Partial Fraction Decomposition: Key Concepts and Repeated Factors

Partial Fraction Decomposition: Key Concepts and Repeated Factors

• The topic is partial fraction decomposition for a rational function f(x)/g(x).
• Key premise: If the degree of the numerator is less than the degree of the denominator, i.e., $\deg f < \deg g$, the rational function is proper and can be decomposed.
• Denominator g(x) is factored into linear factors and/or irreducible quadratics, possibly with repetition.

Basic Decomposition Forms

• For distinct linear factors (x - ai), the decomposition has a term of the form $\frac{A</em>i}{x - a<em>i}$ for each distinct root ai.
• For a repeated linear factor (x - a)^k, include k terms: $\frac{A<em>1}{x - a} + \frac{A</em>2}{(x - a)^2} + \cdots + \frac{A_k}{(x - a)^k}$.
• For an irreducible quadratic factor qj(x) of degree 2, include a term of the form $\frac{B</em>j x + C<em>j}{q</em>j(x)}$.
• For a repeated irreducible quadratic factor, include terms up to the power mj: $\frac{B</em>{j}^{(1)} x + C<em>{j}^{(1)}}{q</em>j(x)^{1}} + \frac{B<em>{j}^{(2)} x + C</em>{j}^{(2)}}{q<em>j(x)^{2}} + \cdots + \frac{B</em>{j}^{(m<em>j)} x + C</em>{j}^{(m<em>j)}}{q</em>j(x)^{m_j}}.$

Method to Determine Coefficients

• Steps:

  1. Ensure the fraction is proper: i.e., $\deg f < \deg g$.
  2. Factor the denominator: $g(x) = \prod (x - a<em>i)^{k</em>i} \cdot \prod q<em>j(x)^{m</em>j}$ where q_j are irreducible quadratics.
  3. Set up the decomposition with unknown constants: for each distinct linear factor add Ai/(x-ai), for each repeated factor add A1/(x-a)^1 + … + Ak/(x-a)^k, and for each irreducible quadratic add (Bj x + Cj)/qj(x)^{mj} terms.
  4. Multiply both sides by the full denominator g(x) to obtain an identity in x.
  5. Solve for the unknowns Ai, Bj, C_j by either:
  • the cover-up method for simple linear factors (if g(x) contains a simple factor (x - a): A_i = f(a)/g'(a)) or
  • equating coefficients of powers of x, or
  • substituting convenient x values (and, for repeated factors, using several x-values or derivatives).

• The cover-up method: If g(x) contains a simple factor (x - a), and g(x) = (x - a) h(x) with h(a) ≠ 0, then the corresponding coefficient is $A = \frac{f(a)}{h(a)}$.

Quick Worked Examples

• Example 1: Distinct linear factors

  • Decompose: $\frac{f(x)}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}$ with $f(x) = 3x + 5, g(x) = (x - 2)(x + 1) = x^2 - x - 2$.
  • Multiply through: $3x + 5 = A(x + 1) + B(x - 2)$.
  • Solve: A + B = 3 and A - 2B = 5 → B = -\tfrac{2}{3}, A = \tfrac{11}{3}.
  • Result: $\frac{3x + 5}{(x - 2)(x + 1)} = \frac{11}{3}\cdot\frac{1}{x - 2} - \frac{2}{3}\cdot\frac{1}{x + 1}$.

• Example 2: Repeated linear factor

  • Decompose: $\frac{f(x)}{(x - 1)^2} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2}$ with a simple choice: let $f(x) = 3x + 2, g(x) = (x - 1)^2$.
  • Multiply through: $3x + 2 = A(x - 1) + B$.
  • Plug x = 1: $3(1) + 2 = B \Rightarrow B = 5.$
  • Compare coefficients: A = 3 (since coefficient of x is A and must equal 3).
  • Result: $\frac{3x + 2}{(x - 1)^2} = \frac{3}{x - 1} + \frac{5}{(x - 1)^2}$.

• Example 3: Mixed repeated and simple factors (brief outline)

  • Suppose $\frac{f(x)}{(x - 1)^2 (x + 3)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 3}$.
  • Multiply by the denominator: $f(x) = A(x - 1)(x + 3) + B(x + 3) + C(x - 1)^2.$
  • Use x = 1 to get B: $f(1) = B(1 + 3) \Rightarrow B = f(1)/4.$
  • Use x = -3 to get C: $f(-3) = C(-4)^2 \Rightarrow C = f(-3)/16.$
  • Use a third value to find A, or compare coefficients to solve for A.

Trigonometry Context (from transcript)

• Values mentioned: $\tan\left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}}, \quad \sec\left( \frac{\pi}{6} \right) = \frac{2}{\sqrt{3}}.$ (notes from the example in the transcript)
• These are tangential notes; not required for partial fractions but appear in the transcript as numeric references.

Connections to Foundations and Real-World Relevance

• Connects to polynomial long division, factorization, and the idea of expressing a rational function as a sum of simpler fractions.
• Foundational for integration of rational functions: integration of A/(x-a) and (Bx+C)/(quadratic) forms.
• Practical: solving systems of linear equations to find coefficients; using substitution/cover-up, coefficient comparison.
• Ethical/practical implications: careful bookkeeping to avoid mistakes; ensure uniqueness of decomposition for a given f/g with deg f < deg g.

Formulas and Key Equations

• Proper fraction condition: $\deg f < \deg g.$
• Denominator factorization: $g(x) = \prod<em>i (x - a</em>i)^{k<em>i} \cdot \prod</em>j q<em>j(x)^{m</em>j},$ where q_j are irreducible quadratics.
• Decomposition form (summary):
  • Distinct linear: $\sum<em>i \frac{A</em>i}{x - a_i}.$
  • Repeated linear: $\sum<em>i \sum</em>{t=1}^{k<em>i} \frac{A</em>{i,t}}{(x - a_i)^t}.$
  • Irreducible quadratics: $\sum<em>j \frac{B</em>j x + C<em>j}{q</em>j(x)},$ and for repeats, add higher powers of q_j in the denominator.
• Coefficient determination: multiply through by g(x) and solve for unknowns by substitution or comparing coefficients.
• Cover-up method (simple factor): if g(x) contains (x - a) as a simple factor, then $A = \frac{f(a)}{h(a)},$ where $g(x) = (x - a)h(x).$

End Notes

• This note is designed to mirror a comprehensive study guide replacing the original source, with step-by-step procedures, worked examples, and key formulas.