# Polynomial Long Division Calculator

> Divide one polynomial by another and get the exact quotient and remainder, with each long-division step and fractional coefficients kept exact.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/polynomial-long-division-calculator

Enter the dividend and the divisor. Each step divides the leading terms, multiplies back and subtracts, and fractions stay exact (1/2, not 0.5).

Also called: Polynomial division calculator, Algebraic long division calculator.

## Inputs
- Dividend P(x) (default x^3 - 2x^2 - 4)
- Divisor D(x) (default x - 3)

## Results
- P(x) ÷ D(x)
- Quotient Q(x)
- Remainder R(x)
- Does D(x) divide P(x)?

## Formula

```
P(x) = D(x)·Q(x) + R(x), with deg R < deg D;  each step: next quotient term = (leading term of remainder) ÷ (leading term of D)
```

Where:
- P: the dividend
- D: the divisor
- Q: the quotient
- R: the remainder

## Worked example
1. x³ ÷ x = x²; subtract x²(x − 3) = x³ − 3x² to leave x² − 4.
2. x² ÷ x = x; subtract x(x − 3) = x² − 3x to leave 3x − 4.
3. 3x ÷ x = 3; subtract 3(x − 3) = 3x − 9 to leave 5, whose degree is below the divisor’s.

**Result:** (x³ − 2x² − 4) ÷ (x − 3) = x² + x + 3 with remainder 5.

## What the result means

A zero remainder means the divisor is a factor of the dividend. The remainder always has a lower degree than the divisor.

## Assumptions and limits
- Coefficients are rational numbers (decimals and fractions are kept exact).
- One variable (x) only; degree up to 50.

## Questions

### When should I use synthetic division instead?

Synthetic division is a shortcut that only works for divisors of the form x − r. Long division works for any divisor, such as x² + 1 or 2x + 3.

### Do I need to include missing powers?

Not when you type an expression; missing powers are treated as zero. When you list coefficients, include a 0 for each missing power.

## Related calculators
- [Synthetic Division Calculator](https://calculators.geekmidst.com/calculators/polynomial-synthetic-division-calculator)
- [Polynomial Roots Calculator](https://calculators.geekmidst.com/calculators/polynomial-roots-calculator)
- [Long Division Calculator](https://calculators.geekmidst.com/calculators/long-division-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/polynomial-long-division-calculator