Polynomial Long & Synthetic Division
Dividing polynomials, and the shortcut when the divisor is linear.
The explanation
Polynomial long division mirrors numerical long division: divide the leading terms, multiply back, subtract, bring down, repeat.
Synthetic division is a much faster shortcut, but only when dividing by x − c.
To divide x³ − 4x² + 2x + 1 by x − 3:
Write c = 3 and the coefficients 1, −4, 2, 1.
Bring down 1. Multiply by 3 → 3, add to −4 → −1. Multiply by 3 → −3, add to 2 → −1. Multiply by 3 → −3, add to 1 → −2.
The result is x² − x − 1 with remainder −2.
Two rules: use c = 3 for the divisor x − 3 (the opposite sign), and include a 0 for every missing power.
The division algorithm for polynomials states that for P(x) and nonzero D(x) there exist unique Q(x) and R(x) with P = D·Q + R and deg R < deg D.
Long division applies to any divisor. Synthetic division is a condensed version valid only for monic linear divisors x − c; it manipulates coefficients alone, which is why placeholder zeros for missing degrees are mandatory.
Two theorems follow directly. The Remainder Theorem: the remainder on dividing by x − c equals P(c) — so synthetic division doubles as a fast evaluation method. The Factor Theorem: x − c is a factor exactly when P(c) = 0, which is the standard tool for reducing a polynomial once one root is known.
For a divisor such as 2x − 3, synthetic division may be used with c = 3/2 provided the resulting quotient is divided by 2 afterwards, since the algorithm assumes a monic divisor.
Worked example
Divide 2x³ + 3x² − 11x − 6 by x + 3 using synthetic division.
- Divisor x + 3 means c = −3.
- Coefficients: 2, 3, −11, −6.
- Bring down 2; 2(−3) = −6, 3 + (−6) = −3; −3(−3) = 9, −11 + 9 = −2; −2(−3) = 6, −6 + 6 = 0.
- Remainder 0, so x + 3 is a factor.
Answer: Quotient 2x² − 3x − 2, remainder 0
Common mistakes
- Using c = 3 when dividing by x + 3.
- Omitting a zero placeholder for a missing power, which shifts every coefficient.