Long division worksheets featuring polynomial division problems.

We are given 6 polynomial long division problems. Let’s solve each one step by step using polynomial long division.

---

Problem 1: (2x² - 5x + 3) ÷ (2x - 1)

Step 1: Divide leading term of dividend by leading term of divisor:
> 2x² ÷ 2x = x

Step 2: Multiply divisor by x:
> x·(2x - 1) = 2x² - x

Step 3: Subtract from dividend:
> (2x² - 5x + 3) - (2x² - x) = (-5x + x) + 3 = -4x + 3

Step 4: Divide -4x by 2x → -2

Step 5: Multiply divisor by -2:
> -2·(2x - 1) = -4x + 2

Step 6: Subtract:
> (-4x + 3) - (-4x + 2) = 3 - 2 = 1

✔ Quotient: x - 2, Remainder: 1

So,
(2x² - 5x + 3) ÷ (2x - 1) = x - 2 + 1/(2x - 1)

---

Problem 2: (m² - 6m + 1) ÷ (m - 4)

Step 1: m² ÷ m = m

Step 2: m·(m - 4) = m² - 4m

Step 3: Subtract:
> (m² - 6m + 1) - (m² - 4m) = -2m + 1

Step 4: -2m ÷ m = -2

Step 5: -2·(m - 4) = -2m + 8

Step 6: Subtract:
> (-2m + 1) - (-2m + 8) = 1 - 8 = -7

✔ Quotient: m - 2, Remainder: -7

So,
(m² - 6m + 1) ÷ (m - 4) = m - 2 - 7/(m - 4)

---

Problem 3: (x² - 9) ÷ (x + 3)

Note: This is a difference of squares:
> x² - 9 = (x - 3)(x + 3)

So, dividing by (x + 3):

✔ Quotient: x - 3, Remainder: 0

Alternatively, via long division:

Step 1: x² ÷ x = x

Step 2: x·(x + 3) = x² + 3x

Step 3: Subtract:
> (x² - 9) - (x² + 3x) = -3x - 9

Step 4: -3x ÷ x = -3

Step 5: -3·(x + 3) = -3x - 9

Step 6: Subtract:
> (-3x - 9) - (-3x - 9) = 0

✔ Quotient: x - 3, Remainder: 0

---

Problem 4: (x² + 2 - 5x) ÷ (x - 3)

First, rewrite dividend in standard form:
> x² - 5x + 2

Step 1: x² ÷ x = x

Step 2: x·(x - 3) = x² - 3x

Step 3: Subtract:
> (x² - 5x + 2) - (x² - 3x) = -2x + 2

Step 4: -2x ÷ x = -2

Step 5: -2·(x - 3) = -2x + 6

Step 6: Subtract:
> (-2x + 2) - (-2x + 6) = 2 - 6 = -4

✔ Quotient: x - 2, Remainder: -4

So,
(x² - 5x + 2) ÷ (x - 3) = x - 2 - 4/(x - 3)

---

Problem 5: (x² + 4) ÷ (x + 1)

Step 1: x² ÷ x = x

Step 2: x·(x + 1) = x² + x

Step 3: Subtract:
> (x² + 4) - (x² + x) = -x + 4

Step 4: -x ÷ x = -1

Step 5: -1·(x + 1) = -x - 1

Step 6: Subtract:
> (-x + 4) - (-x - 1) = 4 + 1 = 5

✔ Quotient: x - 1, Remainder: 5

So,
(x² + 4) ÷ (x + 1) = x - 1 + 5/(x + 1)

---

Problem 6: (5y² - 6y + 7) ÷ (5y - 1)

Step 1: 5y² ÷ 5y = y

Step 2: y·(5y - 1) = 5y² - y

Step 3: Subtract:
> (5y² - 6y + 7) - (5y² - y) = (-6y + y) + 7 = -5y + 7

Step 4: -5y ÷ 5y = -1

Step 5: -1·(5y - 1) = -5y + 1

Step 6: Subtract:
> (-5y + 7) - (-5y + 1) = 7 - 1 = 6

✔ Quotient: y - 1, Remainder: 6

So,
(5y² - 6y + 7) ÷ (5y - 1) = y - 1 + 6/(5y - 1)

---

## ✔ Final Answers:

1. (2x² - 5x + 3) ÷ (2x - 1) = x - 2 + 1/(2x - 1)
2. (m² - 6m + 1) ÷ (m - 4) = m - 2 - 7/(m - 4)
3. (x² - 9) ÷ (x + 3) = x - 3
4. (x² - 5x + 2) ÷ (x - 3) = x - 2 - 4/(x - 3)
5. (x² + 4) ÷ (x + 1) = x - 1 + 5/(x + 1)
6. (5y² - 6y + 7) ÷ (5y - 1) = y - 1 + 6/(5y - 1)

Let me know if you’d like to see any of these visualized with long division layout!