This worksheet provides 10 polynomial division problems to help students master dividing polynomials by binomials.

You’ve uploaded a worksheet titled “Divide the Polynomial”, which contains 10 polynomial division problems — all of the form:
> *(quadratic polynomial) ÷ (linear binomial)*

These are perfect candidates for polynomial long division or synthetic division. Since each divisor is linear and monic (leading coefficient = 1), synthetic division is efficient.

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Let’s solve all 10 problems step by step, using synthetic division where applicable, and provide the quotient and remainder.

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🔢 Problem 1:

(x² - 6x + 10) ÷ (x - 1)

Use synthetic division with root = 1:

```
1 | 1 -6 10
↓ 1 -5
----------------
1 -5 5
```

→ Quotient: x - 5, Remainder: 5

✔ Answer: x - 5 + 5/(x - 1)

---

🔢 Problem 2:

(x² + 5x + 6) ÷ (x + 4)

Root = -4

```
-4 | 1 5 6
↓ -4 -4
----------------
1 1 2
```

→ Quotient: x + 1, Remainder: 2

✔ Answer: x + 1 + 2/(x + 4)

---

🔢 Problem 3:

(x² + 4x + 8) ÷ (x - 1)

Root = 1

```
1 | 1 4 8
↓ 1 5
----------------
1 5 13
```

→ Quotient: x + 5, Remainder: 13

✔ Answer: x + 5 + 13/(x - 1)

---

🔢 Problem 4:

(x² - 3x - 5) ÷ (x + 2)

Root = -2

```
-2 | 1 -3 -5
↓ -2 10
----------------
1 -5 5
```

→ Quotient: x - 5, Remainder: 5

✔ Answer: x - 5 + 5/(x + 2)

---

🔢 Problem 5:

(x² - 2x - 20) ÷ (x - 5)

Root = 5

```
5 | 1 -2 -20
↓ 5 15
----------------
1 3 -5
```

→ Quotient: x + 3, Remainder: -5

✔ Answer: x + 3 - 5/(x - 5)

---

🔢 Problem 6:

(4x² + 24x + 21) ÷ (x + 5)

Root = -5

```
-5 | 4 24 21
↓ -20 -20
----------------
4 4 1
```

→ Quotient: 4x + 4, Remainder: 1

✔ Answer: 4x + 4 + 1/(x + 5)

---

🔢 Problem 7:

(x² - x - 3) ÷ (x - 2)

Root = 2

```
2 | 1 -1 -3
↓ 2 2
----------------
1 1 -1
```

→ Quotient: x + 1, Remainder: -1

✔ Answer: x + 1 - 1/(x - 2)

---

🔢 Problem 8:

(4x² + 3x - 8) ÷ (x + 2)

Root = -2

```
-2 | 4 3 -8
↓ -8 10
----------------
4 -5 2
```

→ Quotient: 4x - 5, Remainder: 2

✔ Answer: 4x - 5 + 2/(x + 2)

---

🔢 Problem 9:

(x² + 2x - 9) ÷ (x - 2)

Root = 2

```
2 | 1 2 -9
↓ 4 12
----------------
1 6 3
```

→ Quotient: x + 6, Remainder: 3

✔ Answer: x + 6 + 3/(x - 2)

---

🔢 Problem 10:

(x² - x - 21) ÷ (x + 4)

Root = -4

```
-4 | 1 -1 -21
↓ -4 20
----------------
1 -5 -1
```

→ Quotient: x - 5, Remainder: -1

✔ Answer: x - 5 - 1/(x + 4)

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## ✔ Final Answers Summary:

| Problem | Quotient | Remainder | Final Answer |
|---------|------------------|----------|----------------------------------|
| 1 | x - 5 | 5 | x - 5 + 5/(x - 1) |
| 2 | x + 1 | 2 | x + 1 + 2/(x + 4) |
| 3 | x + 5 | 13 | x + 5 + 13/(x - 1) |
| 4 | x - 5 | 5 | x - 5 + 5/(x + 2) |
| 5 | x + 3 | -5 | x + 3 - 5/(x - 5) |
| 6 | 4x + 4 | 1 | 4x + 4 + 1/(x + 5) |
| 7 | x + 1 | -1 | x + 1 - 1/(x - 2) |
| 8 | 4x - 5 | 2 | 4x - 5 + 2/(x + 2) |
| 9 | x + 6 | 3 | x + 6 + 3/(x - 2) |
| 10 | x - 5 | -1 | x - 5 - 1/(x + 4) |

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## 📌 Notes:
- All answers are written in the form: Quotient + Remainder/Divisor
- If the remainder is 0, you could write just the quotient — but here, none divide evenly.
- Synthetic division works because all divisors are linear and monic (coefficient of x is 1).
- For non-monic divisors (like 2x - 3), you’d need to adjust — but not needed here.

Let me know if you want these shown with long division steps instead!