Question

Divide using either long or synthetic division

Answer: x³ - 4x² + 15x - 60 + 233/ x+4

3x² - 14x - 5/ x - 5

Answer 1

Let's break down the division of the two polynomials using synthetic division, as it can simplify the process when dividing by a linear factor.

### First Division: \( x^3 - 4x^2 + 15x - 60 \) divided by \( x + 4 \)

**Step 1:** Set up synthetic division for \( x + 4 \). We will use -4 for the synthetic division.

The coefficients of \( x^3 - 4x^2 + 15x - 60 \) are: 1, -4, 15, -60.

```
-4 | 1 -4 15 -60
| -4 32 -60
-------------------------
1 -8 47 -120
```

**Step 2:** Add down the columns.

- Bring down the 1.
- Multiply -4 by 1 to get -4, and add to -4 to get -8.
- Multiply -4 by -8 to get 32, and add to 15 to get 47.
- Multiply -4 by 47 to get -188, and add to -60 to get -120.

The result of the synthetic division is:
- Quotient: \( x^2 - 8x + 47 \)
- Remainder: -120

So, we can write:
\[
\frac{x^3 - 4x^2 + 15x - 60}{x + 4} = x^2 - 8x + 47 - \frac{120}{x + 4}
\]

### Final Result:
The final answer for \( \frac{x^3 - 4x^2 + 15x - 60}{x + 4} \):
\[
= x^2 - 8x + 47 - \frac{120}{x + 4}
\]

---

### Second Division: \( 3x^2 - 14x - 5 \) divided by \( x - 5 \)

**Step 1:** Set up synthetic division for \( x - 5 \). We will use 5 for the synthetic division.

The coefficients of \( 3x^2 - 14x - 5 \) are: 3, -14, -5.

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

**Step 2:** Add down the columns.

- Bring down the 3.
- Multiply 5 by 3 to get 15, and add to -14 to get 1.
- Multiply 5 by 1 to get 5, and add to -5 to get 0.

The result of the synthetic division is:
- Quotient: \( 3x + 1 \)
- Remainder: 0

So, we can write:
\[
\frac{3x^2 - 14x - 5}{x - 5} = 3x + 1
\]

### Final Result:
The final answer for \( \frac{3x^2 - 14x - 5}{x - 5} \):
\[
= 3x + 1
\]

---

Overall, the answers are:
1. \( \frac{x^3 - 4x^2 + 15x - 60}{x + 4} = x^2 - 8x + 47 - \frac{120}{x + 4} \)
2. \( \frac{3x^2 - 14x - 5}{x - 5} = 3x + 1 \)