Synthetic Division Calculator
Divide polynomials by a linear expression using synthetic division. Enter a polynomial and divisor to see the coefficient table, every multiply-and-add step, the exact quotient, and the remainder.
Your Synthetic Division Result
Enter a polynomial and linear divisor, or choose an example to build the synthetic division table.
- Quotient Q(x)
- Remainder R(x)
Final Answer
Factor Check
Remainder Theorem
Synthetic Division Table
Read across: bring down, multiply by c, then add. The last bottom-row value is the remainder.
Synthetic Division Steps
Verification
What Is Synthetic Division?
Synthetic division is a shortened method of polynomial division that uses coefficients instead of writing every polynomial term. It is most commonly used when dividing a polynomial by a linear expression of the form x − c. The method produces a quotient and a remainder through repeated multiplication and addition.
How to Do Synthetic Division Step by Step
Step 1: Find c from the Divisor
For a divisor x − c, use c in the synthetic division table. For x − 2, use 2. For x + 3, use −3. If the divisor is ax + b, first write it as a(x − c), where c = −b/a.
Step 2: List the Polynomial Coefficients
Write the coefficients in descending powers of x. Include zeros for missing terms. For example, x³ − 2x² + 4 has coefficients 1, −2, 0, 4.
Step 3: Bring Down the First Coefficient
Copy the leading coefficient into the bottom row without changing it. This begins the quotient coefficient sequence.
Step 4: Multiply and Add
Multiply the latest bottom-row value by c. Write that product under the next coefficient, then add the two numbers. Repeat until you reach the last coefficient.
Step 5: Read the Answer
All bottom-row values except the last become the quotient coefficients for division by x − c. The last value is the remainder. If the original divisor was ax + b, divide those quotient coefficients by a to obtain the quotient for the original divisor.
Synthetic Division Formula
For a polynomial P(x) divided by x − c, the synthetic division recurrence starts with the leading coefficient and repeatedly multiplies by c and adds the next coefficient.
b₀ = a₀
bᵢ = aᵢ + c bᵢ₋₁
P(x) = (x − c)Q(x) + R
R = P(c)
The final synthetic coefficient gives the remainder. This is also the value of P(c), as stated by the Remainder Theorem. If that value is zero, x − c is a factor of P(x).
Worked Synthetic Division Examples
Use these examples to see how coefficient-based division works, including missing terms and nonzero remainders.
Example 1: Synthetic Division with Zero Remainder
Divide x³ − 6x² + 11x − 6 by x − 2.
Use c = 2 and the coefficients 1, −6, 11, −6.
Bring down 1. Multiply 1 by 2 and add to −6 to get −4. Multiply −4 by 2 and add to 11 to get 3. Multiply 3 by 2 and add to −6 to get 0.
The bottom row is 1, −4, 3, 0.
Quotient: x² − 4x + 3. Remainder: 0.
Since the remainder is zero, x − 2 is a factor.
Example 2: Synthetic Division with Missing Terms
Divide x³ − 2x² + 4 by x − 2.
The coefficient list is 1, −2, 0, 4 because the x term is missing.
Using c = 2 produces the bottom row 1, 0, 0, 4.
Quotient: x². Remainder: 4.
The Remainder Theorem confirms P(2) = 4.
Example 3: Synthetic Division by 2x − 1
Divide 2x³ + 3x² − 5x + 7 by 2x − 1.
Rewrite the divisor as 2(x − 1/2), so c = 1/2.
Synthetic division gives the quotient 2x² + 4x − 3 for the normalized divisor x − 1/2 and a remainder of 11/2.
Because the original divisor has leading coefficient 2, divide the quotient by 2.
Final quotient: x² + 2x − 3/2. Final remainder: 11/2.
Check: (2x − 1)(x² + 2x − 3/2) + 11/2 = 2x³ + 3x² − 5x + 7.
Synthetic Division and the Factor Theorem
The Factor Theorem connects polynomial roots and factors. If P(c) = 0, then x − c is a factor of P(x). Synthetic division makes this easy to test: divide by x − c and inspect the final value in the bottom row. A zero remainder confirms the factor. A nonzero remainder rules it out.
When Should You Use Polynomial Long Division Instead?
Synthetic division is convenient when the divisor is linear. Polynomial long division is more general and works when the divisor contains x², x³, or higher powers. Both methods produce the same mathematical quotient and remainder for a valid linear divisor.
Common Synthetic Division Mistakes
- Using the wrong sign for c: For x + 3, use c = −3, not 3.
- Forgetting zero coefficients: A missing power must still occupy a place in the coefficient row.
- Misreading the last number: The last bottom-row value is the remainder, not another quotient coefficient.
- Forgetting a non-monic divisor: When dividing by ax + b, the synthetic quotient for x − c must be divided by a.
Methodology and Limits
Both calculators use exact rational polynomial arithmetic rather than floating-point approximation. Polynomial long division repeatedly subtracts a multiple of the divisor until the remainder has a lower degree. Synthetic division uses a coefficient recurrence for linear divisors. Both results are checked using the identity P(x) = D(x)Q(x) + R(x). For a linear divisor ax + b, the remainder is also verified by evaluating P(−b/a).
The calculators accept univariate polynomials in x with supported rational coefficients and nonnegative integer exponents. They do not perform general symbolic algebra, multivariable division, or division by the zero polynomial. Degree is limited to 16, input to 1,200 characters and 400 tokens, nesting to 32 levels, numeric literals to 100 digits, and intermediate integers to 4,096 bits. Ambiguous forms such as 1/2x must be written as (1/2)x. Bounds also apply during expansion.
Reviewed: October 2026. Read our methodology and sources.
Frequently Asked Questions
What is synthetic division used for?
Synthetic division is used to divide a polynomial by a linear expression efficiently. It is especially useful for checking factors, evaluating a polynomial at a candidate root, and reducing a polynomial to a lower degree.
When can I use synthetic division?
You can use this calculator when the divisor is a nonzero linear polynomial, such as x − 2, x + 3, or 2x − 1. For a quadratic or higher-degree divisor, use polynomial long division.
How do I find c when the divisor is x + 3?
Rewrite x + 3 as x − (−3). Therefore, c = −3. Using the wrong sign is a common synthetic division error.
What do I do when a polynomial has missing terms?
Insert a zero coefficient for every missing power. For example, x³ − 4x + 1 has coefficients 1, 0, −4, 1.
Can synthetic division work with 2x − 1?
Yes. Rewrite 2x − 1 as 2(x − 1/2). Use c = 1/2 in the synthetic table, then divide the resulting quotient coefficients by 2. The remainder does not change.
What does the last number in synthetic division mean?
The last number in the bottom row is the remainder. The preceding numbers form the quotient coefficients for the normalized divisor.
How is synthetic division related to the Remainder Theorem?
When dividing P(x) by x − c, the remainder equals P(c). Synthetic division calculates that value directly as its final coefficient.
Does a zero remainder mean I found a factor?
Yes. A zero remainder means the linear divisor divides the polynomial evenly. For x − c, this also means P(c) = 0, so c is a root of the polynomial.
Can I use decimal or fractional coefficients?
Yes. The calculator accepts supported finite decimals and rational coefficients and uses exact arithmetic to avoid unnecessary rounding errors.
Does synthetic division find every polynomial root?
No. It can check whether a chosen value is a root and reduce the polynomial when appropriate. Finding all roots may require additional methods.
Sources
OpenStax — Algebra and Trigonometry 2e: Dividing Polynomials; OpenStax — Precalculus 2e: Dividing Polynomials.