Partial Fraction Decomposition Lecture Notes

Introduction to Partial Fraction Decomposition

• Definition: Partial fraction decomposition is the process of breaking a single fraction into multiple smaller fractions.
• Example: Combine ( \frac{2}{3x} + \frac{4}{5y} ) into one fraction.
  • Get a common denominator: ( \frac{10y}{15xy} + \frac{12x}{15xy} = \frac{10y + 12x}{15xy} )
  • Partial fraction decomposition does the reverse, breaking one fraction into multiple fractions.

Basic Concepts

• Rational Function Example: ( \frac{7x - 23}{x^2 - 7x + 10} )
  • Factor the denominator: ( x^2 - 7x + 10 = (x - 2)(x - 5) )
• Linear and Quadratic Factors:
  • Linear factor examples: ( x + 2, 3x - 5 )
  • Quadratic factor examples: ( x^2 + 8x + 3 )
• Repeated Factors:
  • Repeated linear factor: ( x^2 )
  • Repeated quadratic factor: ( (x^2 + 1)^2 )

Decomposition Process

1. Factor Denominator Completely: Write in terms of linear and quadratic factors.
2. Set Up Partial Fractions:
   • Example: ( \frac{7x - 23}{(x - 2)(x - 5)} ) becomes ( \frac{A}{x - 2} + \frac{B}{x - 5} )
3. Solving for Constants (A and B):
   • Multiply both sides by the common denominator.
   • Plug x-values to solve for A and B:
     • Use simple values that eliminate terms.
     • Example values: ( x = 5 ) and ( x = 2 )
   • Result: ( A = 3, B = 4 )

Verification

• Combine fractions back to check against the original:
  • Combine ( \frac{3}{x - 2} + \frac{4}{x - 5} )
  • Should result in ( \frac{7x - 23}{(x - 2)(x - 5)} )

Practice Problem

• Problem: Decompose ( \frac{29 - 3x}{x^2 - x - 6} )
  • Factor denominator: ( x^2 - x - 6 = (x - 3)(x + 2) )
  • Set up: ( \frac{A}{x - 3} + \frac{B}{x + 2} )
  • Solve by plugging suitable x-values:
    • ( x = 3 ) and ( x = -2 )
    • Results: ( A = 4, B = -7 )
• Final Answer: ( \frac{4}{x - 3} - \frac{7}{x + 2} )