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Partial Fraction Decomposition

Definition

Partial fraction decomposition is the process of breaking down a complex rational expression (a ratio of polynomials) into a sum of simpler fractions, called partial fractions. It is the algebraic inverse of finding a common denominator.

Why It Matters

This is the algebraic equivalent of taking a complex machine apart to see how it works. In engineering (Laplace transforms) and calculus (integration), many “monolithic” expressions are impossible to process in their combined form. Decomposition is the “modularization” protocol that breaks these down into solvable, atomic units. Without this skill, you are stuck at the “surface” of complex rational expressions, unable to perform the deep operations required for control theory and advanced physics.

Core Concepts

  • Proper vs. Improper: Decomposition requires a proper rational expression (degree of numerator << degree of denominator). Use long division first if improper.
  • How to read: “The degree of the numerator is strictly less than the degree of the denominator.”
  • Meaning: The fraction must be “top-heavy” enough to split—otherwise divide first.
  • Case 1: Distinct Linear Factors: P(x)(x−a)(x−b)=Ax−a+Bx−b\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}.
  • How to read: “The polynomial P of x divided by the product of the quantities x minus a and x minus b is equal to the constant A divided by the quantity x minus a, plus the constant B divided by the quantity x minus b.”
  • Meaning / when to use: Two different linear factors in the denominator—one constant per factor.
  • Case 2: Repeated Linear Factors: P(x)(x−a)n=A1x−a+A2(x−a)2+⋯+An(x−a)n\frac{P(x)}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_n}{(x-a)^n}.
  • How to read: “The polynomial P divided by the quantity x minus a raised to the nth power is equal to the constant A one divided by the quantity x minus a, plus the constant A two divided by the quantity x minus a squared, continuing up to the constant A n divided by the quantity x minus a raised to the nth power.”
  • Meaning: A repeated root needs one term for each power up to nn.
  • Case 3: Irreducible Quadratic Factors: Factors like (x2+ax+b)(x^2 + ax + b) require a linear numerator: Ax+Bx2+ax+b\frac{Ax + B}{x^2 + ax + b}.
  • How to read: “The linear expression A times x plus B, all divided by the quadratic expression x squared plus a times x plus b.”
  • Meaning: Quadratic factors that don’t factor over reals need a linear (not constant) numerator.
  • Case 4: Repeated Irreducible Quadratic Factors: P(x)(x2+ax+b)n=A1x+B1x2+ax+b+A2x+B2(x2+ax+b)2+⋯+Anx+Bn(x2+ax+b)n\frac{P(x)}{(x^2 + ax + b)^n} = \frac{A_1x + B_1}{x^2 + ax + b} + \frac{A_2x + B_2}{(x^2 + ax + b)^2} + \dots + \frac{A_nx + B_n}{(x^2 + ax + b)^n}.
  • How to read: “The polynomial P of x divided by the irreducible quadratic x squared plus a times x plus b raised to the nth power is equal to a sum of terms, where each term has a linear numerator A k times x plus B k, divided by the quadratic raised to increasing powers up to n.”
  • Meaning: Combines Cases 2 and 3—repeated irreducible quadratics need escalating powers.

Connected Concepts

Connected notes