Math July 13, 2026 · 8 Min Read

Polynomial Division Calculator – Guide & Formulas

Divide polynomials using long division or synthetic division. Enter dividend and divisor to see quotient, remainder, and step-by-step division work for any polynomial.

Divide any two polynomials instantly with our free calculator. Enter dividend and divisor to see quotient, remainder, and step-by-step long division or synthetic division work.

Key Takeaway

Use the free Polynomial Division Calculator to divide polynomials using long division or synthetic division. enter dividend and divisor to see quotient, remainder, and step-by-step division work for any polynomial. Get instant results with step-by-step explanations.

How to Use the Polynomial Division Calculator

  1. Enter the dividend polynomial (the polynomial being divided).
  2. Enter the divisor polynomial (the polynomial dividing by).
  3. Choose between long division and synthetic division (synthetic requires linear divisor).
  4. Review the quotient, remainder, and step-by-step division process.

The Formula

Dividend = Divisor × Quotient + Remainder. The remainder R(x) has degree less than the divisor D(x). If R(x) = 0, the division is exact.

Variable Definitions

  • P(x): The dividend (polynomial being divided)
  • D(x): The divisor (polynomial dividing by)
  • Q(x): The quotient (result of division)
  • R(x): The remainder (what is left over)
  • Synthetic division: A shortcut method when dividing by a linear factor (x - r)

Dividing (2x³ + 3x² - x + 5) by (x + 1)

Perform polynomial long division.

  1. Step 1: Divide first terms: 2x³ ÷ x = 2x². Multiply: 2x²(x + 1) = 2x³ + 2x².
  2. Step 2: Subtract: (2x³ + 3x²) - (2x³ + 2x²) = x². Bring down -x.
  3. Step 3: Divide: x² ÷ x = x. Multiply: x(x + 1) = x² + x. Subtract: (x² - x) - (x² + x) = -2x.
  4. Step 4: Bring down +5. Divide: -2x ÷ x = -2. Multiply: -2(x + 1) = -2x - 2. Subtract: (-2x + 5) - (-2x - 2) = 7.
  5. Step 5: Quotient: 2x² + x - 2. Remainder: 7. Result: 2x² + x - 2 + 7/(x + 1).

Frequently Asked Questions

What is polynomial long division?

Polynomial long division is analogous to numerical long division. You repeatedly divide the leading term of the current remainder by the leading term of the divisor, multiply, subtract, and bring down the next term.

When should I use synthetic division?

Use synthetic division when dividing by a linear factor (x - r). It is faster and involves only the coefficients, avoiding writing out variables. Not suitable for higher-degree divisors.

What does a zero remainder mean?

A zero remainder means the divisor is a factor of the dividend. By the Factor Theorem, if P(r) = 0, then (x - r) is a factor of P(x).

How do I handle missing terms?

Insert terms with coefficient 0 for any missing powers. For example, x³ + 1 should be written as x³ + 0x² + 0x + 1 to maintain proper alignment.

Can I divide by a polynomial of higher degree?

If the divisor has higher degree than the dividend, the quotient is 0 and the remainder is the dividend itself. The division algorithm still applies.

What is the Remainder Theorem?

The Remainder Theorem states that when dividing P(x) by (x - r), the remainder equals P(r). This provides a quick way to evaluate polynomials.

How do I verify my division?

Multiply the quotient by the divisor and add the remainder. The result should equal the original dividend: D(x) × Q(x) + R(x) = P(x).

What are applications of polynomial division?

Applications include: finding polynomial roots, simplifying rational expressions, partial fraction decomposition, checking factors, and analyzing polynomial behavior.