Rational Functions and Partial Fractions

A rational function is defined as a function of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial. The domain of a rational function consists of all real numbers x for which Q(x) is not equal to zero. Understanding the domain is crucial as it identifies points where the function is undefined, often leading to asymptotes or holes.

Rational functions can be classified as proper or improper. A proper rational function has a numerator polynomial with a degree strictly less than the degree of the denominator polynomial (deg(P(x)) < deg(Q(x))). An improper rational function has a numerator polynomial with a degree greater than or equal to the degree of the denominator polynomial (deg(P(x)) >= deg(Q(x))). This distinction is important because improper rational functions often require polynomial long division before further analysis or decomposition.

Asymptotes are lines that the graph of a function approaches as it tends towards infinity. For rational functions, we primarily consider vertical and horizontal asymptotes.

Vertical Asymptotes occur at values of x where the denominator Q(x) is zero and the numerator P(x) is non-zero. If both P(x) and Q(x) are zero at a particular x-value, there might be a 'hole' in the graph instead of a vertical asymptote. To find vertical asymptotes, set the denominator equal to zero and solve for x, ensuring these values do not also make the numerator zero.

Horizontal Asymptotes describe the behavior of the function as x approaches positive or negative infinity. Their existence and location depend on the degrees of P(x) and Q(x):

• If deg(P(x)) < deg(Q(x)), the horizontal asymptote is y = 0.
• If deg(P(x)) = deg(Q(x)), the horizontal asymptote is y = (leading coefficient of P(x)) / (leading coefficient of Q(x)).
• If deg(P(x)) > deg(Q(x)), there is no horizontal asymptote (but there might be an oblique/slant asymptote, which is beyond the scope of basic A-Level but good to be aware of).

Partial fraction decomposition is a technique used to rewrite a complex rational function as a sum of simpler rational functions. This process is particularly useful in calculus for integration and in other areas for finding inverse Laplace transforms or series expansions. The core idea is to reverse the process of adding fractions.

Before applying partial fractions, it is crucial that the rational function is proper. If it is improper, you must first perform polynomial long division to express it as a polynomial plus a proper rational function. For example, if P(x)/Q(x) is improper, we write P(x)/Q(x) = S(x) + R(x)/Q(x), where S(x) is the quotient polynomial and R(x)/Q(x) is the proper remainder fraction.

The denominator Q(x) must be factorized into its simplest linear and irreducible quadratic factors. The form of the partial fraction decomposition depends entirely on the nature of these factors.