Synthetic Division Calculator

This synthetic division calculator divides a polynomial by a linear factor of the form x minus r, a quick shortcut to full long division. Enter the polynomial coefficients, highest power first, and the value r, and the calculator returns the quotient coefficients and the remainder. Synthetic division, also called Ruffini\u2019s rule, only works for dividing by x minus r, but for that case it is fast and tidy. The remainder it gives is exactly the value of the polynomial at r, linking it to the remainder theorem. The method, a worked example and the assumptions are below.

Coefficients (high to low)

Divide by (x - r): r =

Remainder

The formula

To divide by x minus r, bring down the leading coefficient, multiply by r and add to the next coefficient, and repeat. The last number is the remainder, equal to P(r); the others are the quotient coefficients.

Worked example

Dividing x^2 minus 1 by x minus 1 (r = 1): bring down 1, times 1 plus 0 = 1, times 1 plus minus 1 = 0. Quotient is x + 1, remainder 0. Enter 1, 0, -1 and 1 to confirm.

Frequently asked questions

When can I use synthetic division?

When dividing by a linear factor x minus r. For higher degree divisors use polynomial long division.

What does the remainder tell me?

It equals P(r). If it is zero, then x minus r is a factor of the polynomial.

What is r for x + 3?

r is minus 3, because x + 3 is x minus (minus 3).

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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