Synthetic Division Calculator
Divide any polynomial up to degree 5 by a linear factor (x - c) using synthetic division. Enter the coefficients and the divisor value to get the quotient, remainder, and factor check instantly.
Synthetic division is a shortcut for dividing a polynomial by a linear expression of the form (x - c). It is faster than long division and uses only the coefficients.
How Synthetic Division Works:
- Write down the coefficients of the polynomial in order (include zeros for missing terms).
- Write the value c to the left.
- Bring down the first coefficient.
- Multiply it by c and add to the next coefficient.
- Repeat until you reach the last coefficient.
- The final number is the remainder. All preceding numbers are the quotient coefficients.
Worked Example:
Divide x³ - 6x² + 11x - 6 by (x - 2):
Coefficients: 1, -6, 11, -6. Divisor c = 2.
- Bring down 1.
- 1 x 2 = 2, add to -6 to get -4.
- -4 x 2 = -8, add to 11 to get 3.
- 3 x 2 = 6, add to -6 to get 0.
Quotient: x² - 4x + 3. Remainder: 0. Since the remainder is 0, (x - 2) is a factor.
Remainder Theorem: The remainder when dividing a polynomial f(x) by (x - c) equals f(c). If f(c) = 0, then (x - c) is a factor of f(x).