Synthetic Division
Synthetic division is a shorthand method for dividing a polynomial by a linear binomial of the form (x − c), using only the coefficients of the polynomial arranged in a tabular format. It is significantly faster than long division and produces both the quotient polynomial and the remainder directly. The technique is closely linked to the Remainder Theorem (the remainder equals p(c)) and the Factor Theorem (if the remainder is 0, then (x − c) is a factor of the polynomial).
Worked Example
Problem
Use synthetic division to divide p(x) = 2x³ − 3x² + x − 5 by (x − 2).
Solution
Step 1: Write the divisor root: c = 2. Coefficients of p(x): 2, −3, 1, −5. Step 2: Set up the synthetic division table: 2 | 2 −3 1 −5 | 4 2 6 2 1 3 1 Step 3: Bring down 2. Multiply 2×2=4, add to −3: 1. Multiply 2×1=2, add to 1: 3. Multiply 2×3=6, add to −5: 1. Step 4: Quotient coefficients: 2, 1, 3. Remainder: 1. Step 5: Quotient = 2x² + x + 3; Remainder = 1.
Answer
p(x) ÷ (x−2) = 2x² + x + 3, remainder 1; i.e., p(x) = (x−2)(2x²+x+3) + 1
Synthetic Division vs. Polynomial Long Division
| Feature | Synthetic Division | Long Division | Notes |
|---|---|---|---|
| Divisor type | Linear (x − c) only | Any polynomial | Synthetic is more restricted |
| Speed | Faster (coefficients only) | Slower (full terms) | Synthetic preferred for linear |
| Output | Quotient + remainder | Quotient + remainder | Same information |
| Remainder Theorem | Directly gives p(c) | Requires substitution | Key advantage |
| Factor testing | Remainder = 0 means factor | Same conclusion | Synthetic is quicker |
| Complexity | Simple arithmetic | More steps | Synthetic has fewer errors |
Interactive Tools
Wolfram Alpha
Perform polynomial division and verify synthetic division results step by step.
Open ToolKhan Academy — Synthetic Division
Guided lessons on the mechanics and applications of synthetic division.
Open ToolBrilliant.org
In-depth explanations and practice problems on polynomial and synthetic division.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
The term "synthetic" in this context comes from the Greek "synthetikos," meaning "put together" or "combinatory," reflecting the way coefficients are combined in a compact table rather than written out as full polynomial terms. The method is attributed to Paolo Ruffini, who published it in 1804, and is sometimes called "Ruffini's Rule" in European textbooks.