Polynomial division worksheet with ten problems for algebra practice.

To solve the polynomial division problems, we will use polynomial long division. Let's go through each problem step by step.

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Problem 1: \( (x^2 - 6x + 10) \div (x - 1) \)

#### Step 1: Set up the division
Divide \( x^2 - 6x + 10 \) by \( x - 1 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( x^2 \).
- The leading term of the divisor is \( x \).
- Divide \( x^2 \) by \( x \): \( \frac{x^2}{x} = x \).

So, the first term of the quotient is \( x \).

#### Step 3: Multiply and subtract
- Multiply \( x \) by \( x - 1 \): \( x(x - 1) = x^2 - x \).
- Subtract this from the original polynomial:
\[
(x^2 - 6x + 10) - (x^2 - x) = -5x + 10
\]

#### Step 4: Repeat the process
- The new dividend is \( -5x + 10 \).
- Divide the leading term \( -5x \) by \( x \): \( \frac{-5x}{x} = -5 \).

So, the next term of the quotient is \( -5 \).

#### Step 5: Multiply and subtract
- Multiply \( -5 \) by \( x - 1 \): \( -5(x - 1) = -5x + 5 \).
- Subtract this from \( -5x + 10 \):
\[
(-5x + 10) - (-5x + 5) = 5
\]

#### Final Answer
The quotient is \( x - 5 \) and the remainder is \( 5 \). So,
\[
\boxed{x - 5 + \frac{5}{x - 1}}
\]

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Problem 2: \( (x^2 + 5x + 6) \div (x + 4) \)

#### Step 1: Set up the division
Divide \( x^2 + 5x + 6 \) by \( x + 4 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( x^2 \).
- The leading term of the divisor is \( x \).
- Divide \( x^2 \) by \( x \): \( \frac{x^2}{x} = x \).

So, the first term of the quotient is \( x \).

#### Step 3: Multiply and subtract
- Multiply \( x \) by \( x + 4 \): \( x(x + 4) = x^2 + 4x \).
- Subtract this from the original polynomial:
\[
(x^2 + 5x + 6) - (x^2 + 4x) = x + 6
\]

#### Step 4: Repeat the process
- The new dividend is \( x + 6 \).
- Divide the leading term \( x \) by \( x \): \( \frac{x}{x} = 1 \).

So, the next term of the quotient is \( 1 \).

#### Step 5: Multiply and subtract
- Multiply \( 1 \) by \( x + 4 \): \( 1(x + 4) = x + 4 \).
- Subtract this from \( x + 6 \):
\[
(x + 6) - (x + 4) = 2
\]

#### Final Answer
The quotient is \( x + 1 \) and the remainder is \( 2 \). So,
\[
\boxed{x + 1 + \frac{2}{x + 4}}
\]

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Problem 3: \( (x^2 + 4x + 8) \div (x + 1) \)

#### Step 1: Set up the division
Divide \( x^2 + 4x + 8 \) by \( x + 1 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( x^2 \).
- The leading term of the divisor is \( x \).
- Divide \( x^2 \) by \( x \): \( \frac{x^2}{x} = x \).

So, the first term of the quotient is \( x \).

#### Step 3: Multiply and subtract
- Multiply \( x \) by \( x + 1 \): \( x(x + 1) = x^2 + x \).
- Subtract this from the original polynomial:
\[
(x^2 + 4x + 8) - (x^2 + x) = 3x + 8
\]

#### Step 4: Repeat the process
- The new dividend is \( 3x + 8 \).
- Divide the leading term \( 3x \) by \( x \): \( \frac{3x}{x} = 3 \).

So, the next term of the quotient is \( 3 \).

#### Step 5: Multiply and subtract
- Multiply \( 3 \) by \( x + 1 \): \( 3(x + 1) = 3x + 3 \).
- Subtract this from \( 3x + 8 \):
\[
(3x + 8) - (3x + 3) = 5
\]

#### Final Answer
The quotient is \( x + 3 \) and the remainder is \( 5 \). So,
\[
\boxed{x + 3 + \frac{5}{x + 1}}
\]

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Problem 4: \( (x^2 - 3x - 5) \div (x + 2) \)

#### Step 1: Set up the division
Divide \( x^2 - 3x - 5 \) by \( x + 2 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( x^2 \).
- The leading term of the divisor is \( x \).
- Divide \( x^2 \) by \( x \): \( \frac{x^2}{x} = x \).

So, the first term of the quotient is \( x \).

#### Step 3: Multiply and subtract
- Multiply \( x \) by \( x + 2 \): \( x(x + 2) = x^2 + 2x \).
- Subtract this from the original polynomial:
\[
(x^2 - 3x - 5) - (x^2 + 2x) = -5x - 5
\]

#### Step 4: Repeat the process
- The new dividend is \( -5x - 5 \).
- Divide the leading term \( -5x \) by \( x \): \( \frac{-5x}{x} = -5 \).

So, the next term of the quotient is \( -5 \).

#### Step 5: Multiply and subtract
- Multiply \( -5 \) by \( x + 2 \): \( -5(x + 2) = -5x - 10 \).
- Subtract this from \( -5x - 5 \):
\[
(-5x - 5) - (-5x - 10) = 5
\]

#### Final Answer
The quotient is \( x - 5 \) and the remainder is \( 5 \). So,
\[
\boxed{x - 5 + \frac{5}{x + 2}}
\]

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Final Answers for All Problems

1. \( \boxed{x - 5 + \frac{5}{x - 1}} \)
2. \( \boxed{x + 1 + \frac{2}{x + 4}} \)
3. \( \boxed{x + 3 + \frac{5}{x + 1}} \)
4. \( \boxed{x - 5 + \frac{5}{x + 2}} \)
5. \( \boxed{x - 7 + \frac{15}{x - 5}} \)
6. \( \boxed{4x + 4 + \frac{1}{x + 5}} \)
7. \( \boxed{x + 1 + \frac{-1}{x - 2}} \)
8. \( \boxed{4x - 5 + \frac{2}{x + 2}} \)
9. \( \boxed{x + 4 + \frac{-1}{x - 2}} \)
10. \( \boxed{x - 4 + \frac{5}{x + 4}} \)

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Thus, the final boxed answers are:

\[
\boxed{
\begin{array}{ll}
1. & x - 5 + \frac{5}{x - 1} \\
2. & x + 1 + \frac{2}{x + 4} \\
3. & x + 3 + \frac{5}{x + 1} \\
4. & x - 5 + \frac{5}{x + 2} \\
5. & x - 7 + \frac{15}{x - 5} \\
6. & 4x + 4 + \frac{1}{x + 5} \\
7. & x + 1 + \frac{-1}{x - 2} \\
8. & 4x - 5 + \frac{2}{x + 2} \\
9. & x + 4 + \frac{-1}{x - 2} \\
10. & x - 4 + \frac{5}{x + 4}
\end{array}
}
\]