Polynomial Division

🔢 Polynomial Division Calculator – Long Division & Synthetic Division

The Polynomial Division Calculator divides one polynomial by another and returns the quotient, remainder, and the full rational expression in a single click. It supports both polynomial long division (for any divisor degree) and synthetic division (for linear divisors), automatically choosing the most efficient method based on your input.

What Is Polynomial Division?

Polynomial division is the process of dividing one polynomial — the dividend — by another — the divisor — to produce a quotient and a remainder. The result satisfies the fundamental division identity:

Dividend = Divisor × Quotient + Remainder

The remainder is always a polynomial of lower degree than the divisor. When the remainder is zero, the divisor is an exact factor of the dividend — a key result in the Factor Theorem and root-finding.

Polynomial Long Division

Polynomial long division mirrors the familiar digit-by-digit long division algorithm for integers. It works for any divisor of degree 1 or higher.

Example: Divide 2x³ + 5x² − x − 6 by x² + 2x + 3:

Synthetic Division

Synthetic division is a streamlined shortcut for dividing by a linear polynomial of the form x − a or ax + b. Instead of writing out full polynomial expressions at every step, it works with a compact table of coefficients.

Example: Divide 2x³ + 3x² − 5x + 6 by x − 2 (root = 2):

  2 |  2   3  -5   6
      |      4  14  18
      +-----------+--
         2   7   9  24

Quotient: 2x² + 7x + 9   Remainder: 24

The calculator automatically selects synthetic division when the divisor is linear, and displays the full synthetic table alongside the result.

How to Use This Tool

1. Enter your polynomials. Type them as algebraic expressions (e.g. x^3-6x^2+11x-6) or as comma-separated coefficient lists (e.g. 1,-6,11,-6).
2. Choose a method. Auto selects synthetic division for linear divisors and long division otherwise. Override with Long Division or Synthetic if preferred.
3. Review the results. The quotient, remainder, and rational form are displayed instantly. Enable Show Steps for full long-division working or Synthetic Table for the coefficient grid.
4. Verify the answer. Toggle Verify Result to see the reconstructed identity Divisor × Quotient + Remainder = Dividend.

Missing Terms Handling

When a polynomial has skipped powers — for example x⁴ − 5x + 1 which is missing the x³ and x² terms — the calculator automatically inserts zero-coefficient placeholders. This keeps both long-division and synthetic-division layouts correctly aligned without any extra input from you.

The Remainder Theorem

When dividing a polynomial P(x) by x − a, the remainder equals P(a). This means you can evaluate any polynomial at a specific point simply by looking at the remainder from synthetic division — no substitution required.

Common Applications

• Simplifying rational expressions — reduce P(x)/D(x) to a mixed expression quotient + remainder/divisor.
• Finding polynomial roots — use the Remainder and Factor Theorems to test and confirm roots.
• Partial fraction decomposition — long division is the first step when the degree of the numerator ≥ degree of the denominator.
• Polynomial factorization — after finding one root via synthetic division, the quotient is a lower-degree polynomial that can be factored further.
• Calculus — simplify integrands before integration.

Input Format Tips

• Use ^ for exponents: x^3 means x³.
• Spaces are optional: 2x^3 + 5x^2 - x - 6 and 2x^3+5x^2-x-6 are both accepted.
• You can omit coefficients of 1: x^2 means 1x².
• In coefficient mode, always list from the highest degree down to the constant: the polynomial 3x² − x + 7 becomes 3,-1,7.
• Single-letter variable names only (a–z). Default is x.