Synthetic Division Activity Worksheet for students to complete polynomial division using synthetic division method.

Problem: Synthetic Division

The task is to divide the given polynomials using synthetic division and complete the table and quotient by filling in the blanks. Let's solve each problem step by step.

---

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

Given:
\[
\begin{array}{r|rrrr}
3 & 1 & -9 & 27 & -27 \\
& & 3 & -18 & 27 \\
\hline
& 1 & -6 & 9 & 0 \\
\end{array}
\]

- Step 1: Write down the coefficients of the dividend: \(1, -9, 27, -27\).
- Step 2: Bring down the first coefficient: \(1\).
- Step 3: Multiply \(1\) by \(3\) and add to the next coefficient: \((-9 + 3 = -6)\).
- Step 4: Multiply \(-6\) by \(3\) and add to the next coefficient: \((27 + (-18) = 9)\).
- Step 5: Multiply \(9\) by \(3\) and add to the next coefficient: \((-27 + 27 = 0)\).

The quotient is \(x^2 - 6x + 9\) and the remainder is \(0\).

Answer:
\[
x^2 - 6x + 9
\]

---

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

Setup:
\[
\begin{array}{r|rrrr}
2 & 1 & 3 & -6 & -8 \\
& & 2 & 10 & 8 \\
\hline
& 1 & 5 & 4 & 0 \\
\end{array}
\]

- Step 1: Write down the coefficients of the dividend: \(1, 3, -6, -8\).
- Step 2: Bring down the first coefficient: \(1\).
- Step 3: Multiply \(1\) by \(2\) and add to the next coefficient: \((3 + 2 = 5)\).
- Step 4: Multiply \(5\) by \(2\) and add to the next coefficient: \((-6 + 10 = 4)\).
- Step 5: Multiply \(4\) by \(2\) and add to the next coefficient: \((-8 + 8 = 0)\).

The quotient is \(x^2 + 5x + 4\) and the remainder is \(0\).

Answer:
\[
x^2 + 5x + 4
\]

---

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

Setup:
\[
\begin{array}{r|rrrrr}
-1 & 3 & -5 & -5 & 4 & -27 \\
& & -3 & 8 & -3 & -1 \\
\hline
& 3 & -8 & 3 & 1 & -28 \\
\end{array}
\]

- Step 1: Write down the coefficients of the dividend: \(3, -5, -5, 4, -27\).
- Step 2: Bring down the first coefficient: \(3\).
- Step 3: Multiply \(3\) by \(-1\) and add to the next coefficient: \((-5 + (-3) = -8)\).
- Step 4: Multiply \(-8\) by \(-1\) and add to the next coefficient: \((-5 + 8 = 3)\).
- Step 5: Multiply \(3\) by \(-1\) and add to the next coefficient: \((4 + (-3) = 1)\).
- Step 6: Multiply \(1\) by \(-1\) and add to the next coefficient: \((-27 + (-1) = -28)\).

The quotient is \(3x^3 - 8x^2 + 3x + 1\) and the remainder is \(-28\).

Answer:
\[
3x^3 - 8x^2 + 3x + 1 - \frac{28}{x+1}
\]

---

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

Setup:
\[
\begin{array}{r|rrrrr}
-2 & 1 & 0 & 0 & 0 & -16 \\
& & -2 & 4 & -8 & 16 \\
\hline
& 1 & -2 & 4 & -8 & 0 \\
\end{array}
\]

- Step 1: Write down the coefficients of the dividend: \(1, 0, 0, 0, -16\).
- Step 2: Bring down the first coefficient: \(1\).
- Step 3: Multiply \(1\) by \(-2\) and add to the next coefficient: \((0 + (-2) = -2)\).
- Step 4: Multiply \(-2\) by \(-2\) and add to the next coefficient: \((0 + 4 = 4)\).
- Step 5: Multiply \(4\) by \(-2\) and add to the next coefficient: \((0 + (-8) = -8)\).
- Step 6: Multiply \(-8\) by \(-2\) and add to the next coefficient: \((-16 + 16 = 0)\).

The quotient is \(x^3 - 2x^2 + 4x - 8\) and the remainder is \(0\).

Answer:
\[
x^3 - 2x^2 + 4x - 8
\]

---

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

Setup:
\[
\begin{array}{r|rrr}
1 & 2 & -3 & -1 \\
& & 2 & -1 \\
\hline
& 2 & -1 & -2 \\
\end{array}
\]

- Step 1: Write down the coefficients of the dividend: \(2, -3, -1\).
- Step 2: Bring down the first coefficient: \(2\).
- Step 3: Multiply \(2\) by \(1\) and add to the next coefficient: \((-3 + 2 = -1)\).
- Step 4: Multiply \(-1\) by \(1\) and add to the next coefficient: \((-1 + (-1) = -2)\).

The quotient is \(2x - 1\) and the remainder is \(-2\).

Answer:
\[
2x - 1 - \frac{2}{x-1}
\]

---

Final Answers:

1. \( x^2 - 6x + 9 \)
2. \( x^2 + 5x + 4 \)
3. \( 3x^3 - 8x^2 + 3x + 1 - \frac{28}{x+1} \)
4. \( x^3 - 2x^2 + 4x - 8 \)
5. \( 2x - 1 - \frac{2}{x-1} \)

\[
\boxed{x^2 - 6x + 9, \, x^2 + 5x + 4, \, 3x^3 - 8x^2 + 3x + 1 - \frac{28}{x+1}, \, x^3 - 2x^2 + 4x - 8, \, 2x - 1 - \frac{2}{x-1}}
\]