What is Synthetic Division? A Complete Definition

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form (x - c). Instead of writing out the entire long division, you only work with the coefficients. This makes the process faster and less messy. If you have a polynomial like 2x³ + 5x² - 3x + 7 and you want to divide it by (x - 2), synthetic division gives you the quotient and remainder in a few simple steps.

Origin and History

Synthetic division was developed by the Italian mathematician Paolo Ruffini in the early 1800s. He wanted a simpler way to evaluate polynomials and perform division when the divisor is a linear binomial. His method is sometimes called Ruffini's rule. Synthetic division became popular because it saves time and reduces errors compared to polynomial long division — especially when you need to divide many polynomials or when you are learning about roots and factors.

Why It Matters

Synthetic division is not just a classroom trick. It is used in algebra to quickly find the quotient and remainder when dividing by a linear factor. The remainder is especially important because of the Remainder Theorem: if you divide a polynomial P(x) by (x - c), the remainder equals P(c). That means you can evaluate a polynomial at a point by doing synthetic division instead of plugging in a number — faster and less error-prone. Synthetic division also helps in factorizing polynomials: if the remainder is zero, then (x - c) is a factor. This concept is central to solving polynomial equations and graphing. To see the exact formula and why it works, check out our Synthetic Division Formula page.

How It’s Used

Synthetic division is a step-by-step process that uses only the coefficients of the polynomial. For a detailed walkthrough, visit our How to Do Synthetic Division guide. Here is a quick example to show how it works:

Example: Divide 2x³ - 5x² + 3x - 7 by (x - 2).

1. Write the coefficients in descending order: 2, -5, 3, -7
2. Write the value of c (from x - c) to the left: here c = 2
3. Bring down the first coefficient (2) to the bottom row.
4. Multiply that coefficient by c (2 × 2 = 4) and write it under the next coefficient (-5).
5. Add: -5 + 4 = -1. Write -1 below.
6. Multiply -1 by c (2) = -2, write under 3. Add: 3 + (-2) = 1. Write 1.
7. Multiply 1 by c = 2, write under -7. Add: -7 + 2 = -5. This is the remainder.

The bottom row gives the coefficients of the quotient, starting with one degree less than the original: 2, -1, 1 means quotient = 2x² - x + 1, and remainder = -5. So 2x³ - 5x² + 3x - 7 = (x - 2)(2x² - x + 1) - 5. To see more about what the quotient and remainder mean, read our Interpreting Synthetic Division Results page.

Common Misconceptions

• It only works for divisors of form (x - c). If you have (x + 3), you must treat it as (x - (-3)), so c = -3. The sign matters!
• You must include zero coefficients. If a term is missing (e.g., no x² term), write a 0 in its place. Otherwise the division will be wrong.
• Synthetic division can only be used for linear divisors. Yes, it only works when dividing by a linear binomial. For quadratic or higher divisors, you need long division or other methods.
• The remainder is always a number. When dividing by a linear divisor, the remainder is always a constant (could be zero). If the original polynomial has missing terms, the quotient may have missing terms too.
• It’s only for finding quotients. Actually, synthetic division is also used to evaluate polynomials, factor them, and find zeros (roots).

Synthetic division is a powerful tool that makes polynomial division quick and easy. Practice with our Synthetic Division Calculator to see step-by-step solutions and check your work.