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Partial Fraction Decomposition

Breaking down rational expressions into simpler forms

Introduction

  • Partial fraction decomposition breaks down rational expressions
  • Involves breaking down into simpler forms
  • Techniques and principles are used
  • Historical background of Leonard Eu

Classification of Rational Expressions

  • Proper vs improper rational expressions
  • Proper: degree of numerator < degree of denominator
  • Improper: degree of numerator >= degree of denominator
  • Rules for decomposition differ based on classification

Partial Fraction Decomposition - Linear Factors

  • Decomposing rational expressions with linear factors in the denominator
  • Assume constants (a, b, c, etc.) for each factor
  • Use algebraic manipulation and smart substitutions
  • Solve for the values of the constants

Partial Fraction Decomposition - Repeated Linear Factors

  • Decomposing rational expressions with repeated linear factors in the denominator
  • Use the same approach as linear factors
  • Assign a separate constant for each repetition
  • Solve for the values of the constants

Partial Fraction Decomposition - Quadratic Factors

  • Decomposing rational expressions with quadratic factors
  • Quadratic factors that cannot be factored/reduced
  • No real roots, cannot solve for values
  • Express as irreducible quadratics in the form (ax^2 + bx + c)/(dx^2 + ex + f)

Improper Rational Expressions

  • Dealing with improper rational expressions
  • Perform long division to obtain a quotient and remainder
  • The quotient becomes a separate term in the partial fraction decomposition
  • Apply decomposition to the remainder

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