Lesson 6.1

Simplifying Rational Expressions

It's just fraction reduction on steroids. Factor everything, cancel the matches, and watch out for the leftovers.

Introduction

It's just fraction reduction on steroids. Factor everything, cancel the matches, and watch out for the leftovers.

Past Knowledge

You know that simplifies to because you divide both by 3. You calculated and "canceled" the 3s.

Today's Goal

A Rational Function is just a fraction made of polynomials: . We simplify them the exact same way—by finding common factors.

Future Success

In Calculus, you will often get results like . This usually means there is a hidden factor waiting to be canceled so you can actually find the answer (the Limit).

Key Concepts

The "Factor First" Rule

You can ONLY cancel FACTORS (things being multiplied).
You CANNOT cancel TERMS (things being added).
Valid
ILLEGAL

Excluded Values

Simplifying changes the form, but it shouldn't change the domain.

Critical Rule: Even if you cancel a factor like , is STILL not allowed to be 2. The original function was undefined there, so the new one must be too.

This creates a "Hole" in the graph, which we will study in Lesson 6.4.

Worked Examples

Level: Basic

Example 1: GCF Factoring

Simplify .

Step 1: Factor Numerator
Step 2: Cancel Common Factors
Final Answer
excluded value:
Level: Intermediate

Example 2: Quadratic Trinomials

Simplify .

Strategies:
  • Numerator: Difference of Squares
  • Denominator: Reverse Foil (Factor Sum/Product)
Cancel the terms.
Excluded:
Level: Advanced

Example 3: The "Opposite" Trick

Simplify .

Factor the denominator:
Notice: and are almost the same, but the signs are flipped. They are opposites.
.
We factor out a -1 from the denominator to make them match:
Now cancel :
Final Answer

Common Pitfalls

  • Illegal Cancellation:

    Students see and cross out the s to get 5.
    This is wrong because is a TERM, not a FACTOR. It is "married" to the +5.

  • Forgetting the 1:

    When everything cancels out (like ), the answer is 1, not 0!

Real-Life Applications

Photography: Lens Equation

The relationship between focal length (), object distance (), and image distance () is given by .

To solve for , we must combine the right side into a single rational expression:

Then flip both sides: .

Practice Quiz

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