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Partial Fractions definitions

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  • Partial Fraction Decomposition
    A technique for expressing a rational function as a sum of simpler fractions, making integration and algebraic manipulation more manageable.
  • Rational Function
    An expression formed by dividing one polynomial by another, often requiring decomposition for integration.
  • Distinct Linear Factor
    A denominator component of the form (ax + b) that appears only once and is used to set up separate simple fractions.
  • Repeated Linear Factor
    A denominator component of the form (ax + b)^n, requiring a sum of fractions for each power from 1 to n.
  • Irreducible Quadratic Factor
    A quadratic denominator, such as (ax^2 + bx + c), that cannot be factored further over the reals and needs a linear numerator in decomposition.
  • Numerator
    The top part of a fraction, which may be a constant or a linear expression depending on the denominator's structure.
  • Denominator
    The bottom part of a fraction, whose factorization determines the form of the partial fractions.
  • System of Equations
    A set of simultaneous equations used to solve for unknown constants in the decomposition process.
  • Strategic Substitution
    A shortcut method where specific values for the variable are chosen to simplify equations and quickly solve for constants.
  • Common Denominator
    A shared multiple of all denominators, used to clear fractions and equate numerators during decomposition.
  • Improper Rational Function
    A rational function where the numerator's degree is greater than or equal to the denominator's, requiring polynomial division before decomposition.
  • Cover-Up Method
    A shortcut for finding constants in partial fractions by temporarily ignoring certain factors, also known as the Heaviside method.
  • Coefficient Comparison
    A process of matching coefficients of like powers on both sides of an equation to form a system for solving unknowns.
  • Integration
    The process of finding the antiderivative, often simplified by expressing complex rational functions as sums of basic fractions.
  • Polynomial Division
    An initial step for improper rational functions, dividing the numerator by the denominator to simplify before decomposition.