## Performing Synthetic Division: (x³ - 4x + 6) / (x + 3)

Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). Let's apply this method to divide (x³ - 4x + 6) by (x + 3).

### Setting Up the Problem

**Identify the coefficients:**In our dividend (x³ - 4x + 6), the coefficients are 1, 0, -4, and 6. Notice we need a placeholder for the missing x² term.**Identify the divisor:**Our divisor is (x + 3), so 'a' is -3.

Now, let's set up the synthetic division problem:

```
-3 | 1 0 -4 6
_________________
```

### Performing the Calculation

**Bring down the first coefficient:**Bring down the '1' below the line.

```
-3 | 1 0 -4 6
_________________
1
```

**Multiply and add:**Multiply the '1' by -3 (the divisor) and write the result (-3) below the '0'. Add the numbers in the second column (0 + (-3) = -3).

```
-3 | 1 0 -4 6
_________________
1 -3
```

**Repeat the process:**Multiply the -3 by -3, and write the result (9) below the -4. Add the numbers in the third column (-4 + 9 = 5).

```
-3 | 1 0 -4 6
_________________
1 -3 5
```

**Final step:**Multiply the 5 by -3, and write the result (-15) below the 6. Add the numbers in the last column (6 + (-15) = -9).

```
-3 | 1 0 -4 6
_________________
1 -3 5 -9
```

### Interpreting the Result

The numbers below the line (1, -3, 5, and -9) represent the coefficients of the quotient and the remainder. Starting from the left, the coefficients correspond to the powers of x, decreasing by one each time.

**Quotient:**x² - 3x + 5**Remainder:**-9

Therefore, the result of the division (x³ - 4x + 6) / (x + 3) is:

**(x³ - 4x + 6) / (x + 3) = x² - 3x + 5 - 9/(x + 3)**

**Note:** The remainder is expressed as a fraction with the divisor as the denominator.