Problem 12 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{5 x-1}{(x-2)(x+1)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression is \( \frac{4}{x - 2} + \frac{3}{x + 1} \)

Step 1: Formulate the Expression for Decomposition

First, express the given rational expression as the sum of two fractions, each with denominator factors, i.e., \( \frac{5x - 1}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \). Here, \( A \) and \( B \) are constants we need to find.

Step 2: Setup and Solve Equations for Constants

Multiply through by the denominator to clear the fractions which gives: \( 5x - 1 = A(x + 1) + B(x - 2). \) This can be seen as systems of equations, solving for \(A\) and \(B\).

Step 3: Finding the values for A and B

Choose convenient values for \( x \) to make each term vanish in turn. Let's choose \( x = -1 \) which gives: \( A = 4 \). Then choose a value of \( x = 2 \) which results in: \( B = 3 \).

Step 4: Formulate the Final Decomposition

Substitute \( A \) and \( B \) back into the expression. This gives the partial fraction decomposition as: \( \frac{4}{x - 2} + \frac{3}{x + 1} \)