, we found the remainder is 6, just like with Horner's method, but without performing the full division process.
This shows that evaluating is another way to find the remainder of division by .
## Exercise
If is divided by , determine the remainder using the Remainder Theorem.
### Answer Key
According to the Remainder Theorem, the remainder when is divided by is .
So, the remainder is .