Question

Find the quotient of the polynomials using synthetic division: (𝑥 3 − 9) ÷ (𝑥 − 3)

Answer 1

To perform synthetic division of the polynomial \( x^3 - 9 \) by \( x - 3 \), we will follow the steps for synthetic division.

1. **Set up Synthetic Division**:
- Write down the root from \( x - 3 = 0 \), which gives us \( x = 3 \).
- Write the coefficients of the polynomial \( x^3 - 9 \). Since there are missing terms, we represent the polynomial as \( x^3 + 0x^2 + 0x - 9 \). The coefficients we will use are \( 1, 0, 0, -9 \).

2. **Set up the synthetic division**:
```
3 | 1 0 0 -9
|
-----------------
```

3. **Perform the synthetic division step by step**:

- Bring down the leading coefficient (1):
```
3 | 1 0 0 -9
|
-----------------
1
```

- Multiply \( 1 \) by \( 3 \) and write the result under the next coefficient:
```
3 | 1 0 0 -9
| 3
-----------------
1 3
```

- Add: \( 0 + 3 = 3 \).
```
3 | 1 0 0 -9
| 3 9
-----------------
1 3
```

- Multiply \( 3 \) by \( 3 \) (next step):
```
3 | 1 0 0 -9
| 3 9
-----------------
1 3 9
```

- Add: \( 0 + 9 = 9 \).
```
3 | 1 0 0 -9
| 3 9 27
-----------------
1 3 9
```

- Multiply \( 9 \) by \( 3 \):
```
3 | 1 0 0 -9
| 3 9 27
-----------------
1 3 9 18
```

- Add: \( -9 + 27 = 18 \).

4. **Write down the result**:
The numbers at the bottom are the coefficients of the quotient. We have:

- \( 1 \) corresponds to \( x^2 \)
- \( 3 \) corresponds to \( x \)
- \( 9 \) is the remainder.

So, the quotient is \( x^2 + 3x + 9 \) with a remainder of \( 18 \).

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

The quotient is \( x^2 + 3x + 9 \) and the remainder is \( 18 \).

Answer 2

short version

Answer 3

To perform synthetic division of \( x^3 - 9 \) by \( x - 3 \):

1. Set up coefficients: \( 1, 0, 0, -9 \).
2. Use \( x = 3 \) for synthetic division.

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

Result:
- Quotient: \( x^2 + 3x + 9 \)
- Remainder: \( 18 \)

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