Practice worksheet focused on synthetic division, featuring exercises to convert dividends and quotients into polynomial forms, complete synthetic division problems, and find quotients for given polynomial divisions.

Worksheet: Synthetic Division

#### Problem 1: Write the dividend as a polynomial in standard form.
Given synthetic division:
```
-1 | 1 2 -1 -2
1 -1 2
-------------------
1 1 -2 0
```

Solution:
The coefficients of the dividend are \(1, 2, -1, -2\). These correspond to the polynomial:
\[ x^3 + 2x^2 - x - 2 \]

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

---

#### Problem 2: Write the quotient as a polynomial in standard form.
Given synthetic division:
```
4 | 3 -10 0 -5
12 8 32
-------------------
3 2 8 27
```

Solution:
The quotient is given by the coefficients above the horizontal line, excluding the last number (remainder). The coefficients are \(3, 2, 8\), which correspond to the polynomial:
\[ 3x^2 + 2x + 8 \]

Answer:
\[ \boxed{3x^2 + 2x + 8} \]

---

#### Problem 3: Write the dividend as a product of two factors.
Given synthetic division:
```
-2 | 1 2 1 5 6
-2 0 -2 -6
----------------------
1 0 1 3 0
```

Solution:
The synthetic division shows that the remainder is 0, so \(-2\) is a root of the polynomial. The quotient is \(1x^3 + 0x^2 + 1x + 3\), which simplifies to \(x^3 + x + 3\). Therefore, the dividend can be written as:
\[ (x + 2)(x^3 + x + 3) \]

Answer:
\[ \boxed{(x + 2)(x^3 + x + 3)} \]

---

#### Problem 4: Fill in the missing values to complete the synthetic division.
Given:
```
[] | 2 1 [] 10
-4 6 []
-------------------
[] -3 5 []
```

Solution:
1. The first coefficient of the quotient is \(2\), so the divisor is \(-2\).
2. Multiply \(-2\) by \(2\) to get \(-4\), and add it to \(1\) to get \(-3\).
3. Multiply \(-2\) by \(-3\) to get \(6\), and add it to the next coefficient (unknown). Let this unknown coefficient be \(a\). So, \(6 + a = 5\), which gives \(a = -1\).
4. Multiply \(-2\) by \(5\) to get \(-10\), and add it to \(10\) to get \(0\).

Filling in the values:
```
-2 | 2 1 -1 10
-4 6 -10
-------------------
2 -3 5 0
```

Answer:
\[ \boxed{-2, -1, -10, 0} \]

---

#### Problem 5: Fill in the missing values to complete the synthetic division.
Given:
```
[] | [] -4 1 []
3 [] []
-------------------
[] -1 -2 0
```

Solution:
1. The remainder is \(0\), so the divisor is \(1\).
2. The first coefficient of the quotient is \(3\), so the first coefficient of the dividend is \(3\).
3. Multiply \(1\) by \(3\) to get \(3\), and add it to \(-4\) to get \(-1\).
4. Multiply \(1\) by \(-1\) to get \(-1\), and add it to \(1\) to get \(0\).
5. Multiply \(1\) by \(0\) to get \(0\), and add it to the next coefficient (unknown). Let this unknown coefficient be \(b\). So, \(0 + b = 0\), which gives \(b = 0\).

Filling in the values:
```
1 | 3 -4 1 0
3 -1 0
-------------------
3 -1 -2 0
```

Answer:
\[ \boxed{1, 3, -1, 0} \]

---

#### Problem 6: Find the quotient using synthetic division.
\[ (2x^3 + 4x^2 - 3x - 6) \div (x + 3) \]

Solution:
1. Set up the synthetic division with the divisor \(-3\):
```
-3 | 2 4 -3 -6
-6 6 -9
-------------------
2 -2 3 -15
```
2. The quotient is \(2x^2 - 2x + 3\) and the remainder is \(-15\).

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

---

#### Problem 7: Find the quotient using synthetic division.
\[ (x^3 - 6x^2 + 9) \div (x - 4) \]

Solution:
1. Set up the synthetic division with the divisor \(4\):
```
4 | 1 -6 0 9
4 -8 -32
-------------------
1 -2 -8 -23
```
2. The quotient is \(x^2 - 2x - 8\) and the remainder is \(-23\).

Answer:
\[ \boxed{x^2 - 2x - 8} \]

---

#### Problem 8: Find the quotient using synthetic division.
\[ (4x^3 + 27x^2 + 3x + 64) \div (x + 7) \]

Solution:
1. Set up the synthetic division with the divisor \(-7\):
```
-7 | 4 27 3 64
-28 7 -70
-------------------
4 -1 10 -6
```
2. The quotient is \(4x^2 - x + 10\) and the remainder is \(-6\).

Answer:
\[ \boxed{4x^2 - x + 10} \]

---

#### Problem 9: Find the quotient using synthetic division.
\[ (x^4 - 16x^2 - 40x - 25) \div (x - 5) \]

Solution:
1. Set up the synthetic division with the divisor \(5\):
```
5 | 1 0 -16 -40 -25
5 25 45 25
------------------------
1 5 9 5 0
```
2. The quotient is \(x^3 + 5x^2 + 9x + 5\) and the remainder is \(0\).

Answer:
\[ \boxed{x^3 + 5x^2 + 9x + 5} \]

---

Final Answers:

1. \( \boxed{x^3 + 2x^2 - x - 2} \)
2. \( \boxed{3x^2 + 2x + 8} \)
3. \( \boxed{(x + 2)(x^3 + x + 3)} \)
4. \( \boxed{-2, -1, -10, 0} \)
5. \( \boxed{1, 3, -1, 0} \)
6. \( \boxed{2x^2 - 2x + 3} \)
7. \( \boxed{x^2 - 2x - 8} \)
8. \( \boxed{4x^2 - x + 10} \)
9. \( \boxed{x^3 + 5x^2 + 9x + 5} \)