# 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 <InlineMath math="P(x)" /> is divided by <InlineMath math="(x - c)" />, then the remainder is <InlineMath math="S = P(c)" />.

This means that to find the remainder when <InlineMath math="P(x)" /> is divided by <InlineMath math="(x - c)" />, we simply need to evaluate the polynomial <InlineMath math="P(x)" /> at <InlineMath math="x = c" />.

### Why Does the Remainder Theorem Work?

This theorem stems directly from the polynomial division algorithm we already know:

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

Where:

- <InlineMath math="P(x)" /> is the dividend polynomial.
- <InlineMath math="(x - c)" /> is the divisor polynomial (degree 1).
- <InlineMath math="H(x)" /> is the quotient.
- <InlineMath math="S" /> is the remainder (a constant, since the divisor is degree 1).

Now, let's substitute <InlineMath math="x = c" /> into the division algorithm equation:

<div className="flex flex-col gap-4">
  <BlockMath math="P(c) = (c - c) \cdot H(c) + S" />
  <BlockMath math="P(c) = (0) \cdot H(c) + S" />
  <BlockMath math="P(c) = 0 + S" />
  <BlockMath math="P(c) = S" />
</div>

This proves that the value of the polynomial <InlineMath math="P(x)" /> at <InlineMath math="x = c" /> is equal to the remainder <InlineMath math="S" /> when <InlineMath math="P(x)" /> is divided by <InlineMath math="(x - c)" />.

## 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 <InlineMath math="P(x) = 2x^5 + 5x^4 - 10x^3 + 9x^2 - 10" /> is divided by <InlineMath math="x + 4" />.

### Using Horner's Method

The divisor is <InlineMath math="x + 4" />, or <InlineMath math="x - (-4)" />, so <InlineMath math="c = -4" />.

Coefficients of <InlineMath math="P(x)" /> (completing the x term): <InlineMath math="2, 5, -10, 9, 0, -10" />.

<BlockMath
  math="
\begin{array}{c|cccccc}
-4 & 2 & 5 & -10 & 9 & 0 & -10 \\
  &   & -8 & 12 & -8 & -4 & 16 \\
\hline
  & 2 & -3 & 2 & 1 & -4 & \boxed{6} \\
\end{array}
"
/>

From Horner's method, we get:

- Quotient: <InlineMath math="H(x) = 2x^4 - 3x^3 + 2x^2 + x - 4" />
- Remainder: <InlineMath math="S = \boxed{6}" />

### Using the Remainder Theorem

According to the Remainder Theorem, the remainder when <InlineMath math="P(x)" /> is divided by <InlineMath math="x - (-4)" /> is <InlineMath math="P(-4)" />.

Let's calculate <InlineMath math="P(-4)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="P(-4) = 2(-4)^5 + 5(-4)^4 - 10(-4)^3 + 9(-4)^2 - 10" />
  <BlockMath math="P(-4) = 2(-1024) + 5(256) - 10(-64) + 9(16) - 10" />
  <BlockMath math="P(-4) = -2048 + 1280 + 640 + 144 - 10" />
  <BlockMath math="P(-4) = -2048 + 1920 + 144 - 10" />
  <BlockMath math="P(-4) = -128 + 144 - 10" />
  <BlockMath math="P(-4) = 16 - 10" />
  <BlockMath math="P(-4) = 6" />
</div>

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 <InlineMath math="P(c)" /> is another way to find the remainder of division by <InlineMath math="(x-c)" />.

## Exercise

If <InlineMath math="P(x) = 3x^5 - 20x^4 - 6x^3 - 48x - 8" /> is divided by <InlineMath math="x - 7" />, determine the remainder using the Remainder Theorem.

### Answer Key

According to the Remainder Theorem, the remainder when <InlineMath math="P(x)" /> is divided by <InlineMath math="x - 7" /> is <InlineMath math="P(7)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="P(7) = 3(7)^5 - 20(7)^4 - 6(7)^3 - 48(7) - 8" />
  <BlockMath math="P(7) = 3(16807) - 20(2401) - 6(343) - 336 - 8" />
  <BlockMath math="P(7) = 50421 - 48020 - 2058 - 336 - 8" />
  <BlockMath math="P(7) = 2401 - 2058 - 336 - 8" />
  <BlockMath math="P(7) = 343 - 336 - 8" />
  <BlockMath math="P(7) = 7 - 8" />
  <BlockMath math="P(7) = -1" />
</div>

So, the remainder is <InlineMath math="-1" />.
