The Main Idea

• Concept Overview
  • Reverses rational expression addition
  • Breaks down complex fraction into simpler parts
  • Each part has linear denominator
  • Must have distinct linear factors
  • Degree of numerator < degree of denominator
• General Form For [latex]\frac{P(x)}{Q(x)}[/latex] where [latex]Q(x)[/latex] has distinct linear factors: [latex]\frac{P(x)}{(a_1x+b_1)(a_2x+b_2)...(a_nx+b_n)} = \frac{A_1}{a_1x+b_1} + \frac{A_2}{a_2x+b_2} + ... + \frac{A_n}{a_nx+b_n}[/latex]
• Solution Methods
  A. Standard Method:
  • Assign variables to numerators
  • Multiply all terms by common denominator
  • Collect like terms
  • Solve system of equations

  B. Heaviside Method:

  • Substitute values making terms zero
  • Solve for constants one at a time
  • More efficient for some problems
• Key Requirements
  • Linear factors must be distinct
  • Numerator degree < denominator degree
  • All denominators must factor completely
  • System must have unique solution

The Main Idea

• Structure for Repeated Factors
  • Each repeated factor appears multiple times
  • Powers increase from 1 to n (number of repetitions)
  • General form: [latex]\frac{P(x)}{Q(x)} = \frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \frac{A_3}{(ax+b)^3} + ... + \frac{A_n}{(ax+b)^n}[/latex]
• Key Requirements
  • Must identify repeated factors in denominator
  • Numerator degree < denominator degree
  • Each factor gets unique variable
  • Powers must be in increasing order
• Solution Process
  • Factor denominator completely
  • Set up partial fractions with all powers
  • Multiply by common denominator
  • Solve system of equations
• Verification
  • Add all fractions
  • Should match original expression
  • Check power requirements
  • Verify all terms included

The Main Idea

• Key Differences from Linear Factors
  • Quadratic factor cannot be factored further
  • Numerator must be linear ([latex]Ax + B[/latex])
  • Each quadratic factor gets unique linear numerator
  • Can mix with linear factors
• General Form For [latex]P(x)/Q(x)[/latex] with irreducible quadratic factors: [latex]\frac{P(x)}{Q(x)} = \frac{A}{ax+b} + \frac{A_1x + B_1}{a_1x^2 + b_1x + c_1} + \frac{A_2x + B_2}{a_2x^2 + b_2x + c_2} + ...[/latex]
• Identification Strategy
  • Check if quadratic factors are irreducible
  • Identify any linear factors
  • Use constants for linear factors
  • Use linear expressions for quadratic factors
• Multiple Solution Methods
  • Complete system of equations
  • Strategic substitution method
  • Combination of both methods
  • Choose most efficient approach

The Main Idea

• Structure for Repeated Quadratics
  • Each quadratic factor appears multiple times
  • Powers increase from [latex]1[/latex] to [latex]n[/latex]
  • Each factor needs linear numerator ([latex]Ax + B[/latex])
  • General form: [latex]\frac{P(x)}{(ax^2+bx+c)^n} = \frac{A_1x + B_1}{ax^2+bx+c} + \frac{A_2x + B_2}{(ax^2+bx+c)^2} + ... + \frac{A_nx + B_n}{(ax^2+bx+c)^n}[/latex]
• Key Requirements
  • Quadratic must be irreducible
  • Each power gets unique linear numerator
  • Powers must be in increasing order
  • Can mix with linear factors
• Numerator Forms
  • Linear factors: constant numerator ([latex]A[/latex])
  • Quadratic factors: linear numerator ([latex]Ax + B[/latex])
  • Each power needs new variables
  • Degree of [latex]P(x) <[/latex] degree of [latex]Q(x)[/latex]
• Solution Process
  • Factor denominator completely
  • Write decomposition with all powers
  • Multiply by common denominator
  • Solve system of equations