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{Ai}{x - ai} for each distinct root ai.
For a repeated linear factor (x - a)^k, include k terms: \frac{A1}{x - a} + \frac{A2}{(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{Bj x + Cj}{qj(x)}.
For a repeated irreducible quadratic factor, include terms up to the power mj: \frac{B{j}^{(1)} x + C{j}^{(1)}}{qj(x)^{1}} + \frac{B{j}^{(2)} x + C{j}^{(2)}}{qj(x)^{2}} + \cdots + \frac{B{j}^{(mj)} x + C{j}^{(mj)}}{qj(x)^{m_j}}.

Steps:
1. Ensure the fraction is proper: i.e., \deg f < \deg g.
2. Factor the denominator: g(x) = \prod (x - ai)^{ki} \cdot \prod qj(x)^{mj} 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)}.

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.

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.

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.

Proper fraction condition: \deg f < \deg g.
Denominator factorization: g(x) = \prodi (x - ai)^{ki} \cdot \prodj qj(x)^{mj}, where q_j are irreducible quadratics.
Decomposition form (summary):
- Distinct linear: \sumi \frac{Ai}{x - a_i}.
- Repeated linear: \sumi \sum{t=1}^{ki} \frac{A{i,t}}{(x - a_i)^t}.
- Irreducible quadratics: \sumj \frac{Bj x + Cj}{qj(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).

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