Partial fraction decomposition is a mathematical technique used to break down a rational function into simpler fractions. It involves determining the individual fractions that, when combined, make up the original function.
Partial fraction decomposition is commonly used in integration, as it can simplify the integration process by breaking down complex fractions into more manageable ones. It is also used in solving systems of linear equations and in some engineering and physics applications.
The process of partial fraction decomposition involves first factoring the denominator of the rational function. Then, for each distinct factor, a fraction with that factor as its denominator is written. The numerator of each fraction is determined by solving a system of equations using the original function and the fractions written so far.
There are two types of partial fraction decomposition: proper and improper. In proper decomposition, the degrees of the numerators are all less than the degree of the denominator. In improper decomposition, at least one of the numerators has a degree equal to or greater than the degree of the denominator.
One tip is to always start by factoring the denominator completely before writing the fractions. Another is to keep track of the constants in the numerators while solving the system of equations to avoid mistakes. It is also important to check the final solution by adding the fractions back together to make sure it equals the original function.