Synthetic Division Calculator — Step-by-Step Solutions

The Synthetic Division Calculator divides polynomials by linear factors (x - c) using the efficient synthetic division method. It shows a complete step-by-step table, quotient, remainder, and verification.

Enter the polynomial coefficients from highest to lowest degree and the divisor value c. The calculator also demonstrates the Remainder Theorem and Factor Theorem.

Enter the polynomial coefficients (highest degree first, comma-separated) and the divisor c for (x - c).

Coefficients (highest degree first)

For x^3 - 6x^2 + 11x - 6, enter: 1, -6, 11, -6

Divisor c (dividing by x - c)

Your calculations will appear here

How to Use the Synthetic Division Calculator

Enter the coefficients of your polynomial from the highest power to the constant, separated by commas. Include a 0 for every missing power: x^3 - 4x + 1 is entered as 1, 0, -4, 1.
Enter c, where you are dividing by (x - c). To divide by x + 2, enter c = -2, because x + 2 = x - (-2).
Pick a mode: Step-by-Step Division shows each row of the table, Quotient & Remainder gives the answer directly, Verify with Multiplication checks it by multiplying back, and Remainder Theorem shows that the remainder equals the polynomial value at c.
If the remainder is 0, the calculator tells you that (x - c) is a factor, so c is a root of the polynomial.

Formula and Theory

Synthetic division is a shortcut for dividing a polynomial by a linear factor (x - c). It uses only the coefficients and replaces polynomial long division with simple multiply-and-add steps. It only works when the divisor has degree 1 and a leading coefficient of 1.

The procedure: write c to the left and the coefficients in a row. Bring the first coefficient down. Multiply it by c, write the result under the next coefficient and add. Repeat across the row. The last number is the remainder; the numbers before it are the coefficients of the quotient, which has degree one less than the original.

The Remainder Theorem says that dividing P(x) by (x - c) leaves remainder P(c). The Factor Theorem follows: (x - c) is a factor of P(x) exactly when P(c) = 0. This makes synthetic division a quick way to test candidate roots, for example those suggested by the Rational Root Theorem, and then reduce the polynomial to find the rest.

Worked Examples

Example 1: A division with zero remainder

Problem: Divide x^3 - 6x^2 + 11x - 6 by (x - 2).

Solution: Coefficients 1, -6, 11, -6 with c = 2. Bring down 1. Then 1 x 2 = 2 and -6 + 2 = -4. Then -4 x 2 = -8 and 11 - 8 = 3. Then 3 x 2 = 6 and -6 + 6 = 0.

Answer: Quotient x^2 - 4x + 3, remainder 0, so (x - 2) is a factor

Example 2: Dividing by x + 1 with a non-zero remainder

Problem: Divide 2x^3 + 3x^2 - 5x + 7 by (x + 1).

Solution: Write x + 1 as x - (-1), so c = -1. Coefficients 2, 3, -5, 7. Bring down 2. Then 2 x (-1) = -2 and 3 - 2 = 1. Then 1 x (-1) = -1 and -5 - 1 = -6. Then -6 x (-1) = 6 and 7 + 6 = 13. Check with the Remainder Theorem: P(-1) = -2 + 3 + 5 + 7 = 13.

Answer: Quotient 2x^2 + x - 6, remainder 13

Example 3: A polynomial with missing terms

Problem: Divide x^4 - 16 by (x - 2).

Solution: Insert zeros for the missing powers: coefficients 1, 0, 0, 0, -16 with c = 2. Bring down 1. Then 1 x 2 = 2 and 0 + 2 = 2. Then 2 x 2 = 4 and 0 + 4 = 4. Then 4 x 2 = 8 and 0 + 8 = 8. Then 8 x 2 = 16 and -16 + 16 = 0.

Answer: Quotient x^3 + 2x^2 + 4x + 8, remainder 0

Example 4: Using a fractional value of c

Problem: Find the remainder when 2x^3 - x^2 + 4 is divided by (x - 0.5).

Solution: Coefficients 2, -1, 0, 4 with c = 0.5. Bring down 2. Then 2 x 0.5 = 1 and -1 + 1 = 0. Then 0 x 0.5 = 0 and 0 + 0 = 0. Then 0 x 0.5 = 0 and 4 + 0 = 4. Check: P(0.5) = 2(0.125) - 0.25 + 4 = 4.

Answer: Quotient 2x^2, remainder 4

Frequently Asked Questions

How do I do synthetic division?

List the coefficients, bring down the first one, then repeat: multiply by c and add to the next coefficient. The last value is the remainder and the others are the quotient coefficients. The step-by-step mode lays out each row.

How do I divide by x + 3?

Rewrite it as x - (-3) and use c = -3. The most common slip is entering +3 instead of -3.

Why do I need zeros for missing terms?

Each column in the table stands for one power of x. If a power is missing, its coefficient is 0, and leaving it out shifts every later coefficient into the wrong column.

Can I use synthetic division for a divisor like 2x - 1?

Not directly. Synthetic division needs a divisor of the form x - c. For 2x - 1, rewrite it as 2(x - 0.5), divide by (x - 0.5), then divide the quotient by 2.

What does a remainder of 0 tell me?

It means (x - c) divides the polynomial exactly, so c is a root and (x - c) is a factor. You can then keep factoring the smaller quotient.

How is this connected to the Remainder Theorem?

The remainder of the division by (x - c) always equals P(c). So synthetic division is also a fast way to evaluate a polynomial at a point.