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Polynomial Division Calculator

Divide one polynomial by another using long division: (a₃x³ + a₂x² + a₁x + a₀) ÷ (b₁x + b₀).

Reviewed by Christopher FloiedPublished Updated

This free online polynomial division calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

Quotient x² coefficient

0

Quotient x coefficient

1

Quotient constant

0

Remainder

0

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Polynomial Division Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Polynomial Division Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Polynomial Division Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Polynomial Division Calculator

The Polynomial Division Calculator performs long division of polynomials, dividing a polynomial of up to degree 3 by a linear polynomial. Polynomial long division works analogously to integer long division: divide the leading terms, multiply back, subtract, bring down the next term, and repeat. This algorithm is fundamental in algebra for simplifying rational expressions, performing partial fraction decomposition, finding oblique asymptotes of rational functions, and rewriting improper fractions of polynomials. Unlike synthetic division (which only handles divisors of the form x - c), polynomial long division works with any polynomial divisor. The result expresses the dividend P(x) as D(x)·Q(x) + R(x), where the degree of the remainder R(x) is strictly less than the degree of the divisor D(x).

The Math Behind It

Polynomial long division generalizes the familiar algorithm for dividing integers to the ring of polynomials. Given a dividend P(x) of degree n and divisor D(x) of degree m (with m ≤ n), the algorithm produces a quotient Q(x) of degree n - m and a remainder R(x) of degree less than m. The process is iterative: at each step, divide the leading term of the current remainder by the leading term of the divisor to get the next term of the quotient, multiply the entire divisor by this term, subtract from the current remainder, and repeat until the degree of the remainder is less than the degree of the divisor. This algorithm is guaranteed to terminate because the degree decreases at each step. The Division Algorithm for polynomials states that for any polynomials P(x) and D(x) with D(x) ≠ 0, there exist unique polynomials Q(x) and R(x) such that P(x) = D(x)·Q(x) + R(x) with deg(R) < deg(D). This is analogous to the division algorithm for integers. Polynomial division is essential for finding oblique (slant) asymptotes: if deg(P) = deg(D) + 1, then Q(x) is linear and y = Q(x) is the oblique asymptote of P(x)/D(x).

Formula Reference

Division Algorithm

P(x) = D(x)·Q(x) + R(x)

Variables: P = dividend, D = divisor, Q = quotient, R = remainder with deg(R) < deg(D)

Worked Examples

Example 1: Linear divisor

Divide x³ + 2x² - 5x + 3 by (x + 3)

Step 1:x³ ÷ x = x². Multiply: x²(x+3) = x³ + 3x². Subtract: -x² - 5x + 3
Step 2:-x² ÷ x = -x. Multiply: -x(x+3) = -x² - 3x. Subtract: -2x + 3
Step 3:-2x ÷ x = -2. Multiply: -2(x+3) = -2x - 6. Subtract: 9
Step 4:Quotient: x² - x - 2, Remainder: 9

x³ + 2x² - 5x + 3 = (x+3)(x² - x - 2) + 9

Common Mistakes & Tips

  • !Misaligning terms of different degrees during subtraction
  • !Forgetting to include zero coefficients for missing powers of x
  • !Sign errors when subtracting the product of divisor and quotient term
  • !Stopping too early before the remainder has lower degree than the divisor

Related Concepts

Synthetic Division

A streamlined method when the divisor is (x - c)

Partial Fraction Decomposition

Uses polynomial division when the degree of the numerator exceeds the denominator

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Partial Fraction Decomposition Calculator

Synthetic Division Calculator

Frequently Asked Questions

When should I use polynomial long division vs synthetic division?

Use synthetic division when the divisor is a simple linear factor (x - c). Use long division when the divisor has a leading coefficient other than 1 or has degree 2 or higher.

What are oblique asymptotes?

When a rational function P(x)/D(x) has numerator degree exactly one more than denominator degree, polynomial division gives Q(x) = mx + b plus a remainder over D(x). As x → ±∞, the remainder vanishes, so y = mx + b is the oblique (slant) asymptote.

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