Synthetic Division
Synthetic division is a fast shortcut for dividing a polynomial by a linear factor x − c, using only the coefficients.
The method
Write c, then the coefficients of the polynomial (use 0 for missing powers). Bring down the first coefficient, multiply it by c, add to the next coefficient, and repeat. The last number is the remainder and the others are the quotient's coefficients.
Worked example
Divide 2x³ + 3x² − 5x + 6 by x + 2, so c = −2 and the coefficients are 2, 3, −5, 6.
- Bring down 2.
- 2 · (−2) = −4, and 3 + (−4) = −1.
- −1 · (−2) = 2, and −5 + 2 = −3.
- −3 · (−2) = 6, and 6 + 6 = 12.
The quotient is 2x² − x − 3 and the remainder is 12.
Remainder theorem
The remainder equals p(c). Here p(−2) = 2(−8) + 3(4) − 5(−2) + 6 = 12, matching the last row. If the remainder is 0, then x − c is a factor.
Common mistake
Using the wrong sign of c: for x + 2 the value is −2. The shortcut only works for divisors of the form x − c.
Testing roots
Synthetic division is the quickest way to test candidate roots. For p(x) = x³ − 6x² + 11x − 6 try c = 1 with coefficients 1, −6, 11, −6. Bring down 1, then −6 + 1 = −5, then 11 − 5 = 6, then −6 + 6 = 0. The remainder is 0, so x − 1 is a factor and the quotient x² − 5x + 6 factors as (x − 2)(x − 3). The roots are 1, 2 and 3.