Polynomial long division practice questions on a quiz worksheet.

Problem Analysis

The task involves performing polynomial long division for three different problems. Let's solve each problem step by step.

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Problem 1:

Divide \( (x^3 + 6x^2 - x - 30) \div (x - 2) \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\frac{x^3 + 6x^2 - x - 30}{x - 2}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{x^3}{x} = x^2
\]
Write \( x^2 \) above the division bar.

3. Multiply \( x^2 \) by the divisor \( x - 2 \):
\[
x^2 \cdot (x - 2) = x^3 - 2x^2
\]
Subtract this from the original polynomial:
\[
(x^3 + 6x^2 - x - 30) - (x^3 - 2x^2) = 8x^2 - x - 30
\]

4. Repeat the process with the new polynomial \( 8x^2 - x - 30 \):
\[
\frac{8x^2}{x} = 8x
\]
Write \( 8x \) above the division bar next to \( x^2 \).

5. Multiply \( 8x \) by the divisor \( x - 2 \):
\[
8x \cdot (x - 2) = 8x^2 - 16x
\]
Subtract this from \( 8x^2 - x - 30 \):
\[
(8x^2 - x - 30) - (8x^2 - 16x) = 15x - 30
\]

6. Repeat the process with the new polynomial \( 15x - 30 \):
\[
\frac{15x}{x} = 15
\]
Write \( 15 \) above the division bar next to \( 8x \).

7. Multiply \( 15 \) by the divisor \( x - 2 \):
\[
15 \cdot (x - 2) = 15x - 30
\]
Subtract this from \( 15x - 30 \):
\[
(15x - 30) - (15x - 30) = 0
\]

8. Final quotient:
The quotient is \( x^2 + 8x + 15 \), and there is no remainder.

#### Answer:
\[
\boxed{x^2 + 8x + 15}
\]

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Problem 2:

Divide \( (x^3 + 7x^2 - 6x - 72) \div (x + 6) \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\frac{x^3 + 7x^2 - 6x - 72}{x + 6}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{x^3}{x} = x^2
\]
Write \( x^2 \) above the division bar.

3. Multiply \( x^2 \) by the divisor \( x + 6 \):
\[
x^2 \cdot (x + 6) = x^3 + 6x^2
\]
Subtract this from the original polynomial:
\[
(x^3 + 7x^2 - 6x - 72) - (x^3 + 6x^2) = x^2 - 6x - 72
\]

4. Repeat the process with the new polynomial \( x^2 - 6x - 72 \):
\[
\frac{x^2}{x} = x
\]
Write \( x \) above the division bar next to \( x^2 \).

5. Multiply \( x \) by the divisor \( x + 6 \):
\[
x \cdot (x + 6) = x^2 + 6x
\]
Subtract this from \( x^2 - 6x - 72 \):
\[
(x^2 - 6x - 72) - (x^2 + 6x) = -12x - 72
\]

6. Repeat the process with the new polynomial \( -12x - 72 \):
\[
\frac{-12x}{x} = -12
\]
Write \( -12 \) above the division bar next to \( x \).

7. Multiply \( -12 \) by the divisor \( x + 6 \):
\[
-12 \cdot (x + 6) = -12x - 72
\]
Subtract this from \( -12x - 72 \):
\[
(-12x - 72) - (-12x - 72) = 0
\]

8. Final quotient:
The quotient is \( x^2 + x - 12 \), and there is no remainder.

#### Answer:
\[
\boxed{x^2 + x - 12}
\]

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Problem 3:

Divide \( (x^2 + 3x - 18) \div (x - 3) \).

#### Step-by-Step Solution:
1. Set up the division:
\[
\frac{x^2 + 3x - 18}{x - 3}
\]

2. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{x^2}{x} = x
\]
Write \( x \) above the division bar.

3. Multiply \( x \) by the divisor \( x - 3 \):
\[
x \cdot (x - 3) = x^2 - 3x
\]
Subtract this from the original polynomial:
\[
(x^2 + 3x - 18) - (x^2 - 3x) = 6x - 18
\]

4. Repeat the process with the new polynomial \( 6x - 18 \):
\[
\frac{6x}{x} = 6
\]
Write \( 6 \) above the division bar next to \( x \).

5. Multiply \( 6 \) by the divisor \( x - 3 \):
\[
6 \cdot (x - 3) = 6x - 18
\]
Subtract this from \( 6x - 18 \):
\[
(6x - 18) - (6x - 18) = 0
\]

6. Final quotient:
The quotient is \( x + 6 \), and there is no remainder.

#### Answer:
\[
\boxed{x + 6}
\]

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Final Answers:

1. \( \boxed{x^2 + 8x + 15} \)
2. \( \boxed{x^2 + x - 12} \)
3. \( \boxed{x + 6} \)