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Synthetic Division Calculator

Divide a polynomial by a linear factor (x - c) using synthetic division to find the quotient and remainder.

Reviewed by Christopher FloiedPublished Updated

This free online synthetic division calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient of x³

Coefficient of x²

Coefficient of x

If dividing by (x - 3), enter 3. If by (x + 2), enter -2.

Results

Quotient x² coefficient

0

Quotient x coefficient

1

Quotient constant

1

Remainder

1

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Synthetic Division Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Synthetic Division Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Synthetic Division Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Synthetic Division Calculator

The Synthetic Division Calculator performs polynomial division by a linear factor (x - c) using a streamlined tabular algorithm. Synthetic division is a shortcut for polynomial long division that works exclusively with the coefficients, making the process faster and less error-prone. It simultaneously finds the quotient polynomial and the remainder, which by the Remainder Theorem equals P(c). This makes synthetic division a powerful tool for evaluating polynomials, testing potential roots using the Rational Root Theorem, and factoring higher-degree polynomials step by step. The method was developed to simplify the tedious process of polynomial long division, and it is especially efficient when dividing by linear factors. Engineers, scientists, and students use synthetic division extensively when analyzing polynomial functions and transfer functions.

The Math Behind It

Synthetic division leverages the structure of polynomial long division when the divisor is linear (x - c). The algorithm processes coefficients from left to right: bring down the leading coefficient, multiply by c, add to the next coefficient, multiply by c, add to the next, and so on. The last number obtained is the remainder R = P(c) (Remainder Theorem). The preceding numbers are the coefficients of the quotient polynomial, which has degree one less than the original. If the remainder is zero, then (x - c) is a factor of P(x), and c is a root. This connects to the Factor Theorem: (x - c) is a factor of P(x) if and only if P(c) = 0. Synthetic division is the practical implementation of Horner's method for polynomial evaluation, which rewrites P(x) = aₙxⁿ + ... + a₁x + a₀ as P(x) = (...((aₙx + aₙ₋₁)x + aₙ₋₂)x + ... + a₁)x + a₀. This nested form requires only n multiplications and n additions, making it the most efficient way to evaluate a polynomial. Synthetic division can be applied repeatedly to find all roots of a polynomial: once a root c is found, divide by (x - c), then apply synthetic division to the quotient to find the next root.

Formula Reference

Synthetic Division

P(x) = (x - c)·Q(x) + R

Variables: P(x) = original polynomial, Q(x) = quotient, R = remainder = P(c)

Remainder Theorem

R = P(c)

Variables: The remainder when dividing by (x - c) equals P(c)

Worked Examples

Example 1: Division with zero remainder

Divide x³ - 6x² + 11x - 6 by (x - 1)

Step 1:Set up: coefficients [1, -6, 11, -6], c = 1
Step 2:Bring down 1
Step 3:1 × 1 = 1, add to -6: -5
Step 4:1 × -5 = -5, add to 11: 6
Step 5:1 × 6 = 6, add to -6: 0
Step 6:Quotient: x² - 5x + 6, Remainder: 0

x³ - 6x² + 11x - 6 = (x - 1)(x² - 5x + 6)

Example 2: Division with nonzero remainder

Divide 2x³ + 3x² - x + 5 by (x + 2)

Step 1:c = -2 (since divisor is x + 2 = x - (-2))
Step 2:Coefficients: [2, 3, -1, 5]
Step 3:Bring down 2
Step 4:-2 × 2 = -4, add to 3: -1
Step 5:-2 × -1 = 2, add to -1: 1
Step 6:-2 × 1 = -2, add to 5: 3
Step 7:Quotient: 2x² - x + 1, Remainder: 3

2x³ + 3x² - x + 5 = (x + 2)(2x² - x + 1) + 3

Common Mistakes & Tips

  • !Using the wrong sign for c: when dividing by (x + 2), c = -2, not 2
  • !Forgetting to include zero coefficients for missing terms (e.g., x³ + 1 needs coefficients [1, 0, 0, 1])
  • !Confusing the last result number with a quotient coefficient instead of the remainder
  • !Trying to use synthetic division with non-linear divisors (it only works for x - c)

Related Concepts

Polynomial Division

Long division method that works with any polynomial divisor

Rational Zeros

Generate potential rational roots to test with synthetic division

Cubic Equation

Factor cubics by finding a root via synthetic division

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Cubic Equation Calculator

Polynomial Division Calculator

Rational Zeros Calculator

Frequently Asked Questions

Can synthetic division be used with divisors other than (x - c)?

Standard synthetic division only works for linear divisors of the form (x - c). For divisors like (2x - 3), you can divide by (x - 3/2) and adjust. For higher-degree divisors, use polynomial long division.

What is the connection between synthetic division and the Remainder Theorem?

Synthetic division simultaneously computes the quotient Q(x) and the remainder R when dividing P(x) by (x - c). The Remainder Theorem guarantees that R = P(c), so synthetic division also evaluates the polynomial at x = c.

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