Lesson 2.7

Polynomial Long Division

Long division isn't just for numbers. The same "Divide → Multiply → Subtract → Bring Down" algorithm works perfectly for polynomials.

Introduction

Just as , we can divide polynomials and get a quotient and a remainder. This is the "full version" — next lesson we'll learn the shortcut.

Past Knowledge

You know numerical long division: Divide, Multiply, Subtract, Bring Down.

Today's Goal

Apply the same algorithm to divide a polynomial by a linear or quadratic divisor.

Future Success

Long division connects directly to the Factor and Remainder Theorems used to find polynomial zeros.

Key Concepts

1. The Algorithm

Repeat the four-step cycle until the degree of the remainder is less than the degree of the divisor.

1
Divide the leading terms
2
Multiply the result by the divisor
3
Subtract from the dividend
4
Bring down the next term

2. The Result

3. Placeholder Zeros

If the dividend is missing a degree term, you must insert a placeholder. This keeps columns aligned.

Example: is missing and terms.

✅ Check Your Work

You can verify: .

Worked Examples

Example 1: Clean Division (No Remainder)

Basic

Divide .

1 · Divide

2 · Multiply

3 · Subtract

Remove

4 · Bring Down

Bring down

Remainder = 0, so is a factor!

Example 2: Division with a Remainder

Intermediate

Divide .

← Remainder
Reading the answer: Since the remainder is (not zero), we write it as a fraction over the divisor.

Example 3: Missing Terms (Placeholder Zeros)

Advanced

Divide .

⚠️ First: Insert placeholder zeros for the missing and terms:

This is the Difference of Cubes factorization!

Common Pitfalls

Forgetting Placeholder Zeros

If dividing by , you must write it as . Skipping these causes every column to shift and the entire answer to be wrong.

Sign Errors During Subtraction

The "Subtract" step is where most mistakes happen. Remember: subtracting means . Change both signs!

Real-Life Applications

In engineering, a "transfer function" often looks like one polynomial divided by another. Simplifying these by long division reveals the system's behavior at different frequencies — critical for designing stable electronic circuits and control systems.

Practice Quiz

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