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Calculator Answers

Polynomial Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A polynomial calculator is an algebraic utility designed to divide one polynomial expression by another. It implements the polynomial long division algorithm, which divides the terms of highest degree in sequence to compute a quotient polynomial and a remainder polynomial. This tool is widely used in high school and college algebra to simplify rational expressions, find polynomial roots, and trace mathematical steps.

Divide polynomials and solve algebraic expressions. Enter the coefficients of the dividend and divisor to compute the quotient, remainder, and step-by-step division process.

Quick Answer

Divide one polynomial by another using polynomial long division. Enter the coefficients of the dividend and divisor to find the quotient and remainder.

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

Quotient x + 3, Remainder 0

(2x^3 - 4x^2 + x) / (x^2 + 1)

Quotient 2x - 4, Remainder -x + 4

How it works

Polynomial Operations & Division

When dividing a dividend polynomial P(x) by a divisor polynomial D(x), we find the unique quotient Q(x) and remainder R(x) such that:

P(x) = D(x) × Q(x) + R(x)

Where the degree of R(x) is strictly less than the degree of D(x).

For specialized polynomial divisions, you can access our dedicated polynomial long division calculator, synthetic division calculator, or solve quadratic expressions directly using the quadratic formula calculator.

Learn the concept

These guides explain the idea this tool computes, with worked examples and the distinctions readers most often trip over.

What Is a Term in Math? Monomial, binomial, trinomial, and what counts as one term.What Is a Constant in Math? The constant term, and what it does to a graph.

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Frequently asked questions

A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables (e.g., 3x² + 2x − 5).

Polynomial long division is an algorithm for dividing a polynomial by another polynomial of equal or lower degree. It breaks the division down into steps to find a quotient polynomial and a remainder polynomial, satisfying: Dividend = Divisor × Quotient + Remainder.

Synthetic division is a shorthand method for dividing polynomials when the divisor is a linear expression in the form (x − c). For divisors of higher degree (like quadratic or cubic divisors), polynomial long division must be used.

The degree of a polynomial is the highest exponent of the variable in the expression. For example, 4x³ − 2x + 7 has a degree of 3. In division, the degree of the remainder is always strictly less than the degree of the divisor.