## Dividing Polynomials: (x^3-x^2-5x-3) ÷ (x^2+2x+1)

This article will guide you through the process of dividing the polynomial (x^3-x^2-5x-3) by (x^2+2x+1). We'll use the method of **long division** to achieve this.

### 1. Setting Up the Long Division

Start by writing the division problem in the standard long division format:

```
_______
x^2+2x+1 | x^3-x^2-5x-3
```

### 2. Dividing the Leading Terms

**Focus on the leading terms of both the divisor and the dividend:**x^2 (from the divisor) and x^3 (from the dividend).**Divide the leading term of the dividend by the leading term of the divisor:**x^3 ÷ x^2 = x.**Write this quotient (x) above the x^2 term in the dividend:**

```
x
x^2+2x+1 | x^3-x^2-5x-3
```

### 3. Multiply and Subtract

**Multiply the quotient (x) by the entire divisor (x^2+2x+1):**x * (x^2+2x+1) = x^3 + 2x^2 + x.**Write the result below the dividend, aligning terms with their corresponding degrees:**

```
x
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
```

**Subtract the entire expression from the dividend:**(x^3 - x^2 - 5x - 3) - (x^3 + 2x^2 + x) = -3x^2 - 6x - 3.

```
x
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
-----------------
-3x^2-6x-3
```

### 4. Bring Down the Next Term

**Bring down the next term from the dividend (-3) to the bottom row:**

```
x
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
-----------------
-3x^2-6x-3
```

### 5. Repeat Steps 2-4

**Focus on the leading term of the new dividend (-3x^2) and the leading term of the divisor (x^2):**-3x^2 ÷ x^2 = -3.**Write this quotient (-3) next to the x in the quotient:**

```
x - 3
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
-----------------
-3x^2-6x-3
```

**Multiply the new quotient (-3) by the entire divisor:**-3 * (x^2+2x+1) = -3x^2 - 6x - 3.**Write the result below the new dividend:**

```
x - 3
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
-----------------
-3x^2-6x-3
-3x^2-6x-3
```

**Subtract the entire expression:**(-3x^2 - 6x - 3) - (-3x^2 - 6x - 3) = 0.

```
x - 3
x^2+2x+1 | x^3-x^2-5x-3
x^3+2x^2+x
-----------------
-3x^2-6x-3
-3x^2-6x-3
----------
0
```

### 6. The Result

We've reached a remainder of 0, indicating that the division is complete. Therefore, we can conclude:

** (x^3 - x^2 - 5x - 3) ÷ (x^2 + 2x + 1) = x - 3**