Polynomial long division practice questions and answers on a study website.

Let's solve each polynomial long division problem step by step.

---

Problem 1:

Divide using long division.
(x³ + 6x² − x − 30) ÷ (x − 2)

We set up the long division:

```
________________
x - 2 ) x³ + 6x² - x - 30
```

Step 1: Divide the first term of the dividend by the first term of the divisor:
x³ ÷ x = x²

Write x² above the division bar.

Multiply x² by (x - 2):
x² · (x - 2) = x³ - 2x²

Subtract this from the original polynomial:

```
x²
________________
x - 2 ) x³ + 6x² - x - 30
-(x³ - 2x²)
_____________
8x² - x
```

Step 2: Bring down the next term (-x). Now divide 8x² ÷ x = 8x

Write +8x above the division bar.

Multiply 8x by (x - 2):
8x · (x - 2) = 8x² - 16x

Subtract:

```
x² + 8x
________________
x - 2 ) x³ + 6x² - x - 30
-(x³ - 2x²)
_____________
8x² - x
-(8x² - 16x)
___________
15x - 30
```

Step 3: Divide 15x ÷ x = 15

Write +15 above the division bar.

Multiply 15 by (x - 2):
15 · (x - 2) = 15x - 30

Subtract:

```
x² + 8x + 15
________________
x - 2 ) x³ + 6x² - x - 30
-(x³ - 2x²)
_____________
8x² - x
-(8x² - 16x)
___________
15x - 30
-(15x - 30)
_________
0
```

✔ Remainder is 0.

Answer: x² + 8x + 15

✔️ Correct choice: First option — `x² + 8x + 15`

---

Problem 2:

(x³ + 7x² − 6x − 72) ÷ (x + 6)

Set up:

```
________________
x + 6 ) x³ + 7x² - 6x - 72
```

Step 1: x³ ÷ x = x²

Multiply x² · (x + 6) = x³ + 6x²

Subtract:

```
x²
________________
x + 6 ) x³ + 7x² - 6x - 72
-(x³ + 6x²)
_____________
x² - 6x
```

Step 2: x² ÷ x = x

Multiply x · (x + 6) = x² + 6x

Subtract:

```
x² + x
________________
x + 6 ) x³ + 7x² - 6x - 72
-(x³ + 6x²)
_____________
x² - 6x
-(x² + 6x)
___________
-12x - 72
```

Step 3: -12x ÷ x = -12

Multiply -12 · (x + 6) = -12x - 72

Subtract:

```
x² + x - 12
________________
x + 6 ) x³ + 7x² - 6x - 72
-(x³ + 6x²)
_____________
x² - 6x
-(x² + 6x)
___________
-12x - 72
-(-12x - 72)
___________
0
```

✔ Remainder is 0.

Answer: x² + x - 12

✔️ Correct choice: First option — `x² + x - 12`

---

Problem 3:

(x² + 3x − 18) ÷ (x − 3)

Set up:

```
________________
x - 3 ) x² + 3x - 18
```

Step 1: x² ÷ x = x

Multiply x · (x - 3) = x² - 3x

Subtract:

```
x
________________
x - 3 ) x² + 3x - 18
-(x² - 3x)
_____________
6x - 18
```

Step 2: 6x ÷ x = 6

Multiply 6 · (x - 3) = 6x - 18

Subtract:

```
x + 6
________________
x - 3 ) x² + 3x - 18
-(x² - 3x)
_____________
6x - 18
-(6x - 18)
__________
0
```

✔ Remainder is 0.

Answer: x + 6

✔️ Correct choice: First option — `x + 6`

---

## ✔ Final Answers:

1. x² + 8x + 15
2. x² + x - 12
3. x + 6

All divisions are exact (no remainder).