Lesson 5.2

Synthetic Division

The "cheat code" for polynomial division. Faster, cleaner, and the secret to evaluating massive functions instantly.

1

Introduction

Prerequisite Connection: Long division works, but it takes up a lot of space and writing variables is tedious. If we are dividing by a simple linear term , there must be a faster way.

Today's Increment: We introduce Synthetic Division, a collapsed algorithm that uses only coefficients. We also connect it to the Remainder Theorem: dividing by gives you as the remainder.

Why This Matters for Calculus: You will frequently need to verify if a number is a root of a polynomial. Synthetic division is the fastest way to check this by hand.

2

Explanation of Key Concepts

Synthetic Division Steps

Condition: Can ONLY be used when dividing by (coefficient of x is 1).
1. Setup

Write the "box number" (opposite sign of divisor). List coefficients of dividend.

2. "Drop Mama"

Drop the first coefficient straight down below the line. This starts the chain.

3. Multiply & Add

Multiply the bottom number by . Put result in next column. ADD down. Repeat.

The Remainder Theorem

If polynomial is divided by , then the remainder is .

This means synthetic division is a valid way to plug numbers into functions!
3

Worked Examples

Level: Basic

Example 1: Performing the Algorithm

Divide .

2
2
-3
4
-5
↓
4
2
12
2
1
6
7 (R)
The coefficients 2, 1, 6 represent .
The remainder is 7.
Answer: .
Level: Intermediate

Example 2: Evaluation vs. Substitution

Given , find using synthetic division.

You could compute , or you could run the table:
10
1
-10
2
-1
5
↓
10
0
20
190
1
0
2
19
195
Result
. Much faster than calculating powers of 10!
Level: Advanced

Example 3: Forcing a Remainder

Find such that when is divided by , the remainder is 8.

Method: Remainder Theorem
We know .



So the polynomial is .
4

Common Pitfalls

  • Sign Switcheroo:

    If dividing by , you use in the box.
    If dividing by , you use in the box.
    (You solve ).

  • Adding vs Subtracting:

    In Long Division, you subtract rows. In Synthetic Division, you ADD columns. Do not confuse them!

5

Real-World Application

Space Trajectories: Evaluation Speed

In orbital mechanics, spacecraft positions are often approximated by high-degree Taylor polynomials.

Computers use a method called Horner's Method to evaluate these polynomials. Horner's Method is mathematically identical to Synthetic Division! It reduces the number of multiplications needed, saving precious processor cycles on deep space probes.

6

Practice Quiz

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