# What do I get dividing polynomials?

Divides one polynomial by another by long division in exact fractions, showing every step, the quotient and the remainder, and synthetic division when the divisor is x − c.

- Page: https://www.acalculator.org/math/polynomial-division-calculator
- JSON spec: https://www.acalculator.org/math/polynomial-division-calculator.json
- Version: 59862e594dd4

## Default answer

Example with the default inputs (Dividend 6x^3 + 11x^2 - 31x + 15, Divisor 3x - 2): The quotient is 2x^2 + 5x - 7 with remainder 1.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| p | Dividend | The polynomial to divide, in one letter, for example 6x^3 + 11x^2 - 31x + 15. |
| q | Divisor | The polynomial to divide by, in the same letter, not 0. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Quotient | The quotient q(x) of the division. |
| remainder | Remainder | What is left: 0, or a polynomial of lower degree than the divisor. |
| check | Division algorithm | The dividend written as divisor × quotient + remainder. |
| synthetic | Synthetic division | For a divisor x − c: c, then the bottom row of synthetic division (quotient coefficients, remainder). |
| steps | Steps | Each step of the long division. |

## Method

Long division: divide the leading terms, multiply the divisor by that term, subtract, and repeat until the remainder’s degree is below the divisor’s.

## Assumptions

- Coefficients are exact fractions: 0.1 is 1/10.
- One letter, whole powers up to 50.

## Worked examples

1. p = 6x^3 + 11x^2 - 31x + 15, q = 3x - 2 gives result = 2x^2 + 5x - 7, remainder = 1, check = 6x^3 + 11x^2 - 31x + 15 = (3x - 2) × (2x^2 + 5x - 7) + 1. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (6x³ + 11x² − 31x + 15) ÷ (3x − 2) = 2x² + 5x − 7 remainder 1.
2. p = 5x^2 + 3x - 2, q = x + 1 gives result = 5x - 2, remainder = 0, synthetic = c = -1; bottom row: 5, -2, 0. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02): (5x² + 3x − 2) ÷ (x + 1) = 5x − 2, remainder 0.
3. p = x^3 - 1, q = 2x^2 + 1 gives result = x/2, remainder = -x/2 - 1, check = x^3 - 1 = (2x^2 + 1) × x/2 - x/2 - 1. Source: OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials: the Division Algorithm f(x) = d(x)q(x) + r(x) (https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials, retrieved 2026-10-02).

## FAQ

### How do I divide polynomials by long division?

Divide the leading term of the dividend by the leading term of the divisor, multiply the whole divisor by the result, subtract, and repeat with what is left. Stop when what is left has a lower degree than the divisor: that is the remainder.

### What is the division algorithm for polynomials?

For a dividend f(x) and a divisor d(x) that is not 0, there are unique polynomials q(x) and r(x) with f(x) = d(x) q(x) + r(x), where r(x) is 0 or has a lower degree than d(x). The calculator shows this line for your answer.

### What is synthetic division?

A shortcut for dividing by x − c. Write the dividend’s coefficients, bring down the first, then repeatedly multiply by c and add to the next coefficient. The last number is the remainder; the others are the quotient’s coefficients.

### What does a remainder of 0 mean?

The divisor is a factor of the dividend. (5x² + 3x − 2) ÷ (x + 1) leaves 0, so 5x² + 3x − 2 = (x + 1)(5x − 2).

### How does the remainder theorem help?

When you divide by x − c, the remainder equals the dividend’s value at x = c. Dividing −9x⁴ + 10x³ + 7x² − 6 by x − 1 leaves 2, and −9 + 10 + 7 − 6 = 2.

### Can the coefficients be fractions or decimals?

Yes. Every coefficient is an exact fraction (0.5 is 1/2), so the quotient and remainder are exact. x³ − 1 divided by 2x² + 1 gives x/2 with remainder −x/2 − 1.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §5.4 Dividing Polynomials (the Division Algorithm; (5x² + 3x − 2) ÷ (x + 1) = 5x − 2; (6x³ + 11x² − 31x + 15) ÷ (3x − 2) = 2x² + 5x − 7 remainder 1), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-4-dividing-polynomials