Polynomial Long Division Calculator – Step-by-Step Guide (2025)

What Is Polynomial Long Division?

Polynomial long division is the algebraic version of the long division you learned with numbers. You repeatedly divide the highest-degree term of the dividend by the highest-degree term of the divisor, multiply, subtract, and bring down the next term—until the remainder has smaller degree than the divisor.

How to Do Polynomial Long Division (Step by Step)

1. Order terms by descending powers and include any missing degrees with a 0 coefficient (e.g., write x³ + 0x² − 4x + 4).
2. Divide the leading term of the dividend by the leading term of the divisor to get the next quotient term.
3. Multiply the entire divisor by this quotient term.
4. Subtract from the dividend (distribute the minus sign!).
5. Bring down the next term and repeat until the remainder’s degree is less than the divisor’s.
6. Write the answer as Quotient + Remainder/Divisor.

Worked Example #1

Divide: 2x³ + 3x² − 5x + 6 by x − 2

1. 2x³ ÷ x = 2x²
2. Multiply: (x − 2)(2x²) = 2x³ − 4x²
3. Subtract: (2x³ + 3x² − 5x + 6) − (2x³ − 4x²) = 7x² − 5x + 6
4. Next: 7x² ÷ x = 7x
5. Multiply: 7x(x − 2) = 7x² − 14x
6. Subtract: (7x² − 5x) − (7x² − 14x) = 9x (bring down + 6 → 9x + 6)
7. Next: 9x ÷ x = 9
8. Multiply: 9(x − 2) = 9x − 18
9. Subtract: (9x + 6) − (9x − 18) = 24 (remainder)

Result: Quotient = 2x² + 7x + 9,   Remainder = 24

Final answer: 2x² + 7x + 9 + 24/(x − 2)

Worked Example #2 (Mind the Missing Term!)

Divide: x³ − 4x + 4 by x − 1

Rewrite as x³ + 0x² − 4x + 4 to include the missing x² term.

1. x³ ÷ x = x²; multiply x²(x − 1) = x²x − x² = x³ − x²
2. Subtract: (x³ + 0x²) − (x³ − x²) = x²; bring down − 4x → x² − 4x
3. x² ÷ x = x; multiply x(x − 1) = x² − x
4. Subtract: (x² − 4x) − (x² − x) = −3x; bring down + 4 → −3x + 4
5. −3x ÷ x = −3; multiply −3(x − 1) = −3x + 3
6. Subtract: (−3x + 4) − (−3x + 3) = 1 (remainder)

Result: Quotient = x² + x − 3,   Remainder = 1

Final answer: x² + x − 3 + 1/(x − 1)

When to Use Synthetic Division vs. Long Division

• Synthetic division works when dividing by a linear factor of the form x − c.
• Long division works for any divisor (including higher-degree or non-monic factors).

Common Mistakes (and How to Avoid Them)

• Forgetting missing terms. Write zero coefficients to keep columns aligned.
• Sign errors when subtracting. Distribute the negative to every term you’re subtracting.
• Stopping too early. Keep going until the remainder’s degree is smaller than the divisor’s degree.
• Skipping the check. Verify with Dividend = Divisor × Quotient + Remainder.

FAQ: Polynomial Long Division Calculator

Is there an online calculator for polynomial long division?

Many sites offer polynomial solvers. This page focuses on teaching the paper method clearly so you can work any problem and check your results confidently.

How do I know if my answer is correct?

Use the identity Dividend = (Divisor × Quotient) + Remainder. Multiply your quotient by the divisor, add the remainder, and confirm it equals the original dividend.

What if the remainder isn’t zero?

That’s normal. Write the final answer as Quotient + Remainder/Divisor. The remainder’s degree must be smaller than the divisor’s degree.

When should I use synthetic division instead?

If the divisor is exactly x − c (a linear factor), synthetic division can be quicker. Otherwise, use long division.

Do I need to add missing terms like 0x²?

Yes—adding zero-coefficient terms keeps powers aligned and prevents arithmetic mistakes.