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

What Is “Long Division of Polynomials”?

It’s the algebra version of long division. You repeatedly divide → multiply → subtract → bring down, matching like powers at each step. Keep going until the remainder’s degree is lower than the divisor’s degree.

How to Do It (Step by Step)

1. Write terms in descending powers and add any missing powers with coefficient 0 (e.g., x³ + 0x² − 5x + 7).
2. Divide leading terms to get the next quotient term.
3. Multiply the entire divisor by that term.
4. Subtract (distribute the minus to each term!).
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: 4x³ − 9x + 5 by x − 3 (write as 4x³ + 0x² − 9x + 5)

1. 4x³ ÷ x = 4x²; multiply → 4x²(x − 3) = 4x³ − 12x²
2. Subtract: (4x³ + 0x²) − (4x³ − 12x²) = 12x²; bring down − 9x → 12x² − 9x
3. 12x² ÷ x = 12x; multiply → 12x(x − 3) = 12x² − 36x
4. Subtract: (12x² − 9x) − (12x² − 36x) = 27x; bring down + 5 → 27x + 5
5. 27x ÷ x = 27; multiply → 27(x − 3) = 27x − 81
6. Subtract: (27x + 5) − (27x − 81) = 86 (remainder)

Result: Quotient = 4x² + 12x + 27,   Remainder = 86

Final answer: 4x² + 12x + 27 + 86/(x − 3)

Worked Example #2 — Non-Linear Divisor

Divide: x³ + 2x² − x + 1 by x² + 1

1. x³ ÷ x² = x; multiply → x(x² + 1) = x³ + x
2. Subtract: (x³ + 2x² − x) − (x³ + x) = 2x² − 2x; bring down + 1 → 2x² − 2x + 1
3. 2x² ÷ x² = 2; multiply → 2(x² + 1) = 2x² + 2
4. Subtract: (2x² − 2x + 1) − (2x² + 2) = −2x − 1 (remainder; degree 1 < degree 2)

Result: Quotient = x + 2,   Remainder = −2x − 1

Final answer: x + 2 + (−2x − 1)/(x² + 1)

Synthetic vs. Long Division — Which to Use?

• Use synthetic division when dividing by a linear factor of the form x − c.
• Use long division for any other divisor (higher degree, coefficients ≠ 1, or not linear).

Common Mistakes (and Fixes)

• Missing powers. Insert zero-coefficient placeholders to keep columns aligned.
• Sign errors on subtraction. Distribute the minus across all terms before adding.
• Stopping too soon. Continue until remainder’s degree is smaller than the divisor’s.
• Skipping the check. Verify with Dividend = Divisor × Quotient + Remainder.

Practice Problem (with Answer)

Divide: 2x² + 7x + 3 by x + 2

Answer: 2x + 3 − 3/(x + 2) (since the remainder is −3)

FAQ: Calculator for Long Division of Polynomials

Do I have to add missing terms like 0x²?

Yes. Zero-coefficient placeholders keep powers aligned and prevent arithmetic mistakes.

When should I use synthetic division instead?

Use synthetic division only for divisors of the form x − c. For anything else, use long division.

How do I know my result is correct?

Check with Dividend = (Divisor × Quotient) + Remainder. If it reproduces the original dividend, you’re good.

What if my remainder isn’t zero?

That’s common. Express the answer as Quotient + Remainder/Divisor. Make sure the remainder’s degree is lower than the divisor’s.

Can the divisor be quadratic or higher?

Yes. Long division works for any polynomial divisor. Just keep dividing until the remainder’s degree is smaller than the divisor’s degree.