Problem 54 - College Algebra

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

The process of decomposing a rational expression into partial fractions involves factoring the denominator, setting up a system of equations to solve for the variables in the numerators, and then using algebraic manipulations to solve the system.

Step 1 Factor the Denominator

To start with, factor the denominator into distinct linear terms.

Step 2 Set Up the Partial Fractions

After factoring the denominator, rewrite the rational expression as a sum of partial fractions, where each partial fraction has one of the linear terms in the denominator and an undetermined coefficient in the numerator.

Step 3 Clear the Fractions

Multiply both sides of the equation by the denominator of the original rational expression to get an equation without fractions.

Step 4 Set Up a System of Equations

Set the coefficients of like terms equal to each other to obtain a system of equations.

Step 5 Solve the System of Equations

Use one of several techniques (like substitution or the method of elimination) to solve the system of equations, which will yield values for the coefficients in the numerators of the partial fractions.

Step 6 Write the Partial Fraction Decomposition

Using the solved values, write down your final partial fraction decomposition.