Synthetic Division Calculator

Divide a polynomial by (x - c) using synthetic division. Shows the complete table, quotient, and remainder.

Enter Polynomial & Divisor

Enter comma-separated coefficients. Example: "1, 0, -4, 6" represents x^3 - 4x + 6. Use 0 for missing terms.
For (x - 2), enter 2. For (x + 3), enter -3.

Result

Quotient
--
Remainder
--
Polynomial--
Divisor--
f(c) = Remainder--

Step-by-Step Solution

P(x) = (x - c) * Q(x) + R

Understanding Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear divisor of the form (x - c). It is much faster than long division and produces both the quotient polynomial and the remainder in a compact table format.

How It Works

Setup

Write the divisor c on the left and the polynomial coefficients in a row. Include zeros for any missing degree terms.

c | a_n a_{n-1} ... a_0

Process

Bring down the leading coefficient. Multiply by c, add to next coefficient. Repeat until done.

bring down, multiply, add, repeat

Result

The bottom row gives the quotient coefficients and the final value is the remainder.

P(x) = (x-c)*Q(x) + R

Remainder Theorem Connection

By the Remainder Theorem, the remainder when dividing P(x) by (x - c) equals P(c). Synthetic division simultaneously evaluates the polynomial at c and performs the division. If the remainder is 0, then c is a root of the polynomial.

When to Use Synthetic Division

  • Dividing a polynomial by a linear factor (x - c).
  • Evaluating a polynomial at a specific value (Remainder Theorem).
  • Testing potential rational roots using the Rational Root Theorem.
  • Factoring polynomials by successively dividing out known roots.