Synthetic Division interactive worksheet - Free Printable

Let's solve each problem using synthetic division step by step. I'll complete the tables and provide the final quotient and remainder for each.

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1. $(x^3 - 9x^2 + 27x - 27) \div (x - 3)$

This one is already partially filled in:

```
| x³ x² x c
3 | 1 -9 27 -27
| 3 -18 27
-------------------
| 1 -6 9 0
```

- Bring down: 1
- Multiply: $3 \times 1 = 3$, add to next: $-9 + 3 = -6$
- Multiply: $3 \times (-6) = -18$, add to next: $27 + (-18) = 9$
- Multiply: $3 \times 9 = 27$, add to last: $-27 + 27 = 0$

So, the quotient is $x^2 - 6x + 9$, remainder = 0.

✔ Answer: $x^2 - 6x + 9$

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2. $(x^3 + 3x^2 - 6x - 8) \div (x - 2)$

Set up synthetic division with root $2$:

```
| x³ x² x c
2 | 1 3 -6 -8
| 2 10 8
-------------------
| 1 5 4 0
```

Steps:
- Bring down: 1
- $2 \times 1 = 2$, $3 + 2 = 5$
- $2 \times 5 = 10$, $-6 + 10 = 4$
- $2 \times 4 = 8$, $-8 + 8 = 0$

Quotient: $x^2 + 5x + 4$, remainder = 0

✔ Answer: $x^2 + 5x + 4$

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3. $(3x^4 - 5x^3 - 5x^2 + 4x - 27) \div (x + 1)$

Root: $-1$

```
| x⁴ x³ x² x c
-1 | 3 -5 -5 4 -27
| -3 8 -3 1
---------------------------
| 3 -8 3 1 -26
```

Steps:
- Bring down: 3
- $-1 \times 3 = -3$, $-5 + (-3) = -8$
- $-1 \times (-8) = 8$, $-5 + 8 = 3$
- $-1 \times 3 = -3$, $4 + (-3) = 1$
- $-1 \times 1 = -1$, $-27 + (-1) = -26$

Quotient: $3x^3 - 8x^2 + 3x + 1$, remainder = $-26$

✔ Answer: $3x^3 - 8x^2 + 3x + 1 - \dfrac{26}{x+1}$

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4. $(x^4 - 16) \div (x + 2)$

Note: $x^4 - 16 = x^4 + 0x^3 + 0x^2 + 0x - 16$

Root: $-2$

```
| x⁴ x³ x² x c
-2 | 1 0 0 0 -16
| -2 4 -8 16
----------------------------
| 1 -2 4 -8 0
```

Steps:
- Bring down: 1
- $-2 \times 1 = -2$, $0 + (-2) = -2$
- $-2 \times (-2) = 4$, $0 + 4 = 4$
- $-2 \times 4 = -8$, $0 + (-8) = -8$
- $-2 \times (-8) = 16$, $-16 + 16 = 0$

Quotient: $x^3 - 2x^2 + 4x - 8$, remainder = 0

✔ Answer: $x^3 - 2x^2 + 4x - 8$

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5. $(2x^2 - 3x - 1) \div (x - 1)$

Root: $1$

```
| x² x c
1 | 2 -3 -1
| 2 -1
--------------
| 2 -1 -2
```

Steps:
- Bring down: 2
- $1 \times 2 = 2$, $-3 + 2 = -1$
- $1 \times (-1) = -1$, $-1 + (-1) = -2$

Quotient: $2x - 1$, remainder = $-2$

✔ Answer: $2x - 1 - \dfrac{2}{x-1}$

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✔ Final Answers:

1. $x^2 - 6x + 9$
2. $x^2 + 5x + 4$
3. $3x^3 - 8x^2 + 3x + 1 - \dfrac{26}{x+1}$
4. $x^3 - 2x^2 + 4x - 8$
5. $2x - 1 - \dfrac{2}{x-1}$

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🔍 Summary of Table Fill-ins:

#### Problem 1:
- Bottom row: `1`, `-6`, `9`, `0` → Answer: $x^2 -6x + 9$

#### Problem 2:
- Bottom row: `1`, `5`, `4`, `0` → Answer: $x^2 +5x + 4$

#### Problem 3:
- Bottom row: `3`, `-8`, `3`, `1`, `-26` → Answer: $3x^3 -8x^2 +3x +1 -\dfrac{26}{x+1}$

#### Problem 4:
- Fill in missing coefficients:
Original: `1`, `0`, `0`, `0`, `-16`
Then: `-2`, `4`, `-8`, `16`
Bottom: `1`, `-2`, `4`, `-8`, `0` → Answer: $x^3 -2x^2 +4x -8$

#### Problem 5:
- Fill in: `2`, `-1`, `-2` → Answer: $2x -1 -\dfrac{2}{x-1}$

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Let me know if you'd like this formatted as a completed worksheet!