Question

If \[ \frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2} \] then \( f(14) + 2A - B = \) ?

• A. \(5C\)
• B. \(4C\)
• C. \(6C\)
• D. \(7C\)

Hint:

Use algebraic identities and partial fraction decomposition to simplify complex rational expressions.

Answer 1

The Correct Option is A

Step 1: Partial fraction decomposition.
We decompose the given rational function: \[ \frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2} \] By equating coefficients, solving for \( A, B, C \), and evaluating \( f(14) \), we derive: \[ f(14) + 2A - B = 5C \]