Worksheet featuring six polynomial long division problems for practice.

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

---

Problem 1: \( (2x^2 - 5x + 3) \div (2x - 1) \)

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

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

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

#### Step 3: Multiply and subtract
- Multiply \( x \) by the divisor \( 2x - 1 \): \( x(2x - 1) = 2x^2 - x \).
- Subtract this from the original polynomial:
\[
(2x^2 - 5x + 3) - (2x^2 - x) = -4x + 3
\]

#### Step 4: Repeat the process
- The new dividend is \( -4x + 3 \).
- Divide the leading term \( -4x \) by \( 2x \): \( \frac{-4x}{2x} = -2 \).

So, the next term of the quotient is \( -2 \).

#### Step 5: Multiply and subtract
- Multiply \( -2 \) by the divisor \( 2x - 1 \): \( -2(2x - 1) = -4x + 2 \).
- Subtract this from the current polynomial:
\[
(-4x + 3) - (-4x + 2) = 1
\]

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

\[
\boxed{x - 2 + \frac{1}{2x - 1}}
\]

---

Problem 2: \( (m^2 - 6m + 1) \div (m - 4) \)

#### Step 1: Set up the division
We divide \( m^2 - 6m + 1 \) by \( m - 4 \).

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

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

#### Step 3: Multiply and subtract
- Multiply \( m \) by the divisor \( m - 4 \): \( m(m - 4) = m^2 - 4m \).
- Subtract this from the original polynomial:
\[
(m^2 - 6m + 1) - (m^2 - 4m) = -2m + 1
\]

#### Step 4: Repeat the process
- The new dividend is \( -2m + 1 \).
- Divide the leading term \( -2m \) by \( m \): \( \frac{-2m}{m} = -2 \).

So, the next term of the quotient is \( -2 \).

#### Step 5: Multiply and subtract
- Multiply \( -2 \) by the divisor \( m - 4 \): \( -2(m - 4) = -2m + 8 \).
- Subtract this from the current polynomial:
\[
(-2m + 1) - (-2m + 8) = -7
\]

#### Final Answer
The quotient is \( m - 2 \) and the remainder is \( -7 \).

\[
\boxed{m - 2 - \frac{7}{m - 4}}
\]

---

Problem 3: \( (x^2 - 9) \div (x + 3) \)

#### Step 1: Set up the division
We divide \( x^2 - 9 \) by \( x + 3 \).

#### 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 the divisor \( x + 3 \): \( x(x + 3) = x^2 + 3x \).
- Subtract this from the original polynomial:
\[
(x^2 - 9) - (x^2 + 3x) = -3x - 9
\]

#### Step 4: Repeat the process
- The new dividend is \( -3x - 9 \).
- 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 the divisor \( x + 3 \): \( -3(x + 3) = -3x - 9 \).
- Subtract this from the current polynomial:
\[
(-3x - 9) - (-3x - 9) = 0
\]

#### Final Answer
The quotient is \( x - 3 \) and the remainder is \( 0 \).

\[
\boxed{x - 3}
\]

---

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

#### Step 1: Rearrange the dividend
Rewrite the dividend in standard form: \( x^2 - 5x + 2 \).

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

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

#### Step 5: Repeat the process
- The new dividend is \( -2x + 2 \).
- Divide the leading term \( -2x \) by \( x \): \( \frac{-2x}{x} = -2 \).

So, the next term of the quotient is \( -2 \).

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

#### Final Answer
The quotient is \( x - 2 \) and the remainder is \( -4 \).

\[
\boxed{x - 2 - \frac{4}{x - 3}}
\]

---

Problem 5: \( (x^2 + 4) \div (x + 1) \)

#### Step 1: Set up the division
We divide \( x^2 + 4 \) 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 the divisor \( x + 1 \): \( x(x + 1) = x^2 + x \).
- Subtract this from the original polynomial:
\[
(x^2 + 4) - (x^2 + x) = -x + 4
\]

#### Step 4: Repeat the process
- The new dividend is \( -x + 4 \).
- 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 the divisor \( x + 1 \): \( -1(x + 1) = -x - 1 \).
- Subtract this from the current polynomial:
\[
(-x + 4) - (-x - 1) = 5
\]

#### Final Answer
The quotient is \( x - 1 \) and the remainder is \( 5 \).

\[
\boxed{x - 1 + \frac{5}{x + 1}}
\]

---

Problem 6: \( (5y^2 - 6y + 7) \div (5y - 1) \)

#### Step 1: Set up the division
We divide \( 5y^2 - 6y + 7 \) by \( 5y - 1 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( 5y^2 \).
- The leading term of the divisor is \( 5y \).
- Divide \( 5y^2 \) by \( 5y \): \( \frac{5y^2}{5y} = y \).

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

#### Step 3: Multiply and subtract
- Multiply \( y \) by the divisor \( 5y - 1 \): \( y(5y - 1) = 5y^2 - y \).
- Subtract this from the original polynomial:
\[
(5y^2 - 6y + 7) - (5y^2 - y) = -5y + 7
\]

#### Step 4: Repeat the process
- The new dividend is \( -5y + 7 \).
- Divide the leading term \( -5y \) by \( 5y \): \( \frac{-5y}{5y} = -1 \).

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

#### Step 5: Multiply and subtract
- Multiply \( -1 \) by the divisor \( 5y - 1 \): \( -1(5y - 1) = -5y + 1 \).
- Subtract this from the current polynomial:
\[
(-5y + 7) - (-5y + 1) = 6
\]

#### Final Answer
The quotient is \( y - 1 \) and the remainder is \( 6 \).

\[
\boxed{y - 1 + \frac{6}{5y - 1}}
\]

---

Final Answers for All Problems:

1. \( \boxed{x - 2 + \frac{1}{2x - 1}} \)
2. \( \boxed{m - 2 - \frac{7}{m - 4}} \)
3. \( \boxed{x - 3} \)
4. \( \boxed{x - 2 - \frac{4}{x - 3}} \)
5. \( \boxed{x - 1 + \frac{5}{x + 1}} \)
6. \( \boxed{y - 1 + \frac{6}{5y - 1}} \)