Long Division of Polynomials Solver – Step-by-Step (2025)

What Is Long Division of Polynomials?

It’s the polynomial version of numeric long division. At each step, match the highest powers, then repeat: divide → multiply → subtract → bring down. Continue until the remainder’s degree is less than the divisor’s degree.

Step-by-Step Rules (Quick Reference)

1. Order terms in descending powers and insert 0-coefficient placeholders for any missing powers.
2. Divide leading terms (dividend ÷ divisor) to get the next quotient term.
3. Multiply the whole divisor by that quotient term.
4. Subtract (distribute the minus across all terms).
5. Bring down the next term; repeat until the remainder’s degree is smaller than the divisor’s.
6. Write the result as Quotient + Remainder/Divisor.

Worked Example #1 — Linear Divisor

Divide: 3x^3 − 2x^2 + x − 8 by x − 3

1. 3x^3 ÷ x = 3x^2; multiply → 3x^2(x − 3) = 3x^3 − 9x^2
2. Subtract: (3x^3 − 2x^2) − (3x^3 − 9x^2) = 7x^2; bring down + x → 7x^2 + x
3. 7x^2 ÷ x = 7x; multiply → 7x(x − 3) = 7x^2 − 21x
4. Subtract: (7x^2 + x) − (7x^2 − 21x) = 22x; bring down − 8 → 22x − 8
5. 22x ÷ x = 22; multiply → 22(x − 3) = 22x − 66
6. Subtract: (22x − 8) − (22x − 66) = 58 (remainder)

Result: Quotient = 3x^2 + 7x + 22,   Remainder = 58

Final answer: 3x^2 + 7x + 22 + 58/(x − 3)

Worked Example #2 — Quadratic Divisor

Divide: 2x^4 − 5x^3 + 0x^2 + 6x − 3 by x^2 − x + 2

1. 2x^4 ÷ x^2 = 2x^2; multiply → 2x^2(x^2 − x + 2) = 2x^4 − 2x^3 + 4x^2
2. Subtract: (2x^4 − 5x^3 + 0x^2) − (2x^4 − 2x^3 + 4x^2) = −3x^3 − 4x^2; bring down + 6x → −3x^3 − 4x^2 + 6x
3. −3x^3 ÷ x^2 = −3x; multiply → −3x(x^2 − x + 2) = −3x^3 + 3x^2 − 6x
4. Subtract: (−3x^3 − 4x^2 + 6x) − (−3x^3 + 3x^2 − 6x) = −7x^2 + 12x; bring down − 3 → −7x^2 + 12x − 3
5. −7x^2 ÷ x^2 = −7; multiply → −7(x^2 − x + 2) = −7x^2 + 7x − 14
6. Subtract: (−7x^2 + 12x − 3) − (−7x^2 + 7x − 14) = 5x + 11 (remainder, degree 1 < degree 2)

Result: Quotient = 2x^2 − 3x − 7,   Remainder = 5x + 11

Final answer: 2x^2 − 3x − 7 + (5x + 11)/(x^2 − x + 2)

Synthetic vs. Long Division

• Use synthetic division for linear, monic divisors: x − c.
• Use long division for anything else (higher degree, leading coefficient ≠ 1, or not linear).

Common Mistakes (and Fixes)

• Missing terms: Add 0-coefficients so powers stay aligned.
• Sign errors on subtraction: Distribute the minus sign across every term.
• Stopping early: Keep dividing until remainder’s degree is smaller than the divisor’s.
• Skipping the check: Verify with Dividend = Divisor × Quotient + Remainder.

Practice Problem

Divide: x^3 + 4x^2 − 7x + 10 by x + 2

Answer: x^2 + 2x − 11 + 32/(x + 2)

FAQ: Long Division of Polynomials Solver

How do I know when to stop dividing?

When the remainder’s degree is lower than the divisor’s degree. Then write the result as Quotient + Remainder/Divisor.

Do I need to add zero coefficients for missing powers?

Yes—this keeps the columns aligned and prevents arithmetic mistakes.

Can I use synthetic division instead?

Only if the divisor is linear and monic (x − c). Otherwise, use long division.

How can I check that my answer is correct?

Compute Divisor × Quotient + Remainder and confirm it equals the original dividend. If the divisor is x − c, the remainder should also equal f(c).

What if my remainder is zero?

Then the divisor is a factor of the dividend and the quotient is the exact result—no fraction part needed.