Question

If the equivalent partial fraction of \[ \frac{x^3}{(2x - 1)(x + 2)(x - 3)} \] is of the form \[ A + \frac{B}{2x - 1} + \frac{C}{x + 2} + \frac{D}{x - 3} \] then the value of \( A + B + C = \) ?

• A. \(-\frac{8}{25}\)
• B. \(\frac{4}{25}\)
• C. \(-\frac{1}{50}\)
• D. \(\frac{1}{2}\)

Hint:

For partial fractions, either expand and compare coefficients, or substitute convenient \( x \) values to find individual constants. To compute a sum of constants, choose values that eliminate other terms.

Answer 1

The Correct Option is B

We are given: \[ \frac{x^3}{(2x - 1)(x + 2)(x - 3)} = A + \frac{B}{2x - 1} + \frac{C}{x + 2} + \frac{D}{x - 3} \] To find \( A + B + C \), use the technique of: - Matching coefficients, - Substituting convenient values. But since only \( A + B + C \) is asked, we can substitute convenient values or evaluate using known tools. Alternatively, we expand the expression or use software (or compare numerators after common denominators). After evaluation and solving (either manually or via systems), we get: \[ A + B + C = \frac{4}{25} \]