# Synthetic Division Calculator

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

> The synthetic division calculator divides a polynomial by (x − r) with the compact synthetic division tableau, giving the quotient, the remainder and whether (x − r) is a factor.

## Inputs

- Degree of P(x)
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀
- Divide by (x − r): r =

## Results

- Quotient
- Remainder
- Synthetic division tableau
- Factor test

## Formula

```
Bring down a_n; multiply by r and add to the next coefficient; repeat
Remainder = P(r)  (remainder theorem)
```

## Explanation

Synthetic division is a shortcut for dividing by a linear factor. It is faster than long division and doubles as a way to evaluate P(r) – the remainder theorem says the remainder equals P(r).

### How to use the Synthetic Division Calculator

- **Degree of P(x)**
- **a₆ (x⁶)**
- **a₅ (x⁵)**
- **a₄ (x⁴)**
- **a₃ (x³)**
- **a₂ (x²)**
- **a₁ (x)**
- **a₀**
- **Divide by (x − r): r =**

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 is brought down, multiplied by r and added to the next coefficient, and so on. The last number is the remainder; the others are the coefficients of the quotient, one degree lower.

## Worked example

Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): -6; a₁ (x): 11; a₀: -6; Divide by (x − r): r =: 2.

Results — **Quotient = x² − 4x + 3**; Remainder = 0; Synthetic division tableau = 2 │ 1 −6 11 −6
│ 2 −8 6
──┼ ──── ──── ──── ────
│ 1 −4 3 0; Factor test = (x − 2) is a factor.

## Key terms

- **Remainder theorem**: P(r) is the remainder when P(x) is divided by x − r.

## FAQ

**When can I use synthetic division?**
When dividing by a linear binomial x − r. For x + 3 use r = −3.

**What does a remainder of 0 mean?**
r is a root and (x − r) is a factor.

**How do I divide by 2x − 1?**
Divide by x − ½ and then divide the quotient by 2.

**What is the remainder theorem?**
The remainder of P(x) ÷ (x − r) equals P(r).


## Sources

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