Synthetic Division Calculator

This synthetic division calculator divides a polynomial by a linear factor of the form x minus r, giving you the quotient and remainder in seconds without working through long division by hand. Synthetic division, sometimes called Ruffini's rule, is a streamlined shortcut that only works when the divisor is x minus r, but for that common case it is much faster and less error-prone than the full long-division method. You enter the polynomial's coefficients from the highest power down to the constant term, separated by commas or spaces, along with the value of r you are dividing by (remember that dividing by x plus 3 means r equals minus 3). The calculator brings down the leading coefficient, multiplies it by r, adds the result to the next coefficient, and repeats this down the row, exactly as you would by hand. It then returns the quotient as a list of coefficients one degree lower than your original polynomial, plus the remainder and the r value you used. The remainder is not just a leftover number; by the remainder theorem it equals the polynomial evaluated at r, so a remainder of zero tells you that x minus r is a factor. It is ideal for algebra students checking homework, finding roots, or factorising polynomials quickly, though it only handles linear divisors; for anything of higher degree you will need polynomial long division.

Coefficients (high to low)

Divide by (x - r): r =

Quotient: [1, 1]

Remainder0

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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