Problem 54 - Algebra and Trigonometry

Question

Explain how to find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator.

Step-by-Step Solution

Answer

To find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator, first express the rational expression as a proper fraction, then factorize the denominator, set up the partial fractions and solve for the constants in the numerators of the partial fractions.

Step 1: Express the Fraction as a Proper Fraction

The first step in finding the partial fraction decomposition of a given rational expression is to express it as a proper fraction if it's not one already. If needed, divide the numerator by the denominator to get a polynomial, plus a proper fraction.

Step 2: Factor the Denominator

Factor the denominator into its simplest forms. This may include linear factors, irreducible quadratic factors or other more complex forms. In this case, only distinct linear factors are mentioned so only those need to be considered.

Step 3: Set Up Partial Fractions

The next step is to write the original expression as a sum of simpler fractions. Each fraction (or 'partial fraction') has one part of the factored denominator and an undetermined numerator. Each of these numerators are typically represented by letters, such as A, B, C, etc.

Step 4: Solve for the Constants

Equating the numerators of both sides, create a system of linear equations and solve for the undetermined coefficients A, B, C, etc.