Using the Synthetic Division Calculator
Use this to test a possible linear factor or reduce the degree of a polynomial. Enter degree-one through degree-eight coefficients in descending order and retain missing powers as zeros. The divisor must be monic and linear.
Worked example
Use the example inputs below to reproduce this result. These are the form's starting values, not recommended targets. Changing your inputs updates your result above; this worked example stays fixed for comparison. Results are rounded for display.
Example inputs
- Polynomial coefficients, highest power first
- 1, -6, 11, -6
- r in divisor x − r
- 2
Example result
- Quotient
- x^2 − 4x + 3
- Remainder
- 0
- Factor theorem
- x − (2) is a factor.
- Verification
- P(2) = 0 (remainder theorem).
- Steps
- Divide x^3 − 6x^2 + 11x − 6 by x − (2). Bring down the first coefficient: 1. Multiply 1 by 2, then add -6: -4. Multiply -4 by 2, then add 11: 3. Multiply 3 by 2, then add -6: 0. The last entry is the remainder; preceding entries form the quotient. P(x) = (x − (2)) · (x^2 − 4x + 3) + (0).
Method and assumptions
Apply synthetic division using Horner’s recurrence. The last entry is the remainder P(r). Supports degree 1–8 and integer coefficients from −10,000 to 10,000; integer r from −100 to 100. Operations outside the safe-integer range are rejected.
Reference: openstax.org — method and further reading
Check the answer in the original problem
After solving, substitute the result into the original equation or relationship where possible. This can catch a sign error or a coefficient entered in the wrong position. Keep the exact inputs available when comparing with another tool, since changing a coefficient or rounding an intermediate value can change the problem being solved.
Common question
What does a nonzero remainder mean?
The polynomial is not divisible by x − r without a remainder. The final value is P(r), and the identity is P(x) = (x − r)Q(x) + remainder.