Lesson 2.8

Synthetic Division

Synthetic division is the "shortcut" version of polynomial long division. It's faster, but it only works when dividing by a linear factor of the form .

Introduction

Long division works, but it's slow. When your divisor is linear (), synthetic division does the same thing in a fraction of the time using only the coefficients.

Past Knowledge

You just learned polynomial long division (Lesson 2.7).

Today's Goal

Divide polynomials quickly using the synthetic division shortcut.

Future Success

Synthetic division is the fastest way to test if a number is a zero of a polynomial — the backbone of root-finding.

Key Concepts

1. When Can You Use It?

Only when dividing by a linear expression of the form .

→ use
→ use (since )
→ NOT linear; use long division

2. The Process

Set up a table with the coefficients and the value .

1
Write the value c on the left and the coefficients of the dividend in a row
2
Bring down the first coefficient
3
Multiply by c and write the result under the next coefficient
4
Add the column, then repeat Multiply → Add until done
5
The last number is the remainder; the others are the quotient coefficients

💡 Key Insight

The quotient polynomial is always one degree less than the dividend. If you start with degree 3, the quotient is degree 2.

Worked Examples

Example 1: Basic Synthetic Division

Basic

Divide .

1

Set Up

Divisor is , so . Write the coefficients: .

2

Run the Process

214−1−10
21222
161112
3

Read the Answer

Bottom row = quotient coefficients + remainder. Degree drops by 1.

Example 2: Dividing by (x + 3)

Intermediate

Divide .

!

Identify

, so . Don't use !

1

Run Synthetic Division

−321−136
−615−6
2−520
✓

Answer

Remainder = 0, so is a factor!

Example 3: Missing Coefficients

Advanced

Divide .

!

Insert Placeholder Zeros

is missing and terms. Coefficients: .

1

Run Synthetic Division with

21000−16
24816
12480
✓

Answer

Remainder = 0 confirms is a zero of .

Common Pitfalls

Using the Wrong Sign for

For , students often use . It's because . The sign is always opposite of what you see.

Forgetting Placeholder Zeros

Missing terms must get a coefficient. For , the coefficients are , NOT .

Real-Life Applications

In computer science, algorithms that evaluate polynomials at specific values (like Horner's Method) use the exact same multiply-then-add pattern as synthetic division. It's the most efficient way to compute — faster than plugging in directly.

Practice Quiz

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