Problem 54 - College Algebra Essentials

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 partial fraction decomposition of a rational expression with distinct linear factors in the denominator involves factoring the denominator, setting up the partial fraction decomposition structure, equating it to the original fraction, solving for the unknowns, and then substitifying the resolved values back into the partial fraction structure.

Step 1: Factoring the denominator

Factoring the denominator of the rational expression or identify the distinct linear factors if the expression is already simplified. In general, any rational function will be in the form: \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials in \(x\). Factoring \(Q(x)\) will give distinct linear factors.

Step 2: Writing the partial fraction decomposition

Setting up the partial fraction decomposition involves writing the expression \(\frac{P(x)}{Q(x)}\) as the sum of simpler fractions, where each denominator factor is in the denominator of one of these fractions. The form for a decomposition for distinct linear factors in the denominator will be: \(\frac{P(x)}{Q(x)} = \frac{A}{a} + \frac{B}{b} + \frac{C}{c}+ ... + \frac{N}{z}\), depending on how many factors you have.

Step 3: Equating to Original Fraction

First, get rid of the fractions by multiplying through by the common denominator, \(Q(x)\): \( P(x) = AQ(x/a) + BQ(x/b) + CQ(x/c) + ... + NQ(x/z)\). Then, choose convenient values for \(x\) that will allow you to easily solve for the variables \(A,B,C,...,N\). These can usually be the roots of the linear factors in the denominator \(Q(x)\).

Step 4: Solving for the unknowns

Solve for the variables \(A,B,C,...,N\) through substituting the chosen \(x\) values into the equation from Step 3.

Step 5: Writing the Final Solution

Substitute the solved values of \(A, B, C, ..., N\) into partial fraction decomposition established in step 2. This will give the partial fraction decomposition of the rational expression.