Partial Fractions

A fraction is defined as the indicated quotient of two polynomials, expressed in the algebraic form $\frac{N(x)}{D(x)}$, where $D(x) \neq 0$. Before resolving any algebraic fraction into partial fractions, it is vital to determine whether it is proper or improper. A proper fraction has the degree of the numerator $N(x)$ strictly less than the degree of the denominator $D(x)$ (i.e., $\text{deg}(N) < \text{deg}(D)$). Conversely, an improper fraction has the degree of the numerator greater than or equal to the degree of the denominator ($\text{deg}(N) \ge \text{deg}(D)$). For BISE board exams, always remember that partial fraction resolution is only directly applicable to proper fractions. If you encounter an improper fraction, you must first use algebraic long division to convert it into the sum of a polynomial and a proper fraction before proceeding.

To score full marks in the short and long questions of your board exam, you must memorize the four standard forms of partial fractions based on the factorization of the denominator $D(x)$:

1. Non-repeated Linear Factors: When $D(x)$ consists of distinct linear terms like $(ax + b)$, each factor gets a constant numerator, e.g., $\frac{A}{ax + b}$.
2. Repeated Linear Factors: When a linear factor is repeated $n$ times, such as $(ax + b)^n$, it generates a sum of $n$ fractions with increasing powers in the denominators, e.g., $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_n}{(ax + b)^n}$.
3. Non-repeated Quadratic Factors: When $D(x)$ contains irreducible quadratic factors like $(ax^2 + bx + c)$, the corresponding partial fraction takes a linear expression in the numerator, e.g., $\frac{Ax + B}{ax^2 + bx + c}$.
4. Repeated Quadratic Factors: For repeated quadratic factors such as $(ax^2 + bx + c)^2$, the terms expand similarly with linear numerators, e.g., $\frac{Ax + B}{ax^2 + bx + c} + \frac{Cx + D}{(ax^2 + bx + c)^2}$.

When attempting board exam questions, the standard methodology involves clearing the denominators by multiplying the entire equation by the LCD (Least Common Denominator), resulting in an identity equation. To find the unknown constants ($A, B, C$, etc.), examiners look for two primary techniques: the Substitution Method (substituting suitable real values of $x$, usually the roots obtained by setting linear factors to zero) and the Coefficient Comparison Method (expanding both sides, grouping powers of $x$, and equating the coefficients of like powers on both sides). Long questions in the board exams frequently test either repeated linear factors or quadratic factors, making accuracy in simultaneous linear equations crucial for the final steps.