## Solving the Polynomial Division: (x^4 - 5x^3 - 8x^2 + 13x - 12) / (x - 6)

This article aims to demonstrate the process of dividing the polynomial **x^4 - 5x^3 - 8x^2 + 13x - 12** by the binomial **x - 6**. We will utilize the **long division method** to find the quotient and remainder.

### Step 1: Setting Up the Long Division

Begin by setting up the problem as a long division problem:

```
____________
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
```

### Step 2: Dividing the Leading Terms

- The leading term of the divisor (x - 6) is
**x**. - The leading term of the dividend (x^4 - 5x^3 - 8x^2 + 13x - 12) is
**x^4**. - Divide x^4 by x, which gives us
**x^3**. Write this above the line in the quotient space.

```
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
```

### Step 3: Multiplying the Quotient Term

- Multiply the quotient term (x^3) by the divisor (x - 6). This gives us
**x^4 - 6x^3**.

```
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
```

### Step 4: Subtracting the Result

- Subtract the result (x^4 - 6x^3) from the dividend. Remember to change the signs of the terms being subtracted.

```
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3
```

### Step 5: Bringing Down the Next Term

- Bring down the next term of the dividend (-8x^2).

```
x^3
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
```

### Step 6: Repeating the Process

- Repeat steps 2-5 with the new dividend (x^3 - 8x^2).
- Divide the leading term of the new dividend (x^3) by the leading term of the divisor (x), which gives us
**x^2**. - Write this term above the line in the quotient space.

```
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
```

- Multiply the new quotient term (x^2) by the divisor (x - 6), resulting in
**x^3 - 6x^2**. - Subtract this product from the current dividend, remembering to change the signs.

```
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2
```

- Bring down the next term of the dividend (+13x).

```
x^3 + x^2
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
```

### Step 7: Continuing the Division

- Repeat the process with the new dividend (-2x^2 + 13x).
- Divide the leading term of the dividend (-2x^2) by the leading term of the divisor (x), which gives us
**-2x**. - Write this term above the line in the quotient space.

```
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
```

- Multiply the new quotient term (-2x) by the divisor (x - 6), resulting in
**-2x^2 + 12x**. - Subtract this product from the current dividend, remembering to change the signs.

```
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x
```

### Step 8: Final Steps

- Bring down the last term of the dividend (-12).

```
x^3 + x^2 - 2x
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
```

- Divide the leading term of the new dividend (x) by the leading term of the divisor (x), which gives us
**1**. - Write this term above the line in the quotient space.

```
x^3 + x^2 - 2x + 1
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
x - 6
```

- Multiply the new quotient term (1) by the divisor (x - 6), resulting in
**x - 6**. - Subtract this product from the current dividend, remembering to change the signs.

```
x^3 + x^2 - 2x + 1
x - 6 | x^4 - 5x^3 - 8x^2 + 13x - 12
x^4 - 6x^3
---------
x^3 - 8x^2
x^3 - 6x^2
---------
-2x^2 + 13x
-2x^2 + 12x
---------
x - 12
x - 6
--------
-6
```

### Step 9: The Result

The remainder of the division is **-6**. Therefore, the final result of the polynomial division can be expressed as:

**(x^4 - 5x^3 - 8x^2 + 13x - 12) / (x - 6) = x^3 + x^2 - 2x + 1 - 6/(x - 6)**

This can also be written as:

**(x^4 - 5x^3 - 8x^2 + 13x - 12) = (x - 6)(x^3 + x^2 - 2x + 1) - 6**