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Polynomial Calculator
Last updated: May 31, 2026
Written by Blake Boege
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
| Step | Quotient term | Subtract (term x divisor) |
|---|---|---|
| 1 | 1x^2 | 1, -2 |
| 2 | 2x | 2, -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.
Quotient
x^2 + 2x - 3
Remainder: 0
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.
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 equation calculator.
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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.
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