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Synthetic Division Calculator

Divide a polynomial by x - c using synthetic division

Polynomial coefficients
Divide by x - c, c =

Synthetic Division Rule

Bring down, multiply by c, add, repeat

Synthetic division is a compact shortcut for dividing a polynomial by a linear factor of the form x-c. It uses only the polynomial coefficients, so the written work is shorter than full polynomial long division. The process begins by bringing down the leading coefficient, then repeatedly multiplying by c and adding the next coefficient. The final number is the remainder, and the earlier numbers form the quotient coefficients.

⚠Note: This method applies directly to divisors with leading coefficient 1, such as x-2 or x+3.

What Is Synthetic Division?

Synthetic division is a shortened version of polynomial long division for divisors like x-c. Instead of writing variables at every step, it works only with coefficients, making the process faster and less error-prone.

Bring Down

The first coefficient becomes the first quotient coefficient.

Multiply

Multiply the current bottom-row value by c and place it under the next coefficient.

Add

Add the column to get the next bottom-row value.

Remainder

The last bottom-row number is the remainder, equal to P(c).

💡 Example: Divide x³-3x²+2x+4 by x-1. Synthetic division with c=1 gives quotient x²-2x and remainder 4.

Applications of Synthetic Division

Polynomial DivisionRemainder TheoremFactor Testing

Frequently Asked Questions

What is a synthetic division calculator?▼
A synthetic division calculator divides a polynomial by a linear divisor of the form x-c using a compact coefficient method.
When can I use synthetic division?▼
Use synthetic division when the divisor is x-c, such as x-2 or x+3.
How do I enter x+3 as a divisor?▼
Since x+3 equals x-(-3), enter c=-3.
What does the last number mean?▼
The last synthetic division number is the remainder. It also equals f(c) by the remainder theorem.
Is synthetic division the same as long division?▼
It gives the same quotient and remainder as polynomial long division, but only for linear divisors in the x-c form.

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