# Synthetic Division Calculator

Divide polynomials using synthetic division. Enter coefficients and divisor to find the quotient and remainder. Free step-by-step polynomial division calculator.

## What this calculates

Divide any polynomial up to degree 5 by a linear factor (x - c) using synthetic division. Enter the coefficients and the divisor value to get the quotient, remainder, and factor check instantly.

## Inputs

- **Polynomial Degree** — options: Degree 2 (ax² + bx + c), Degree 3 (ax³ + bx² + cx + d), Degree 4 (ax⁴ + bx³ + cx² + dx + e), Degree 5 (ax⁵ + ... + f) — Select the degree of the polynomial to divide.
- **x⁵ coefficient**
- **x⁴ coefficient**
- **x³ coefficient**
- **x² coefficient**
- **x coefficient**
- **Constant term**
- **Divisor c (dividing by x - c)** — The value c in the divisor (x - c). For (x + 3), enter -3.

## Outputs

- **Original Polynomial** — formatted as text — The polynomial being divided.
- **Divisor** — formatted as text — The divisor expression (x - c).
- **Quotient** — formatted as text — The resulting polynomial after division.
- **Remainder** — The remainder of the division.
- **Factor Check** — formatted as text — Whether (x - c) is a factor of the polynomial.

## Details

Synthetic division is a shortcut for dividing a polynomial by a linear expression of the form (x - c). It is faster than long division and uses only the coefficients.

**How Synthetic Division Works:**

1. Write down the coefficients of the polynomial in order (include zeros for missing terms).
2. Write the value c to the left.
3. Bring down the first coefficient.
4. Multiply it by c and add to the next coefficient.
5. Repeat until you reach the last coefficient.
6. The final number is the remainder. All preceding numbers are the quotient coefficients.

**Worked Example:**

Divide x³ - 6x² + 11x - 6 by (x - 2):

Coefficients: 1, -6, 11, -6. Divisor c = 2.

- Bring down 1.
- 1 x 2 = 2, add to -6 to get -4.
- -4 x 2 = -8, add to 11 to get 3.
- 3 x 2 = 6, add to -6 to get 0.

Quotient: x² - 4x + 3. Remainder: 0. Since the remainder is 0, (x - 2) is a factor.

**Remainder Theorem:** The remainder when dividing a polynomial f(x) by (x - c) equals f(c). If f(c) = 0, then (x - c) is a factor of f(x).

## Frequently Asked Questions

**Q: What is synthetic division?**

A: Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x - c). Instead of the full long division process, you work only with the coefficients, making it faster and less error-prone. It works for any polynomial degree divided by a linear term.

**Q: When can I use synthetic division?**

A: Synthetic division only works when the divisor is a linear expression of the form (x - c), where the coefficient of x is 1. For divisors like (2x - 3), you would need to factor out the 2 first or use polynomial long division instead.

**Q: What is the Remainder Theorem?**

A: The Remainder Theorem states that when you divide polynomial f(x) by (x - c), the remainder equals f(c). So evaluating f(2) for f(x) = x³ - 6x² + 11x - 6 gives f(2) = 8 - 24 + 22 - 6 = 0. Since the remainder is 0, (x - 2) is a factor.

**Q: How do I handle missing terms in the polynomial?**

A: If a power of x is missing from the polynomial, use 0 as its coefficient. For example, x³ + 5x - 2 is missing the x² term, so its coefficients are 1, 0, 5, -2. Forgetting to include the zero is the most common mistake in synthetic division.

**Q: What does it mean when the remainder is zero?**

A: A remainder of zero means (x - c) divides the polynomial evenly, so (x - c) is a factor. The quotient then gives the other factor. For example, if x³ - 6x² + 11x - 6 divided by (x - 1) gives quotient x² - 5x + 6 with remainder 0, then the polynomial factors as (x - 1)(x² - 5x + 6) = (x - 1)(x - 2)(x - 3).

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Source: https://vastcalc.com/calculators/math/synthetic-division
Category: Math
Last updated: 2026-04-08