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Calculator Answers

Polynomial Long Division Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A polynomial long division calculator is an algebraic tool that divides a dividend polynomial by a divisor polynomial of any degree. Applying an algorithm analogous to integer long division, the calculator recursively divides leading terms, multiplies, subtracts, and brings down remaining terms. It handles polynomials with missing degree terms by inserting zero placeholders. The calculator outputs the step-by-step long division work, the final quotient, and any remainder. Algebra and calculus students use this tool to simplify rational functions and identify asymptotes.

Enter coefficients of the dividend and the divisor in descending order. The calculator runs polynomial long division and returns the quotient, the remainder, and the per-step working in a clear table.

See also: Long Division Calculator, Synthetic Division Calculator, and Polynomial Calculator.

Quick Answer

Perform polynomial long division. Enter the dividend and divisor polynomials to find the quotient and remainder with complete step-by-step calculations.

Coefficients

Enter coefficients in descending order, including zeros for missing powers. For example, x^3 - 7x + 6 is 1, 0, -7, 6.

e.g. 1, 0, -7, 6

e.g. 1, -2 (for x - 2)

Step trace

StepQuotient termSubtract (term x divisor)
11x^21, -2
22x2, -4
3-3-3, 6

Each step picks a quotient term so the leading coefficient of the working remainder cancels, then subtracts the product. The process stops once the working polynomial has degree less than the divisor.

Result

Quotient

x^2 + 2x - 3

Remainder: 0

Dividendx^3 - 7x + 6
Divisorx - 2
Quotient coefs1, 2, -3
Remainder coefs0

Dividend = Divisor x Quotient + Remainder, with degree of Remainder less than degree of Divisor. The full identity holds term by term for any pair of polynomials.

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Examples

(x^3 - 7x + 6) / (x - 2)

Quotient x^2 + 2x - 3, Remainder 0

(2x^3 + 3x^2 - x + 1) / (x^2 - 1)

Quotient 2x + 3, Remainder x + 4

(x^4 + 1) / (x^2 + 1)

Quotient x^2 - 1, Remainder 2

(x^2 + 3x + 2) / (x^3 + 1)

Quotient 0, Remainder x^2 + 3x + 2

How it works

At each step the algorithm picks the quotient term that cancels the leading coefficient of the working dividend, multiplies it by the divisor, and subtracts. The new working dividend has a lower degree, and the process repeats.

Identity · dividend = divisor x quotient + remainder

Step rule · qᵢ = lead(working) / lead(divisor)

Stops when the working polynomial's degree is less than the divisor's.

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Long Division CalculatorDivide two whole numbers and see the quotient, remainder, decimal result, and mixed-number form. Handles decimal inputs as a fall-through decimal answer.Synthetic Division CalculatorDivide a polynomial by x - c using synthetic division. Enter the polynomial coefficients and the divisor value c; the calculator returns the quotient coefficients, the remainder, and the full synthetic table step by step.Polynomial CalculatorPerform polynomial long division, find quotient and remainder, and work with polynomial expressions step by step.Taylor Polynomial CalculatorCalculate the nth-degree Taylor polynomial (or Maclaurin series) of any differentiable function about a center point with step-by-step derivatives.

Frequently asked questions

Polynomial long division is the standard algorithm for dividing one polynomial by another. It produces a quotient polynomial and a remainder, just like integer long division. The identity is dividend = divisor x quotient + remainder, with the remainder of strictly lower degree than the divisor.

List the coefficients of each polynomial in descending order of power, separated by commas. Include zeros for any missing terms. The polynomial 2x^3 - 5x + 1 has coefficients 2, 0, -5, 1.

Use synthetic division when the divisor is exactly x - c (linear with leading coefficient 1) for the fastest path. Use polynomial long division for any other divisor, including 2x - 5, x^2 + 1, x^3 + x - 1, and so on.

Then division stops immediately: the quotient is 0 and the remainder is the dividend itself. The calculator reports both.

That is fine. Polynomial long division works over the rationals (and over the reals); coefficients may be any real number. The calculator shows up to 6 decimal places and integers when the value is exact.

If the remainder had degree at least equal to the divisor, you could keep dividing one more step. The algorithm stops exactly when no further reduction is possible, which is the definition of polynomial remainder.