Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the expression. $$ \frac{x+1}{3(x-2)^{2}} $$

Step-by-Step Solution

Answer

So, the partial fraction decomposition of the given expression is: \( \frac{3}{x-2}+ \frac{7}{(x-2)^2}\)

Step 1: Identify the form of decomposition

The given fraction has a denominator containing a repeated linear factor \((x-2)^{2}\). Therefore our decomposition form will be: \( \frac{A}{x-2}+ \frac{B}{(x-2)^2}\) where A and B are coefficients we need to find.

Step 2: Clear out fractions

To make it easier, multiply each term of the equation by \((x-2)^2\) to clear out the fractions. You get: \(x + 1 = A*(x-2) + B\).

Step 3: Solve for A and B

Now, this equation must hold for all x. Pick appropriate x values to turn off the different terms and solve for A and B. Let x = 2, then we get A = 3. Replace A back into equation, we get: \(x + 1 = 3*(x-2) + B\). Now, let x = 0 to find B. This yields B = 7.