Partial fraction decomposition is a method used to break down a rational function into simpler fractions. It involves representing the function as a sum of simpler fractions with distinct denominators.
Partial fraction decomposition is often used when integrating rational functions, as it can simplify the integration process. It is also used in solving systems of linear equations and in solving differential equations.
Partial fraction decomposition involves finding the partial fraction form of a rational function by equating its coefficients to the coefficients of simpler fractions. The coefficients can be found by using algebraic manipulation or by using the method of undetermined coefficients.
In proper partial fractions, the degree of the numerator is less than the degree of the denominator. In improper partial fractions, the degree of the numerator is equal to or greater than the degree of the denominator. Improper partial fractions can be converted to proper fractions by long division before performing partial fraction decomposition.
Partial fraction decomposition can only be used for rational functions, where the numerator and denominator are both polynomials. It cannot be used for functions with non-polynomial terms, such as trigonometric or logarithmic functions.