## Synthetic Division: A Step-by-Step Guide for (x^3 - 6x^2 + 9) / (x - 4)

Synthetic division is a simplified method for dividing polynomials, particularly when the divisor is a linear expression (like x - a). Let's walk through the process for the example of (x^3 - 6x^2 + 9) / (x - 4).

### Step 1: Set up the division problem

- Write the coefficients of the dividend (x^3 - 6x^2 + 9) in a row:
**1 -6 0 9**(Notice we include a zero for the missing x term). - Write the constant term of the divisor (x - 4) as a number outside the row:
**4**

```
4 | 1 -6 0 9
-----------------
```

### Step 2: Bring down the first coefficient

- Bring down the first coefficient (1) below the line.

```
4 | 1 -6 0 9
-----------------
1
```

### Step 3: Multiply and add

- Multiply the number you just brought down (1) by the divisor (4) and write the result (4) under the next coefficient (-6).
- Add the numbers in the second column (-6 + 4 = -2).

```
4 | 1 -6 0 9
-----------------
1 -2
```

### Step 4: Repeat the process

- Repeat steps 2 and 3 for the remaining coefficients:
- Multiply -2 by 4 and write the result (-8) under the 0.
- Add 0 and -8 to get -8.
- Multiply -8 by 4 and write the result (-32) under the 9.
- Add 9 and -32 to get -23.

```
4 | 1 -6 0 9
-----------------
1 -2 -8 -23
```

### Step 5: Interpret the result

- The last number (-23) is the remainder.
- The other numbers (1, -2, -8) are the coefficients of the quotient polynomial. Since the dividend was a cubic polynomial (x^3), the quotient is a quadratic:

**Therefore, the result of (x^3 - 6x^2 + 9) / (x - 4) is:**

**Quotient:**x^2 - 2x - 8**Remainder:**-23

We can express this as:
**(x^3 - 6x^2 + 9) / (x - 4) = x^2 - 2x - 8 - 23/(x-4)**