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URL: https://nakafa.com/en/subjects/mathematics/polynomial/division-polynomial
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Learn polynomial division algorithm with clear examples. Understand quotient, remainder concepts, and verify results using fractional & multiplication forms.

---

## 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 $$4$$:

Visible text: For example, when dividing by :

- $$\frac{7}{4}$$ can be written as $$1 \frac{3}{4}$$ ($$1$$
  remainder $$3$$)
- This can also be written as $$7 = 4 \cdot 1 + 3$$

Visible text: - can be written as (
 remainder )
- This can also be written as

Here:

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

Visible text: - is the **dividend**
- is the **divisor**
- is the **quotient**
- 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 $$P(x)$$ (the dividend) and $$Q(x)$$ (the divisor) are two polynomials, with $$Q(x) \neq 0$$, then there exist unique polynomials $$H(x)$$ (the quotient) and $$S(x)$$ (the remainder) such that:

Visible text: If (the dividend) and (the divisor) are two polynomials, with , then there exist unique polynomials (the quotient) and (the remainder) such that:

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

or it can be written as:

```math
\frac{P(x)}{Q(x)} = H(x) + \frac{S(x)}{Q(x)}
```

with the condition that the degree of $$S(x)$$ is less than the degree of $$Q(x)$$, or $$S(x) = 0$$ (zero remainder).

Visible text: with the condition that the degree of is less than the degree of , or (zero remainder).

**Terminology:**

- $$P(x)$$: Dividend Polynomial
- $$Q(x)$$: Divisor Polynomial
- $$H(x)$$: Quotient Polynomial
- $$S(x)$$: Remainder Polynomial

Visible text: - : Dividend Polynomial
- : Divisor Polynomial
- : Quotient Polynomial
- : Remainder Polynomial

**Illustrative Example:**

The division of $$x^3 + 4x^2 + 5x + 8$$ by $$x + 3$$ yields:

Visible text: The division of by yields:

- Quotient $$H(x) = x^2 + x + 2$$
- Remainder $$S(x) = 2$$

Visible text: - Quotient 
- Remainder

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

1.  Fractional form:

    
    
    ```math
    \frac{x^3 + 4x^2 + 5x + 8}{x + 3} = x^2 + x + 2 + \frac{2}{x + 3}
    ```

2.  Multiplication form:

    
    
    ```math
    x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2
    ```

Visible text: 1. Fractional form:

 
 

2. Multiplication form:

Note that the degree of the remainder ($$S(x)=2$$, degree $$0$$) is less than the degree of the divisor ($$x+3$$, degree $$1$$).

Visible text: Note that the degree of the remainder (, degree ) is less than the degree of the divisor (, degree ).

### 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 $$x^3 + 4x^2 + 5x + 8 = (x + 3)(x^2 + x + 2) + 2$$.

Visible text: Prove that .

Let's expand the right side:

Component: MathContainer
Children:

```math
(x + 3)(x^2 + x + 2) + 2
```

```math
= [x(x^2 + x + 2) + 3(x^2 + x + 2)] + 2 \quad \text{(Distribute } (x+3))
```

```math
= [x^3 + x^2 + 2x + 3x^2 + 3x + 6] + 2 \quad \text{(Distribute } x \text{ and} 3)
```

```math
= x^3 + (x^2 + 3x^2) + (2x + 3x) + 6 + 2 \quad \text{(Group like terms)}
```

```math
= x^3 + 4x^2 + 5x + 8 \quad \text{(Final result)}
```

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