Partial Fractions

A rational fraction is an algebraic expression of the form $N(x)/D(x)$, where $N(x)$ and $D(x)$ are polynomials and $D(x) \neq 0$. In board examinations, fractions are strictly categorized into proper fractions (degree of numerator is less than the degree of denominator, $\text{deg}(N) < \text{deg}(D)$) and improper fractions ($\text{deg}(N) \ge \text{deg}(D)$). Remember that partial fraction resolution can only be applied directly to proper fractions. If you encounter an improper fraction in the exam, you must first use long division to convert it into the sum of a polynomial and a proper fraction before proceeding.

To resolve a proper rational fraction into partial fractions, the form of the expansion depends entirely on the factorization of the denominator $D(x)$. You must memorize the standard forms tested in BISE exams:

1. Non-repeated linear factors: Each distinct linear factor $(ax + b)$ contributes a single partial fraction of the form $\frac{A}{ax + b}$.
2. Repeated linear factors: A linear factor repeated $n$ times, $(ax + b)^n$, generates $n$ partial fractions: $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_n}{(ax + b)^n}$.
3. Non-repeated quadratic factors: An irreducible quadratic factor $(ax^2 + bx + c)$ contributes a term with a linear numerator: $\frac{Ax + B}{ax^2 + bx + c}$.
4. Repeated quadratic factors: A repeated irreducible quadratic factor $(ax^2 + bx + c)^n$ expands into terms with increasing powers in the denominator, each having a linear numerator.

To solve for the unknown constants ($A, B, C$, etc.), two primary methods are used in board problem-solving: the Substitution Method (substituting convenient real roots of the denominator to quickly isolate constants) and the Comparing Coefficients Method (equating the coefficients of like powers of $x$ on both sides to form a system of simultaneous linear equations). Mastering both techniques ensures accuracy and saves valuable time during the mathematics paper.