A partial fraction expansion is a method used in algebra to simplify a rational function by expressing it as a sum of simpler fractions. It is often used to solve integration problems and to find the inverse of a Laplace transform.
Partial fraction expansion is necessary when the degree of the numerator is greater than or equal to the degree of the denominator in a rational function. It is also used when the denominator can be factored into linear and quadratic factors.
To find the partial fraction expansion, you first need to factor the denominator of the rational function into linear and quadratic factors. Then, you set up a system of equations using the coefficients of the fractions and the original function. Finally, you solve for the unknown coefficients using algebraic methods.
Yes, partial fraction expansion can also be used for improper fractions. In this case, you first need to perform long division to convert the improper fraction into a polynomial plus a proper fraction. Then, you can use the same method as before to find the partial fraction expansion.
Partial fraction expansion has various applications in science, particularly in the fields of physics, engineering, and mathematics. It is used in solving differential equations, finding the inverse Laplace transform, and solving integration problems in calculus. It is also used in signal processing and control systems.