Synthetic division practice worksheet with four polynomial division problems.

The task involves solving polynomial division problems using Synthetic Division. Synthetic division is a shorthand method of dividing polynomials by binomials of the form \( x - c \). Let's solve each problem step by step.

---

Problem 1:

Divide \( x^4 - 3x^3 + 4x^2 - x + 3 \) by \( x + 2 \).

#### Steps:
1. Set up the synthetic division:
- The divisor is \( x + 2 \), so \( c = -2 \).
- Write down the coefficients of the dividend: \( 1, -3, 4, -1, 3 \).

```
-2 | 1 -3 4 -1 3
| -2 10 -28 58
---------------------
1 -5 14 -29 61
```

2. Perform the synthetic division:
- Bring down the first coefficient: \( 1 \).
- Multiply \( 1 \times (-2) = -2 \) and add to the next coefficient: \( -3 + (-2) = -5 \).
- Multiply \( -5 \times (-2) = 10 \) and add to the next coefficient: \( 4 + 10 = 14 \).
- Multiply \( 14 \times (-2) = -28 \) and add to the next coefficient: \( -1 + (-28) = -29 \).
- Multiply \( -29 \times (-2) = 58 \) and add to the next coefficient: \( 3 + 58 = 61 \).

3. Interpret the results:
- The numbers on the bottom row are the coefficients of the quotient polynomial, and the last number is the remainder.
- Quotient: \( x^3 - 5x^2 + 14x - 29 \)
- Remainder: \( 61 \)

#### Final Answer:
\[
\boxed{x^3 - 5x^2 + 14x - 29 + \frac{61}{x + 2}}
\]

---

Problem 2:

Divide \( x^3 - 2x^2 - 3x + 7 \) by \( x - 2 \).

#### Steps:
1. Set up the synthetic division:
- The divisor is \( x - 2 \), so \( c = 2 \).
- Write down the coefficients of the dividend: \( 1, -2, -3, 7 \).

```
2 | 1 -2 -3 7
| 2 0 -6
-----------------
1 0 -3 1
```

2. Perform the synthetic division:
- Bring down the first coefficient: \( 1 \).
- Multiply \( 1 \times 2 = 2 \) and add to the next coefficient: \( -2 + 2 = 0 \).
- Multiply \( 0 \times 2 = 0 \) and add to the next coefficient: \( -3 + 0 = -3 \).
- Multiply \( -3 \times 2 = -6 \) and add to the next coefficient: \( 7 + (-6) = 1 \).

3. Interpret the results:
- The numbers on the bottom row are the coefficients of the quotient polynomial, and the last number is the remainder.
- Quotient: \( x^2 + 0x - 3 = x^2 - 3 \)
- Remainder: \( 1 \)

#### Final Answer:
\[
\boxed{x^2 - 3 + \frac{1}{x - 2}}
\]

---

Problem 3:

Divide \( x^3 - x + 1 \) by \( x + i \).

#### Steps:
1. Set up the synthetic division:
- The divisor is \( x + i \), so \( c = -i \).
- Write down the coefficients of the dividend: \( 1, 0, -1, 1 \).

```
-i | 1 0 -1 1
| -i i 1+i
------------------
1 -i i-1 2+i
```

2. Perform the synthetic division:
- Bring down the first coefficient: \( 1 \).
- Multiply \( 1 \times (-i) = -i \) and add to the next coefficient: \( 0 + (-i) = -i \).
- Multiply \( -i \times (-i) = -i^2 = 1 \) and add to the next coefficient: \( -1 + 1 = 0 \).
- Multiply \( 0 \times (-i) = 0 \) and add to the next coefficient: \( 1 + 0 = 1 \).

3. Interpret the results:
- The numbers on the bottom row are the coefficients of the quotient polynomial, and the last number is the remainder.
- Quotient: \( x^2 - ix + (i - 1) \)
- Remainder: \( 2 + i \)

#### Final Answer:
\[
\boxed{x^2 - ix + (i - 1) + \frac{2 + i}{x + i}}
\]

---

Problem 4:

Divide \( -3x^3 + 2x^2 + x \) by \( x - 1 \).

#### Steps:
1. Set up the synthetic division:
- The divisor is \( x - 1 \), so \( c = 1 \).
- Write down the coefficients of the dividend: \( -3, 2, 1, 0 \).

```
1 | -3 2 1 0
| -3 -1 0
-----------------
-3 -1 0 0
```

2. Perform the synthetic division:
- Bring down the first coefficient: \( -3 \).
- Multiply \( -3 \times 1 = -3 \) and add to the next coefficient: \( 2 + (-3) = -1 \).
- Multiply \( -1 \times 1 = -1 \) and add to the next coefficient: \( 1 + (-1) = 0 \).
- Multiply \( 0 \times 1 = 0 \) and add to the next coefficient: \( 0 + 0 = 0 \).

3. Interpret the results:
- The numbers on the bottom row are the coefficients of the quotient polynomial, and the last number is the remainder.
- Quotient: \( -3x^2 - x \)
- Remainder: \( 0 \)

#### Final Answer:
\[
\boxed{-3x^2 - x}
\]

---

Final Answers:

1. \( \boxed{x^3 - 5x^2 + 14x - 29 + \frac{61}{x + 2}} \)
2. \( \boxed{x^2 - 3 + \frac{1}{x - 2}} \)
3. \( \boxed{x^2 - ix + (i - 1) + \frac{2 + i}{x + i}} \)
4. \( \boxed{-3x^2 - x} \)