Synthetic Division Calculator

How it is calculated

List the coefficients from the highest power down to the constant term, separated by commas or spaces, and include a 0 for every missing power (x³ − 8 is 1, 0, 0, −8). At least two coefficients are needed. The second field is r, and the divisor is (x − r), so dividing by x + 2 means entering r = −2. The engine brings down the first coefficient and then repeats carry = next coefficient + carry × r. The carries before the last one are the coefficients of the quotient, which has one degree less than the polynomial, and the final carry is the remainder. By the Remainder Theorem that remainder equals f(r). Dots are the decimal separator; a comma always separates coefficients. Non-numeric entries, a single coefficient, or intermediate values beyond 2⁵³ (where digits stop being exact) show no result. Results are listed with trailing zeros removed.

Common mistakes

• Entering the sign of the root wrongly. The divisor is (x − r), so for (x + 2) you must type −2, not 2.
• Leaving out zero coefficients. x³ − 8 must be entered as 1, 0, 0, −8; typing 1, −8 describes a different polynomial.
• Entering the coefficients from the lowest power upward. The first number must belong to the highest power.
• Writing a decimal coefficient with a comma, such as 2,5. The comma separates coefficients, so it is read as two numbers, 2 and 5; use a dot (2.5).
• Using it to divide by a quadratic or by (ax − b) directly. Only (x − r) with a leading coefficient of 1 works, so rewrite a divisor like 2x − 6 as 2(x − 3) first.