Polynomial Long Division Calculator

Perform polynomial long division with step‑by‑step guidance. Get quotient, remainder, and detailed intermediate steps. Now with fraction coefficients and enhanced visualisation.

Standard form: Divide P(x) by Q(x). Example: (x³ - 2x² + 3x - 4) ÷ (x - 2)

Use ^ for exponent, * for multiplication (optional). Missing terms are automatically filled with zero coefficients.

(x³-2x²+3x-4)/(x-2)
(x⁴-1)/(x-1)
(2x³+3x²-4x+5)/(x+1)
(x⁵-1)/(x-1)
(6x⁴+5x³-3x²+2x+1)/(2x²+1)

Understanding Polynomial Long Division

Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. It is analogous to arithmetic long division and is used to simplify rational expressions, factor polynomials, and solve equations.

General Form: Given polynomials P(x) (dividend) and D(x) (divisor), we find Q(x) (quotient) and R(x) (remainder) such that:

P(x) = Q(x) · D(x) + R(x)

where degree(R) < degree(D) (or R = 0).

Why It Works (Analogy to Integer Division)

Just as 17 ÷ 5 gives quotient 3 and remainder 2 (because 17 = 3·5 + 2), polynomial division repeatedly subtracts multiples of the divisor to reduce the degree. The quotient terms are found by dividing leading terms, similar to how you divide digits in arithmetic.

Step‑by‑Step Algorithm

1

Arrange: Write both polynomials in descending powers of x, including missing terms with coefficient 0. (Our calculator automatically inserts zero coefficients.)

2

Divide leading terms: Divide the leading term of the dividend by the leading term of the divisor – this gives the first term of the quotient.

3

Multiply & Subtract: Multiply the divisor by that quotient term, subtract from the current remainder, and bring down the next term.

4

Repeat: Continue until the degree of the remainder is less than the degree of the divisor.

Key Theorems

  • Remainder Theorem: When dividing by (x – c), the remainder is P(c).
  • Factor Theorem: (x – c) is a factor of P(x) iff P(c) = 0.
  • Rational Root Theorem: Helps find possible linear factors with integer coefficients.

Applications

  • Factoring polynomials: If remainder = 0, the divisor is a factor.
  • Simplifying rational expressions: Used in integration (partial fractions) and asymptote analysis.
  • Root finding: Synthetic division (a shortcut for linear divisors) helps evaluate polynomials quickly.
  • Signal processing: Polynomial division appears in digital filter design (z‑transforms).

Historical Note

The method of polynomial long division has been known since ancient times, but was formalized in the 16th and 17th centuries alongside the development of symbolic algebra. It is essentially the same as integer long division, adapted for polynomials.

Special Cases

  • Divisor degree > dividend degree: Quotient = 0, remainder = dividend.
  • Leading coefficient not 1: Quotient terms may be fractions – our calculator displays them as simplified fractions.
  • Missing terms: Always insert 0 coefficients to align columns correctly.

Calculator enhancements (v2.1):

  • ✓ Fraction coefficients for exact results
  • ✓ Step formatting with superscripts and clear alignment
  • ✓ Automatic detection and display of missing‑term zero coefficients
  • ✓ Interactive legend on graph (click to toggle curves)
  • ✓ Handles any polynomial (including non‑linear divisors)

Frequently Asked Questions

The calculator automatically inserts terms with coefficient 0. For example, x³ + 1 is interpreted as x³ + 0x² + 0x + 1. The steps will show these zero coefficients to help alignment.

Yes, the divisor can be any polynomial (linear, quadratic, etc.). The algorithm works for any degree, as long as the divisor is not zero.

The calculator uses nerdamer's symbolic engine to convert decimals to simplified fractions (e.g., 2.5 → 5/2). This gives exact results.

The graph plots the original dividend P(x) and the product Q(x)·divisor. When the remainder is zero, the two curves coincide. You can click the legend to show/hide each curve.