Medium

Synthetic Division

Simple Explanation

Synthetic division is a fast shortcut for dividing a polynomial by a linear divisor (x − a): write only the coefficients, bring down the first one, then repeatedly multiply by a and add, skipping all the variable-tracking that ordinary long division requires.

Why Do We Need It?

Synthetic division does the exact same job as long division for linear divisors, but much faster — the standard tool used throughout this chapter for testing possible roots quickly.

Formula

The Polynomial Division Identity

P(x) = D(x) · Q(x) + R(x), where the degree of R(x) is less than the degree of D(x)

Dividing a polynomial P(x) by a divisor D(x) always produces a quotient Q(x) and a remainder R(x), related by this identity — the same "dividend = divisor × quotient + remainder" pattern used for ordinary numbers.

P(x): — the dividend — the polynomial being divided
D(x): — the divisor — the polynomial being divided by
Q(x): — the quotient
R(x): — the remainder (degree strictly less than D(x), or R(x)=0 if it divides evenly)

When to use it: Whenever you divide one polynomial by another and need to check or express the result correctly.

Worked Example

Use synthetic division

Divide x³ − 4x² + x + 6 by (x − 3) using synthetic division.

Why Does This Work?

Synthetic division is exactly polynomial long division for a divisor of the form (x − a), with all the repeated x-terms stripped away — since the divisor is always the same simple form, only the numerical coefficients need to be tracked, which is what makes the shortcut possible.

Real-Life Example

Quickly testing candidate roots by hand

A student needs to check several possible roots of a cubic polynomial to find one that works, before fully factoring it.

Synthetic division lets each candidate be tested in just a few quick lines, far faster than repeating full long division for every guess.

Practice

Use synthetic division to divide x² − x − 6 by (x − 3). What is the remainder?

Medium

Common mistake

Using the wrong sign for a — for a divisor (x − a), the value placed at the left of the synthetic division is +a, not −a (e.g. dividing by (x+5) means a = −5, not 5).

Quick Review

Synthetic division only works for linear divisors of the form (x − a).
Bring down the first coefficient, then repeatedly multiply by a and add.
The last number is the remainder; the rest give the quotient, one degree lower.