Long division worksheets featuring polynomial division problems for practice.

We are given 6 polynomial long division problems. I’ll solve each one step by step, showing the long division process and the final answer (quotient + remainder, if any).

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Problem 1:

(2x² - 5x + 3) ÷ (2x - 1)

Step 1: Divide leading terms:
(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 next term:
(-4x) ÷ (2x) = -2

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

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

✔ Final Answer:
Quotient: x - 2, Remainder: 1
→ So, (2x² - 5x + 3) ÷ (2x - 1) = x - 2 + 1/(2x - 1)

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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) = (-6m + 4m) + 1 = -2m + 1

Step 4: -2m ÷ m = -2

Step 5: -2 × (m - 4) = -2m + 8

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

✔ Final Answer:
Quotient: m - 2, Remainder: -7
→ So, (m² - 6m + 1) ÷ (m - 4) = m - 2 - 7/(m - 4)

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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

✔ Final Answer:
x - 3

*(You can verify via long division if needed — same result)*

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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) = (-5x + 3x) + 2 = -2x + 2

Step 4: -2x ÷ x = -2

Step 5: -2 × (x - 3) = -2x + 6

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

✔ Final Answer:
Quotient: x - 2, Remainder: -4
→ So, (x² - 5x + 2) ÷ (x - 3) = x - 2 - 4/(x - 3)

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Problem 5:

(x² + 4) ÷ (x + 1)

Step 1: x² ÷ x = x

Step 2: x × (x + 1) = x² + x

Step 3: Subtract:
(x² + 0x + 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

✔ Final Answer:
Quotient: x - 1, Remainder: 5
→ So, (x² + 4) ÷ (x + 1) = x - 1 + 5/(x + 1)

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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

✔ Final Answer:
Quotient: y - 1, Remainder: 6
→ So, (5y² - 6y + 7) ÷ (5y - 1) = y - 1 + 6/(5y - 1)

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## ✔ Final Answers Summary:

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 written out vertically as traditional long division!