Lesson 17.3

Partial Fraction Decomposition (Basic)

Breaking apart complex fractions into a sum of simpler ones—the reverse of adding fractions.

Introduction

Breaking apart complex fractions into a sum of simpler ones—the reverse of adding fractions.

Past Knowledge

You can add fractions by finding a common denominator and factor polynomials.

Today's Goal

We reverse the process—splitting one fraction into a sum of simpler ones.

Future Success

In Calculus, this technique makes integration of rational functions possible.

Key Concepts

The Goal

Find the constants and

Rule for Distinct Linear Factors

The Cover-Up Method (Heaviside)

1

To find , "cover" and substitute

2

Evaluate the remaining fraction at that value

3

Repeat for each factor

Worked Examples

Example 1: Two Linear Factors (Basic)

Decompose:

Step 1: Set up the form

Step 2: Multiply both sides by LCD

Step 3: Use strategic substitution

:

:

Example 2: Three Linear Factors (Intermediate)

Decompose:

Setup

Cover-up method

:

:

:

Example 3: Improper Fraction (Advanced)

Decompose:

Step 1: Check degrees

Numerator degree (2) ≥ denominator degree (2) → IMPROPER. Must divide first!

Step 2: Long division

Step 3: Decompose the proper part

Common Pitfalls

Forgetting to check for improper fractions

If degree of numerator ≥ denominator, you MUST divide first.

Sign errors with negative roots

When substituting into , you get , not .

Not factoring the denominator completely

You must factor fully before setting up partial fractions.

Real-World Application

Integration in Calculus

In Calculus, you'll need to integrate rational functions. Partial fractions turn one hard integral into several easy ones:

Practice Quiz

Practice Quiz

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