Synthetic Division Calculator
Enter a polynomial's coefficients below to instantly divide by (x − c), with every step shown.
How Synthetic Division Works
Bring down the first coefficient, multiply it by c, add that to the next coefficient, and repeat. The last number produced is the remainder; everything before it is the quotient's coefficients.
Worked example
Divide x³ − 6x² + 11x − 6 by (x − 1).
1, then −6 + 1(1) = −5, then 11 + (−5)(1) = 6, then −6 + 6(1) = 0
Quotient: x² − 5x + 6, remainder 0
A second example
Divide x³ + 2x² − 5x − 6 by (x − 2).
1, then 2 + 1(2) = 4, then −5 + 4(2) = 3, then −6 + 3(2) = 0
Quotient: x² + 4x + 3, remainder 0
Keep going
- Synthetic division divides a polynomial by (x − c); the polynomial calculator evaluates that same polynomial directly at a chosen value of x. Polynomial Calculator
- A zero remainder in synthetic division means you've found a root; the factoring calculator uses roots like that to break the polynomial into factors. Factoring Calculator
Frequently Asked Questions
What is synthetic division?
Synthetic division is a shortcut for dividing a polynomial by a linear factor (x − c), using only the coefficients instead of writing out the full long-division process.
When can you use synthetic division?
Only when dividing by a linear expression of the form x − c. For any other divisor, you need regular polynomial long division instead.
What does a remainder of 0 tell you?
A remainder of 0 means (x − c) divides evenly into the polynomial, which means c is a root of the polynomial and (x − c) is one of its factors.
How does synthetic division connect to the remainder theorem?
The Remainder Theorem states that dividing a polynomial f(x) by (x − c) leaves a remainder equal to f(c). That means synthetic division doubles as a fast way to evaluate a polynomial at a specific value, without direct substitution.
What's a common mistake with synthetic division?
Trying to use it on a divisor that isn't a simple (x − c), like (2x − 3). Synthetic division only works when the coefficient on x in the divisor is 1. For a divisor like (2x − 3), you'd need regular polynomial long division instead, or divide by (x − 3/2) and then adjust the quotient for the leading coefficient.
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