Partial fractions are a mathematical technique used to decompose a complex rational function into smaller, simpler fractions.
Partial fractions are commonly used in integration problems, where they can simplify the integration process. They can also be used in solving linear differential equations and in finding the inverse Laplace transform.
To solve using partial fractions, you must first factor the denominator of the rational function into linear and irreducible quadratic factors. Then, you set up a system of equations using the coefficients of the fractions and solve for the unknowns.
Yes, there are a few restrictions to keep in mind when using partial fractions. The factors in the denominator must be distinct, the degree of the numerator must be less than the degree of the denominator, and all the coefficients must be real numbers.
Yes, partial fractions can also be used for improper rational functions, where the degree of the numerator is greater than or equal to the degree of the denominator. In these cases, the partial fractions will include a polynomial term in addition to the simpler fractions.