# Polynomial Division Calculator

URL: https://speedycalculator.com/calculators/math/polynomial-division/
Last updated: 2026-10-30

> The polynomial division calculator divides P(x) by Q(x) (degrees up to 6) using long division and returns the quotient and remainder, with a check.

## Inputs

- Degree of P(x)
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀
- Degree of Q(x)
- b₆ (x⁶)
- b₅ (x⁵)
- b₄ (x⁴)
- b₃ (x³)
- b₂ (x²)
- b₁ (x)
- b₀

## Results

- Quotient S(x)
- Remainder R(x)
- Dividend P(x)
- Divisor Q(x)
- Check

## Formula

```
P(x) = Q(x) · S(x) + R(x),  deg R < deg Q
```

## Explanation

Polynomial long division works like numerical long division: divide the leading terms, multiply back, subtract and bring down the next term. It is used to simplify rational functions, find oblique asymptotes and factor polynomials once a root is known.

### How to use the Polynomial Division Calculator

- Choose the **degree** of each polynomial – the matching coefficient fields appear.
- Enter the **coefficients** from the highest power down; leave a field at 0 for a missing power.
- Read the result, written with powers of x in descending order.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The leading coefficient of the remainder is repeatedly divided by the leading coefficient of Q(x) until the remainder's degree is less than Q's degree. A zero remainder means Q(x) is a factor of P(x).

## Worked example

Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): -6; a₁ (x): 11; a₀: -6; Degree of Q(x): 1; b₁ (x): 1; b₀: -1.

Results — **Quotient S(x) = x² − 5x + 6**; Remainder R(x) = 0; Dividend P(x) = x³ − 6x² + 11x − 6; Divisor Q(x) = x − 1; Check = P(x) = (x − 1)(x² − 5x + 6).

## Key terms

- **Quotient**: The polynomial S(x) in P = QS + R.
- **Remainder**: The polynomial R(x) left over, of lower degree than Q.

## FAQ

**What does a zero remainder mean?**
The divisor is a factor of the dividend.

**When can I use synthetic division instead?**
When the divisor is linear, x − r.

**What is the degree of the quotient?**
deg P − deg Q.

**How do I write the answer?**
P/Q = S + R/Q.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [Wolfram MathWorld – Long division](https://mathworld.wolfram.com/LongDivision.html)