Question:
If
\[
\frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2}
\]
then \( f(14) + 2A - B = \) ?
If
\[
\frac{x^4}{(x^2+1)(x-2)} = f(x) + \frac{Ax+B}{x^2+1} + \frac{C}{x-2}
\]
then \( f(14) + 2A - B = \) ?
Show Hint
Use algebraic identities and partial fraction decomposition to simplify complex rational expressions.
Updated On: Mar 11, 2025
- \(5C\)
- \(4C\)
- \(6C\)
- \(7C\)
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The Correct Option is A
Solution and Explanation
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: \[ f(14) + 2A - B = 5C \]
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: \[ f(14) + 2A - B = 5C \]
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