Synthetic Division — Mathematics Knowledge System

Overview

Synthetic division is a streamlined shorthand for dividing a polynomial by a linear factor (x - c), using only the coefficients arranged in a compact array. It produces the same quotient and remainder as polynomial long division but with far less writing, making it the standard tool for quickly testing candidate roots.

Formal Definition

Definition

To divide f(x) = aₙxⁿ+...+a₀ by (x - c), write only the coefficients aₙ,...,a₀ in a row, bring down the leading coefficient, then repeatedly multiply by c and add:

f(x) = (x-c)\,q(x) + r, \quad r = f(c)

Synthetic division computes q(x)'s coefficients and the remainder r = f(c) directly

Properties

Applicability

\text{Synthetic division applies only when dividing by a linear factor } (x-c)

Remainder equals f(c)

r = f(c) \text{ (Remainder Theorem)}

Degree drop

\text{If } f \text{ has degree } n, \text{ the quotient } q(x) \text{ has degree } n-1

Worked Examples

Write coefficients 2, -3, 4, -5 and c=2. Bring down the 2.
$2$
Multiply 2×2=4, add to -3: get 1. Multiply 1×2=2, add to 4: get 6. Multiply 6×2=12, add to -5: get 7.
$2,\ 1,\ 6 \mid 7$

Answer: Quotient: 2x² + x + 6, remainder 7

Practice Problems

Difficulty 3/10

Use synthetic division to divide f(x) = x³ + 4x² - 5x - 20 by (x + 4), and state whether (x+4) is a factor.

Common Mistakes

Common Mistake

Using synthetic division to divide by a non-linear or non-monic-linear divisor like (2x-3) without adjusting.

Synthetic division in its basic form requires a divisor of the form (x-c). For (2x-3), first solve for c=3/2, do synthetic division by (x-3/2), then divide the quotient's coefficients by 2 to account for the original leading coefficient.

Common Mistake

Using +c instead of the root c when the divisor is (x+c), e.g. using c=4 for divisor (x+4).

Write (x+4) as (x-(-4)), so the synthetic division constant is c=-4, not +4 — a sign error here silently produces a wrong quotient and remainder.

Summary

Synthetic division is a compact coefficient-only shortcut for dividing by (x-c).
The last number produced is the remainder, which equals f(c) by the Remainder Theorem.
It only works directly for linear divisors of the form (x-c); a zero remainder confirms (x-c) is a factor.
Always convert (x+c) to (x-(-c)) before reading off the value to use in the division.

References

WebsiteWikipedia — Synthetic division