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Partial fractions is a method of breaking down a complex rational function into simpler fractions. It involves decomposing the function into smaller fractions with known denominators and coefficients.
Partial fractions are often used in integration, where it is necessary to break down a complex function into simpler fractions in order to integrate it. It is also used in algebraic manipulation and solving differential equations.
The following are the basic steps for resolving a rational function into partial fractions:1. Factor the denominator of the rational function.2. Decompose the function into simpler fractions with unknown coefficients.3. Set up and solve a system of equations to find the unknown coefficients.4. Write the partial fraction decomposition of the original function.
To confirm the correctness of a partial fraction decomposition, we can either:1. Add the decomposed fractions back together and see if it simplifies back to the original function.2. Substitute values for the independent variable into the original function and the partial fraction decomposition and compare the results.
Some common mistakes to avoid when working with partial fractions are:1. Making errors while factoring the denominator.2. Inconsistencies or errors in setting up and solving the system of equations.3. Forgetting to include all necessary terms in the partial fraction decomposition.4. Not checking the correctness of the decomposition before using it in further calculations.
