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Learn the Remainder Theorem to find polynomial division remainders without long division, with examples and proof.

---

## 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 $$P(x)$$ is divided by $$(x - c)$$, then the remainder is $$S = P(c)$$.

Visible text: If a polynomial is divided by , then the remainder is .

This means that to find the remainder when $$P(x)$$ is divided by $$(x - c)$$, we simply need to evaluate the polynomial $$P(x)$$ at $$x = c$$.

Visible text: 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:

```math
P(x) = (x - c) \cdot H(x) + S
```

Where:

- $$P(x)$$ is the dividend polynomial.
- $$(x - c)$$ is the divisor polynomial (degree $$1$$).
- $$H(x)$$ is the quotient.
- $$S$$ is the remainder (a constant, since the divisor is degree $$1$$).

Visible text: - is the dividend polynomial.
- is the divisor polynomial (degree ).
- is the quotient.
- is the remainder (a constant, since the divisor is degree ).

Now, let's substitute $$x = c$$ into the division algorithm equation:

Visible text: Now, let's substitute into the division algorithm equation:

Component: MathContainer
Children:

```math
P(c) = (c - c) \cdot H(c) + S
```

```math
P(c) = (0) \cdot H(c) + S
```

```math
P(c) = 0 + S
```

```math
P(c) = S
```

This proves that the value of the polynomial $$P(x)$$ at $$x = c$$ is equal to the remainder $$S$$ when $$P(x)$$ is divided by $$(x - c)$$.

Visible text: 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 $$P(x) = 2x^5 + 5x^4 - 10x^3 + 9x^2 - 10$$ is divided by $$x + 4$$.

Visible text: Find the remainder if is divided by .

### Using Horner's Method

The divisor is $$x + 4$$, or $$x - (-4)$$, so $$c = -4$$.

Visible text: The divisor is , or , so .

Coefficients of $$P(x)$$ (completing the x term): $$2, 5, -10, 9, 0, -10$$.

Visible text: Coefficients of (completing the x term): .

```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: $$H(x) = 2x^4 - 3x^3 + 2x^2 + x - 4$$
- Remainder: $$S = \boxed{6}$$

Visible text: - Quotient: 
- Remainder:

### Using the Remainder Theorem

According to the Remainder Theorem, the remainder when $$P(x)$$ is divided by $$x - (-4)$$ is $$P(-4)$$.

Visible text: According to the Remainder Theorem, the remainder when is divided by is .

Let's calculate $$P(-4)$$:

Visible text: Let's calculate :

Component: MathContainer
Children:

```math
P(-4) = 2(-4)^5 + 5(-4)^4 - 10(-4)^3 + 9(-4)^2 - 10
```

```math
P(-4) = 2(-1024) + 5(256) - 10(-64) + 9(16) - 10
```

```math
P(-4) = -2048 + 1280 + 640 + 144 - 10
```

```math
P(-4) = -2048 + 1920 + 144 - 10
```

```math
P(-4) = -128 + 144 - 10
```

```math
P(-4) = 16 - 10
```

```math
P(-4) = 6
```

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.

Visible text: The result is the same! Using the Remainder Theorem, we found the remainder is , just like with Horner's method, but without performing the full division process.

This shows that evaluating $$P(c)$$ is another way to find the remainder of division by $$(x-c)$$.

Visible text: This shows that evaluating is another way to find the remainder of division by .

## Exercise

If $$P(x) = 3x^5 - 20x^4 - 6x^3 - 48x - 8$$ is divided by $$x - 7$$, determine the remainder using the Remainder Theorem.

Visible text: If is divided by , determine the remainder using the Remainder Theorem.

### Answer Key

According to the Remainder Theorem, the remainder when $$P(x)$$ is divided by $$x - 7$$ is $$P(7)$$.

Visible text: According to the Remainder Theorem, the remainder when is divided by is .

Component: MathContainer
Children:

```math
P(7) = 3(7)^5 - 20(7)^4 - 6(7)^3 - 48(7) - 8
```

```math
P(7) = 3(16807) - 20(2401) - 6(343) - 336 - 8
```

```math
P(7) = 50421 - 48020 - 2058 - 336 - 8
```

```math
P(7) = 2401 - 2058 - 336 - 8
```

```math
P(7) = 343 - 336 - 8
```

```math
P(7) = 7 - 8
```

```math
P(7) = -1
```

So, the remainder is $$-1$$.

Visible text: So, the remainder is .