Final answer:
Partial fraction decomposition is a technique used to decompose rational functions into simpler fractions. It is applicable when the denominator can be factored into linear and irreducible quadratic factors.
Step-by-step explanation:
Partial fraction decomposition is a technique used to decompose a rational function into simpler fractions. It is typically used to integrate rational functions, where the degree of the numerator is less than the degree of the denominator. This technique is applicable when the denominator can be factored into linear and irreducible quadratic factors.
For example, consider the rational function (3x^2 + 2x + 5) / ((x + 1)(x - 2)). We can decompose it into two partial fractions: A / (x + 1) + B / (x - 2), where A and B are constants. By finding the values of A and B, we can integrate the function.
Another example is the rational function (x^3 + 2x^2 + x + 3) / ((x - 1)^2(x + 2)). In this case, the denominator can be factored into (x - 1)^2 and (x + 2). The partial fraction decomposition would be A / (x - 1) + B / (x - 1)^2 + C / (x + 2), where A, B, and C are constants.