Question

Let \( x \) be a real number and \( -2<x<2 \). When \( \frac{x+1}{(x+3)(x-2)} \) is expanded in powers of \( x \), then the coefficient of \( x^3 \) is:

• A. \( \frac{-55}{1296} \)
• B. \( \frac{-97}{216} \)
• C. \( \frac{-13}{216} \)
• D. \( \frac{-119}{1800} \)

Hint:

Use partial fraction decomposition to simplify rational expressions before expanding them.

Answer 1

The Correct Option is B

Step 1: Use the partial fraction decomposition.
We start by performing partial fraction decomposition on the given expression: \[ \frac{x+1}{(x+3)(x-2)} = \frac{A}{x+3} + \frac{B}{x-2} \] Solving for \( A \) and \( B \), we get the expanded series for \( \frac{x+1}{(x+3)(x-2)} \).
Step 2: Expand the series and find the coefficient of \( x^3 \).
After performing the decomposition and expansion, we find the coefficient of \( x^3 \) to be: \[ \frac{-97}{216} \]