Yes, partial fractions can be applied to rational functions with complex roots in the denominator. However, in most standard calculus courses, the focus is on real coefficients. When dealing with complex roots, they always appear in conjugate pairs. The decomposition will involve terms with both the complex root and its conjugate. While the algebra can be more challenging with complex numbers, the fundamental principles of partial fractions remain the same.

When the denominator has repeated roots, the decomposition becomes slightly more involved. For a repeated linear factor of the form (ax + b)ⁿ, the decomposition will have a term for each power of the factor up to n. This means that the rational function will be expressed as a sum of fractions with denominators (ax + b), (ax + b)², ..., (ax + b)ⁿ. Each of these terms will have its coefficient, which needs to be determined. The presence of repeated roots requires additional terms in the decomposition, making the process more intricate.

In higher-level maths, particularly in differential equations and control systems, the Laplace Transform is a powerful tool. When finding the inverse Laplace Transform of a function, the method of partial fractions is often employed to break down complex rational functions into simpler terms. This decomposition makes it easier to identify and apply known inverse transforms. Thus, the technique of partial fractions serves as a bridge between algebraic manipulations and the application of Laplace Transforms in solving real-world mathematical problems.

While the basic principles of partial fractions remain consistent, there are some strategies that can make the process more efficient. One common method is to use the cover-up rule for distinct linear factors. This involves covering up the factor in the original fraction and substituting the root of the covered factor into the remaining expression to find the coefficient. Another strategy is to equate coefficients of like terms on both sides, which can sometimes reduce the amount of algebraic manipulation required. However, it's essential to understand the underlying principles before relying on shortcuts.

Partial fractions decomposition is specifically designed for "proper" rational functions, where the degree of the numerator is less than the degree of the denominator. If the rational function is "improper" (the degree of the numerator is greater than or equal to the degree of the denominator), it must first be divided to express it as a polynomial plus a proper rational function. Only then can the proper rational function be decomposed using partial fractions. This limitation ensures that the decomposition process is manageable and results in simpler fractions that are easier to work with.