Synthetic Division Calculator

Your details

Choose how to enter the dividend. Coefficient mode is faster; polynomial notation is more natural.
Comma-separated, in descending powers. Include 0 for any missing term, e.g. x^3 + 2 enter: 1, 0, 0, 2.
The value r in the divisor (x - r). To divide by (x + 3) use r = -3. For a non-monic divisor like 2x - 6, use the Divisor leading coefficient field below.
Enable if your divisor is (ax - b) where a is not 1, e.g. 2x - 3. The algorithm divides by (x - b/a) then adjusts the quotient.
Lists all p/q candidates from the Rational Root Theorem. Useful for finding roots to try next.
Division resultr is a root (remainder = 0)
Quotient: x^2 - 5x + 6 | Remainder: 0
Quotient Q(x)x^2 - 5x + 6
Remainder R0
Full equation P(x) = D(x)Q(x) + Rx^3 - 6x^2 + 11x - 6 = (x - 1) * (x^2 - 5x + 6) + 0
P(r) by Remainder Theorem0

The remainder is zero, so (x - r) divides the polynomial exactly.

  • The quotient is x^2 - 5x + 6, one degree lower than the original.
  • By the Remainder Theorem, the remainder equals P(r), giving you a fast polynomial evaluation at any point.
  • Since the remainder is zero, the Factor Theorem confirms (x - r) is a factor. Apply synthetic division again to the quotient to find the next root.
  • The bottom row of the tableau is the only arithmetic you need: every number except the last is a quotient coefficient.

Next stepRun synthetic division again on the quotient to find the remaining roots, or use the quadratic formula if the quotient is degree 2.

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