Partial Fractions & Equations – Complete Guide

This comprehensive guide covers all topics from the textbook chapter on Partial Fractions and Equations. Each topic includes key concepts and memorization tips to help you master the material.

Partial Fractions

Definition: Expressing a single rational function as a sum of two or more simpler rational functions.

Example:

1/(x-1)(x-2) = A/(x-1) + B/(x-2)

(x+3)/(x²-1) = A/(x-1) + B/(x+1)

Memorization Tip

Partial fractions “split” complex fractions into simpler parts that are easier to integrate or manipulate. Think of it as “fraction decomposition”.

Types of Equations

Two main types:

Conditional Equation: True only for specific values of the variable
Identity: True for ALL values of the variable

Examples:

Conditional: x² – 5x + 6 = 0 (true only for x=2,3)

Identity: (a+b)² = a² + 2ab + b² (true for all a,b)

Quick Tip

Identities use ≡ symbol or =, and partial fractions are identities since they hold for all x (except where denominator is zero).

Rational Fractions

Definition: Quotient of two polynomials P(x)/Q(x) with no common factors.

Two types:

Proper: Degree(P) < Degree(Q)
Improper: Degree(P) ≥ Degree(Q)

Examples:

Proper: (x+1)/(x²+2x+3) [degree 1 < degree 2]

Improper: (x³+1)/(x²+1) [degree 3 ≥ degree 2]

Memory Aid

For improper fractions: Divide first to get Polynomial + Proper Fraction. Always convert improper to proper before partial fractions.

Linear Factors (Proper Fractions)

For distinct linear factors in denominator:

Formula:

P(x)/[(x-a)(x-b)] = A/(x-a) + B/(x-b)

Method: Multiply through by denominator, then substitute convenient x-values or compare coefficients.

Technique

Use the “cover-up” method: To find A, cover (x-a) in original, substitute x=a in what remains. Repeat for other constants.

Repeated Linear Factors

For repeated linear factors (x-a)ⁿ in denominator:

Formula:

P(x)/(x-a)ⁿ = A₁/(x-a) + A₂/(x-a)² + … + Aₙ/(x-a)ⁿ

Each power gets its own term in the partial fraction decomposition.

Remember

Don’t forget all the intermediate powers! For (x-a)³, you need terms for 1/(x-a), 1/(x-a)², AND 1/(x-a)³.

Quadratic Factors (Proper Fractions)

For irreducible quadratic factors in denominator:

Formula:

P(x)/[(x²+px+q)(x-a)] = (Ax+B)/(x²+px+q) + C/(x-a)

Quadratic factors get numerator of form Ax+B (linear), not just a constant.

Key Point

Check if quadratic is factorable first! Only use Ax+B form for irreducible quadratics (discriminant < 0).

Repeated Quadratic Factors

For repeated irreducible quadratic factors (x²+px+q)ⁿ:

Formula:

P(x)/(x²+px+q)ⁿ = (A₁x+B₁)/(x²+px+q) + (A₂x+B₂)/(x²+px+q)² + … + (Aₙx+Bₙ)/(x²+px+q)ⁿ

Memory Aid

Similar to repeated linear factors but with linear numerators (Ax+B) for each term instead of just constants.

Important Notes

Key points to remember:

Partial fractions are identities (true for all x)
Improper fraction = Polynomial + Proper fraction
Always check if fraction is proper before partial fractions
Identity can be shown by ≡ or =
Conditional equations are true only for specific values

Common Mistake

Forgetting to convert improper fractions to proper fractions first! Always check degrees of numerator and denominator.

Study Guidelines for Students

How to Master Partial Fractions

Understand the difference: Know the distinction between identities (always true) and conditional equations (true for specific values only).
Check fraction type first: Always determine if a rational function is proper or improper before attempting partial fractions.
Memorize the forms: Learn the different partial fraction forms for linear factors, repeated factors, quadratic factors, and repeated quadratic factors.
Practice the cover-up method: For distinct linear factors, the cover-up method is the fastest way to find constants.
Use convenient values: When solving for constants, substitute values that make some terms zero to simplify equations.
Compare coefficients: When convenient values don’t work, compare coefficients of like terms on both sides.
Verify your answer: Always combine your partial fractions back together to check if you get the original fraction.
Handle improper fractions: Remember: Improper fraction = Polynomial + Proper fraction. Always do polynomial division first if needed.
Recognize irreducible quadratics: Check discriminant to determine if a quadratic factor can be broken into linear factors.
Practice with various denominators: Work problems with different combinations of factor types.

Step-by-Step Process

For Partial Fraction Decomposition:

Check if fraction is proper (degree numerator < degree denominator)
If improper, do polynomial division first
Factor the denominator completely
Set up the partial fraction form based on factor types
Multiply through by the common denominator
Solve for the unknown constants (use substitution or coefficient comparison)
Write the final partial fraction decomposition
Verify by combining back to original

Common Mistakes to Avoid

1. Not checking if fraction is proper first
2. Forgetting terms for repeated factors
3. Using constant numerator for quadratic factors (should be Ax+B)
4. Not fully factoring the denominator
5. Algebraic errors when solving for constants
6. Forgetting to include the polynomial part from improper fractions

Partial Fractions & Equations – Complete Guide