Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear)

Partial fractions is a technique used to break down a complex rational function into a sum of simpler fractions. This is particularly useful in calculus for integration, in solving differential equations, and in simplifying algebraic expressions. In the Edexcel A-Level Mathematics syllabus, you are required to decompose rational functions where the denominator is a product of linear factors (including repeated linear factors) and quadratic factors that do not factorise further, provided the numerator is of a lower degree than the denominator. The denominators you will encounter are not more complicated than squared linear terms (e.g., (x+1)^2) and you will handle at most three terms in the decomposition.

Mastering partial fractions is essential because it allows you to integrate rational functions that would otherwise be difficult or impossible to integrate directly. It also appears in further mathematics topics such as differential equations and series expansions. The skill requires a solid understanding of algebraic manipulation, solving linear equations, and handling polynomial division when necessary. In exams, you will typically be asked to express a given rational function in partial fractions, often as a stepping stone to integration or binomial expansion.

The process involves factoring the denominator completely, then writing the rational function as a sum of fractions with unknown constants in the numerators. For each linear factor (ax+b), you include a term of the form A/(ax+b). For a repeated linear factor (ax+b)^2, you include two terms: A/(ax+b) + B/(ax+b)^2. For an irreducible quadratic factor (ax^2+bx+c), you include a term of the form (Ax+B)/(ax^2+bx+c). You then multiply through by the denominator to clear fractions and solve for the constants by equating coefficients or substituting suitable values of x.