Question

Divide using either long or synthetic division

3x² - 14x - 5/ x - 5

Answer 1

To divide \( 3x^2 - 14x - 5 \) by \( x - 5 \) using synthetic division, we can use the root of the divisor \( x - 5 = 0 \) which gives \( x = 5 \).

Here are the steps for synthetic division:

1. Set up the synthetic division. Write down the coefficients of the polynomial \( 3x^2 - 14x - 5 \), which are \( 3, -14, -5 \). Since we are dividing by \( x - 5 \), we will use \( 5 \).

2. The setup will look like this:

```
5 | 3 -14 -5
|
```

3. Bring down the leading coefficient (the first number):

```
5 | 3 -14 -5
|
|_________________
3
```

4. Multiply \( 5 \) (the number on the left) by \( 3 \) (the number just brought down) and write the result under the next coefficient:

```
5 | 3 -14 -5
| 15
|_________________
3 1
```

5. Add the column:

```
5 | 3 -14 -5
| 15
|_________________
3 1 -5
```

6. Multiply \( 5 \) by \( 1 \) and write the result under the next coefficient:

```
5 | 3 -14 -5
| 15 5
|_________________
3 1 0
```

7. Add the column again:

```
5 | 3 -14 -5
| 15 5
|_________________
3 1 0
```

Now, the synthetic division process is complete. The result at the bottom \( 3, 1, 0 \) corresponds to the quotient.

The quotient is \( 3x + 1 \) and the remainder is \( 0 \).

Therefore, the result of \( \frac{3x^2 - 14x - 5}{x - 5} \) is

\[
3x + 1
\]

with a remainder of \( 0 \).

Final answer:

\[
\frac{3x^2 - 14x - 5}{x - 5} = 3x + 1
\]