Synthetic Division Practice Worksheet - Divide polynomials using synthetic division and complete the solutions.

Let's solve each problem using synthetic division, step by step, and fill in the blanks as required.

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🔷 Problem 1: Already Solved

Given:
$$
(m^3 - 36m - 3) \div (m - 6)
$$

We are given the completed synthetic division:

```
m³ m² m c
6 | 1 0 -36 -3
| 6 36 0
-------------------
1 6 0 -3
```

Answer: $ m^2 + 6m - 3 $ with remainder $-3$, so:
$$
\boxed{m^2 + 6m - 3 - \frac{3}{m - 6}}
$$

✔ Correct!

---

🔷 Problem 2:

$$
(4b^3 - 4b^2 - 15b - 16) \div (b + 1)
$$

Step 1: Use root from divisor: $ b + 1 = 0 \Rightarrow b = -1 $

Set up synthetic division:

```
b³ b² b c
-1 | 4 -4 -15 -16
| -4 8 7
---------------------
4 -8 -7 -9
```

Explanation:

1. Bring down 4.
2. Multiply: $-1 \times 4 = -4$ → add to next: $-4 + (-4) = -8$
3. Multiply: $-1 \times (-8) = 8$ → add to next: $-15 + 8 = -7$
4. Multiply: $-1 \times (-7) = 7$ → add to last: $-16 + 7 = -9$

So, quotient is: $ 4b^2 - 8b - 7 $, remainder: $-9$

Now fill in boxes:

```
b³ b² b c
-1 | 4 -4 -15 -16
| -4 8 7
---------------------
4 -8 -7 -9
```

Answer:
$$
\boxed{4b^2 - 8b - 7 - \frac{9}{b+1}}
$$

> Note: The remainder is negative, but we write it as $-\frac{9}{b+1}$, so the sign is already accounted for.

✔ Fill in:
- First box: 4
- Second: -8
- Third: -7
- Fourth: -9

So answer line:
Answer: 4 b² – 8 b – 7 – 9 / b+1

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🔷 Problem 3:

$$
(k^5 + 14k^4 + 35k^3 - 41k^2 + 36k - 46) \div (k + 5)
$$

Root: $ k = -5 $

Set up:

```
k⁵ k⁴ k³ k² k c
-5 | 1 14 35 -41 36 -46
| -5 -45 50 -45 45
--------------------------------
1 9 -10 9 -9 -1
```

Steps:

1. Bring down 1.
2. $-5 × 1 = -5$ → $14 + (-5) = 9$
3. $-5 × 9 = -45$ → $35 + (-45) = -10$
4. $-5 × (-10) = 50$ → $-41 + 50 = 9$
5. $-5 × 9 = -45$ → $36 + (-45) = -9$
6. $-5 × (-9) = 45$ → $-46 + 45 = -1$

Quotient: $ k^4 + 9k^3 - 10k^2 + 9k - 9 $, remainder: $-1$

So:
$$
\boxed{k^4 + 9k^3 - 10k^2 + 9k - 9 - \frac{1}{k+5}}
$$

Fill in boxes under the line:
- 1, 9, -10, 9, -9, -1

Answer line:
Answer: k⁴ + 9 k³ – 10 k² + 9 k – 9 – 1 / k+5

---

🔷 Problem 4:

$$
(-2y^2 - 9y - 7) \div (y + 1)
$$

Root: $ y = -1 $

Set up:

```
y² y c
-1 | -2 -9 -7
| 2 7
-------------
-2 -7 0
```

Steps:

1. Bring down $-2$
2. $-1 × (-2) = 2$ → $-9 + 2 = -7$
3. $-1 × (-7) = 7$ → $-7 + 7 = 0$

Quotient: $-2y - 7$, remainder: 0

So:
$$
\boxed{-2y - 7}
$$

Fill in:

```
y² y c
-1 | -2 -9 -7
| 2 7
-------------
-2 -7 0
```

Answer: –2 y – 7

---

🔷 Problem 5:

$$
(a^4 + 12a^3 + 32a^2 + 0a + 0) \div (a + 8)
$$

Note: No $a$ or constant term → coefficients: 1, 12, 32, 0, 0

Root: $ a = -8 $

Set up:

```
a⁴ a³ a² a c
-8 | 1 12 32 0 0
| -8 -32 0 0
-------------------------
1 4 0 0 0
```

Steps:

1. Bring down 1
2. $-8 × 1 = -8$ → $12 + (-8) = 4$
3. $-8 × 4 = -32$ → $32 + (-32) = 0$
4. $-8 × 0 = 0$ → $0 + 0 = 0$
5. $-8 × 0 = 0$ → $0 + 0 = 0$

Quotient: $ a^3 + 4a^2 + 0a + 0 = a^3 + 4a^2 $, remainder: 0

So:
$$
\boxed{a^3 + 4a^2}
$$

Fill in:

```
a⁴ a³ a² a c
-8 | 1 12 32 0 0
| -8 -32 0 0
-------------------------
1 4 0 0 0
```

Answer: a³ + 4 a²

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## ✔ Final Answers Summary:

2)

Answer: $ 4b^2 - 8b - 7 - \frac{9}{b+1} $
Boxes: 4, -8, -7, -9

3)

Answer: $ k^4 + 9k^3 - 10k^2 + 9k - 9 - \frac{1}{k+5} $
Boxes: 1, 9, -10, 9, -9, -1

4)

Answer: $ -2y - 7 $
Boxes:
- Top row: -2, -9, -7
- Bottom row: -2, -7, 0

5)

Answer: $ a^3 + 4a^2 $
Boxes:
- Middle row: -8, -32, 0, 0
- Bottom row: 1, 4, 0, 0, 0

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📝 How to Fill in the Worksheet:

#### Problem 2:
- Below the line: 4, -8, -7, -9
- Answer: 4 b² – 8 b – 7 – 9 / b+1

#### Problem 3:
- Below the line: 1, 9, -10, 9, -9, -1
- Answer: k⁴ + 9 k³ – 10 k² + 9 k – 9 – 1 / k+5

#### Problem 4:
- Top row: -2, -9, -7
- Bottom row: -2, -7, 0
- Answer: –2 y – 7

#### Problem 5:
- Middle row: -8, -32, 0, 0
- Bottom row: 1, 4, 0, 0, 0
- Answer: a³ + 4 a²

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💡 I Learned That...

You might write something like:
> "I learned that synthetic division is a fast way to divide polynomials when dividing by a linear factor of the form (x - c). It’s especially useful for finding remainders and factoring polynomials."

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