Lesson 3.4

Dividing Rational Expressions

Dividing by a fraction means multiplying by its reciprocal. The classic mantra: Keep-Change-Flip.

Introduction

Dividing rational expressions is almost identical to multiplying them — with one extra first step. You flip the second fraction (take its reciprocal), then proceed exactly as in Lesson 3.3.

Past Knowledge

Multiplying rational expressions (Lesson 3.3) and all factoring techniques.

Today's Goal

Apply Keep-Change-Flip to convert division into multiplication, then simplify.

Future Success

Complex fractions (Lesson 3.8) use division of rational expressions as a core step.

Key Concepts

Keep-Change-Flip

KEEP

First fraction stays

CHANGE

÷ becomes ×

FLIP

Second fraction flips

Full Process

1

Keep-Change-Flip to rewrite as multiplication

2

Factor all numerators and denominators

3

Cancel common factors

4

Multiply remaining factors

Worked Examples

Example 1: Basic Division

Basic

Divide .

1

Keep-Change-Flip

2

Cancel and simplify

Example 2: Factoring Required

Intermediate

Divide .

1

Keep-Change-Flip

2

Factor and cancel

Example 3: Both Need Factoring

Advanced

Divide .

1

Keep-Change-Flip

2

Factor and cancel

Common Pitfalls

Flipping the Wrong Fraction

Only flip the second fraction (the divisor). The first fraction stays exactly as-is.

Forgetting New Restrictions

After flipping, the old numerator becomes a denominator — check it for new domain restrictions too.

Real-Life Applications

In chemistry, concentration problems often involve dividing one rate expression by another. The Keep-Change-Flip technique streamlines these calculations, especially when dealing with complex rate laws.

Practice Quiz

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