Synthetic Division Calculator

Coefficients — highest degree first, commas or spaces (e.g. 1, -6, 11, -6)

Divide by x −

Quotient —

Remainder —

How to Do Synthetic Division

Enter the polynomial coefficients and the root r of the divisor x − r.

Enter the coefficients

List every coefficient from the highest power down to the constant, including zeros for any missing terms — for example 1, 0, −4 for x² − 4.

Enter the divisor

Synthetic division divides by a linear factor x − r. Enter r; to divide by x + 1, use r = −1.

Read the quotient and remainder

The quotient is one degree lower than the input. A remainder of 0 means x − r is a factor.

Frequently Asked Questions

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by a linear factor x − r using only the coefficients, with no x symbols written down. It gives the same quotient and remainder as polynomial long division in far fewer steps. Dividing x³ − 6x² + 11x − 6 by x − 1 gives the quotient x² − 5x + 6 with a remainder of 0. It only works for a linear divisor, so anything of higher degree needs long division.

How do I divide by x + 1?

In synthetic division, write the divisor as x − r. Since x + 1 = x − (−1), use r = −1. The calculator handles the sign for you.

What does the remainder tell me?

By the remainder theorem, the remainder equals the polynomial evaluated at r. So dividing by x − 1 and getting a remainder of 0 tells you that P(1) = 0. A zero remainder therefore means r is a root and x − r is a factor, which is the factor theorem. A non-zero remainder still gives you P(r) for free: dividing x³ − 6x² + 11x − 6 by x − 2 leaves 0, while dividing by x − 4 leaves 6.

Is my data sent anywhere?

Nothing you enter into the synthetic division calculator leaves your device. The synthetic division calculator does all of its arithmetic client-side in JavaScript, so your coefficients and divisor are never uploaded to a server. No account is needed and no calculation is stored or logged. Once the page has loaded it works with no network connection at all.

How to Do Synthetic Division

Enter the coefficients

Enter the divisor

Read the quotient and remainder

Frequently Asked Questions

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