## Long Division of Polynomials: (w^3 - 2w^2 - 2w + 1) / (w - 1)

Long division of polynomials is a method used to divide one polynomial by another, similar to the long division of numbers. This process involves a series of steps to find the quotient and remainder.

Let's illustrate this with the example of dividing (w^3 - 2w^2 - 2w + 1) by (w - 1):

**Step 1: Set up the division.**

```
_______
w - 1 | w^3 - 2w^2 - 2w + 1
```

**Step 2: Divide the leading terms.**

- The leading term of the divisor (w - 1) is 'w'.
- The leading term of the dividend (w^3 - 2w^2 - 2w + 1) is 'w^3'.
- Divide 'w^3' by 'w' to get 'w^2'. This is the first term of the quotient.

```
w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1
```

**Step 3: Multiply the quotient term by the divisor.**

- Multiply 'w^2' by (w - 1) to get 'w^3 - w^2'.

```
w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
```

**Step 4: Subtract the result from the dividend.**

- Subtract 'w^3 - w^2' from the dividend.

```
w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w
```

**Step 5: Bring down the next term.**

- Bring down the next term of the dividend, which is '-2w'.

```
w^2 _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
```

**Step 6: Repeat steps 2-5.**

- Now, the leading term of the new dividend is '-w^2'.
- Divide '-w^2' by 'w' to get '-w'. This is the next term of the quotient.

```
w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
-w^2 + w
```

- Multiply '-w' by (w - 1) to get '-w^2 + w'.
- Subtract this result from the previous line.

```
w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
-w^2 + w
-------
-3w + 1
```

- Bring down the next term, '1'.

```
w^2 - w _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
-w^2 + w
-------
-3w + 1
```

**Step 7: Repeat steps 2-5 again.**

- Divide '-3w' by 'w' to get '-3'. This is the final term of the quotient.

```
w^2 - w - 3 _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
-w^2 + w
-------
-3w + 1
-3w + 3
```

- Multiply '-3' by (w - 1) to get '-3w + 3'.
- Subtract this result from the previous line.

```
w^2 - w - 3 _______
w - 1 | w^3 - 2w^2 - 2w + 1
w^3 - w^2
-------
-w^2 - 2w + 1
-w^2 + w
-------
-3w + 1
-3w + 3
-------
-2
```

**Step 8: Identify the quotient and remainder.**

- The quotient is the polynomial obtained on top:
**w^2 - w - 3**. - The remainder is the final result after the last subtraction:
**-2**.

Therefore, the result of dividing (w^3 - 2w^2 - 2w + 1) by (w - 1) can be expressed as:

**(w^3 - 2w^2 - 2w + 1) / (w - 1) = w^2 - w - 3 - 2/(w - 1)**