Question

find the quotient of the polynomials using synthetic division (x^3-9) divided (x-3)

Answer 1

To perform synthetic division of \(x^3 - 9\) by \(x - 3\), we can follow these steps:

1. **Identify the coefficients**: For the polynomial \(x^3 - 9\), we can rewrite it as \(x^3 + 0x^2 + 0x - 9\). The coefficients are:

- \(1\) (for \(x^3\))
- \(0\) (for \(x^2\))
- \(0\) (for \(x\))
- \(-9\) (constant term)

2. **Set up synthetic division**: We will use \(3\) (the root of \(x - 3\)) for synthetic division. Write \(3\) to the left and the coefficients to the right:

```
3 | 1 0 0 -9
|
|_________________
```

3. **Perform the synthetic division**:

- Bring down the \(1\):

```
3 | 1 0 0 -9
|
|_________________
1
```

- Multiply \(1\) by \(3\) and add to the next coefficient \(0\):

```
3 | 1 0 0 -9
| 3
|_________________
1 3
```

- Multiply \(3\) by \(3\) and add to \(0\):

```
3 | 1 0 0 -9
| 3 9
|_________________
1 3 9
```

- Multiply \(9\) by \(3\) and add to \(-9\):

```
3 | 1 0 0 -9
| 3 9 27
|_________________
1 3 9 18
```

4. **Write the result**: The bottom row \(1, 3, 9\) represents the coefficients of the quotient polynomial, which is \(x^2 + 3x + 9\). The final value \(18\) is the remainder.

Thus, the result of dividing \(x^3 - 9\) by \(x - 3\) is:

\[
x^2 + 3x + 9 \quad \text{remainder } 18
\]

In summary:

\[
\frac{x^3 - 9}{x - 3} = x^2 + 3x + 9 + \frac{18}{x - 3}
\]