Synthetic Division Calculator

About this calculator

Synthetic division is a compact shortcut for dividing a polynomial by a linear factor (x − r): instead of writing out full polynomial long division, you bring down the leading coefficient and repeatedly multiply by r and add into the next coefficient, working straight across a single row of numbers. The result is the same as long division — a quotient polynomial one degree lower, plus a remainder — but reached with far less writing.

The technique is closely tied to the Remainder Theorem: the remainder from dividing by (x − r) equals f(r), so a remainder of zero instantly tells you r is a root of the polynomial, which is why synthetic division is a standard way to test candidate roots and factor polynomials in algebra courses. It is also, under the hood, the same computation as Horner's method for efficiently evaluating a polynomial at a point.

Enter your polynomial's coefficients and a value r to divide by (x − r); the calculator returns the quotient's coefficients and the remainder.