Synthetic Division activity - Free Printable

Problem: Synthetic Division

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

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#### 1) \( (m^3 - 36m - 3) \div (m - 6) \)

- Dividend: \( m^3 + 0m^2 - 36m - 3 \)
- Divisor: \( m - 6 \)
- Root of divisor: \( r = 6 \)

Synthetic Division Steps:
1. Write down the coefficients of the dividend: \( 1, 0, -36, -3 \).
2. Bring down the first coefficient: \( 1 \).
3. Multiply \( 1 \times 6 = 6 \) and add to the next coefficient: \( 0 + 6 = 6 \).
4. Multiply \( 6 \times 6 = 36 \) and add to the next coefficient: \( -36 + 36 = 0 \).
5. Multiply \( 0 \times 6 = 0 \) and add to the next coefficient: \( -3 + 0 = -3 \).

Result:
- Quotient: \( m^2 + 6m + 0 \)
- Remainder: \( -3 \)

Answer: \( m^2 + 6m - \frac{3}{m-6} \)

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

- Dividend: \( 4b^3 - 4b^2 - 15b - 16 \)
- Divisor: \( b + 1 \)
- Root of divisor: \( r = -1 \)

Synthetic Division Steps:
1. Write down the coefficients of the dividend: \( 4, -4, -15, -16 \).
2. Bring down the first coefficient: \( 4 \).
3. Multiply \( 4 \times (-1) = -4 \) and add to the next coefficient: \( -4 + (-4) = -8 \).
4. Multiply \( -8 \times (-1) = 8 \) and add to the next coefficient: \( -15 + 8 = -7 \).
5. Multiply \( -7 \times (-1) = 7 \) and add to the next coefficient: \( -16 + 7 = -9 \).

Result:
- Quotient: \( 4b^2 - 8b - 7 \)
- Remainder: \( -9 \)

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

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

- Dividend: \( k^5 + 14k^4 + 35k^3 - 41k^2 + 36k - 46 \)
- Divisor: \( k + 5 \)
- Root of divisor: \( r = -5 \)

Synthetic Division Steps:
1. Write down the coefficients of the dividend: \( 1, 14, 35, -41, 36, -46 \).
2. Bring down the first coefficient: \( 1 \).
3. Multiply \( 1 \times (-5) = -5 \) and add to the next coefficient: \( 14 + (-5) = 9 \).
4. Multiply \( 9 \times (-5) = -45 \) and add to the next coefficient: \( 35 + (-45) = -10 \).
5. Multiply \( -10 \times (-5) = 50 \) and add to the next coefficient: \( -41 + 50 = 9 \).
6. Multiply \( 9 \times (-5) = -45 \) and add to the next coefficient: \( 36 + (-45) = -9 \).
7. Multiply \( -9 \times (-5) = 45 \) and add to the next coefficient: \( -46 + 45 = -1 \).

Result:
- Quotient: \( k^4 + 9k^3 - 10k^2 + 9k - 9 \)
- Remainder: \( -1 \)

Answer: \( k^4 + 9k^3 - 10k^2 + 9k - 9 - \frac{1}{k+5} \)

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#### 4) \( (-2y^2 - 9y - 7) \div (y + 1) \)

- Dividend: \( -2y^2 - 9y - 7 \)
- Divisor: \( y + 1 \)
- Root of divisor: \( r = -1 \)

Synthetic Division Steps:
1. Write down the coefficients of the dividend: \( -2, -9, -7 \).
2. Bring down the first coefficient: \( -2 \).
3. Multiply \( -2 \times (-1) = 2 \) and add to the next coefficient: \( -9 + 2 = -7 \).
4. Multiply \( -7 \times (-1) = 7 \) and add to the next coefficient: \( -7 + 7 = 0 \).

Result:
- Quotient: \( -2y - 7 \)
- Remainder: \( 0 \)

Answer: \( -2y - 7 \)

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#### 5) \( (a^4 + 12a^3 + 32a^2) \div (a + 8) \)

- Dividend: \( a^4 + 12a^3 + 32a^2 + 0a + 0 \)
- Divisor: \( a + 8 \)
- Root of divisor: \( r = -8 \)

Synthetic Division Steps:
1. Write down the coefficients of the dividend: \( 1, 12, 32, 0, 0 \).
2. Bring down the first coefficient: \( 1 \).
3. Multiply \( 1 \times (-8) = -8 \) and add to the next coefficient: \( 12 + (-8) = 4 \).
4. Multiply \( 4 \times (-8) = -32 \) and add to the next coefficient: \( 32 + (-32) = 0 \).
5. Multiply \( 0 \times (-8) = 0 \) and add to the next coefficient: \( 0 + 0 = 0 \).
6. Multiply \( 0 \times (-8) = 0 \) and add to the next coefficient: \( 0 + 0 = 0 \).

Result:
- Quotient: \( a^3 + 4a^2 \)
- Remainder: \( 0 \)

Answer: \( a^3 + 4a^2 \)

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Final Answers:

1. \( m^2 + 6m - \frac{3}{m-6} \)
2. \( 4b^2 - 8b - 7 - \frac{9}{b+1} \)
3. \( k^4 + 9k^3 - 10k^2 + 9k - 9 - \frac{1}{k+5} \)
4. \( -2y - 7 \)
5. \( a^3 + 4a^2 \)

\[
\boxed{
\begin{aligned}
1. & \quad m^2 + 6m - \frac{3}{m-6} \\
2. & \quad 4b^2 - 8b - 7 - \frac{9}{b+1} \\
3. & \quad k^4 + 9k^3 - 10k^2 + 9k - 9 - \frac{1}{k+5} \\
4. & \quad -2y - 7 \\
5. & \quad a^3 + 4a^2
\end{aligned}
}
\]