Question:
If the equivalent partial fraction of
\[
\frac{x^3}{(2x-1)(x+2)(x-3)}
\]
is given by
\[
A+\frac{B}{2x-1}+\frac{C}{x+2}+\frac{D}{x-3},
\]
then the value of \(C\) is
If the equivalent partial fraction of
\[
\frac{x^3}{(2x-1)(x+2)(x-3)}
\]
is given by
\[
A+\frac{B}{2x-1}+\frac{C}{x+2}+\frac{D}{x-3},
\]
then the value of \(C\) is
Show Hint
To find the coefficient of a simple linear factor in partial fractions, multiply by that factor and substitute the value that makes it zero.
Updated On: Jun 26, 2026
- \(\dfrac{1}{2}\)
- \(-\dfrac{1}{50}\)
- \(-\dfrac{8}{25}\)
- \(\dfrac{27}{25}\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Write the given partial fraction form.
We have \[ \frac{x^3}{(2x-1)(x+2)(x-3)} = A+\frac{B}{2x-1}+\frac{C}{x+2}+\frac{D}{x-3} \]
Step 2: Isolate the coefficient \(C\).
Since \(C\) is the coefficient of \[ \frac{1}{x+2}, \] multiply both sides by \[ x+2 \] \[ (x+2)\frac{x^3}{(2x-1)(x+2)(x-3)} = (x+2)A+\frac{B(x+2)}{2x-1}+C+\frac{D(x+2)}{x-3} \]
Step 3: Substitute \(x=-2\).
To eliminate all terms except \(C\), put \[ x=-2 \] Then, \[ C=\left.\frac{x^3}{(2x-1)(x-3)}\right|_{x=-2} \]
Step 4: Evaluate the numerator.
At \[ x=-2, \] we get \[ x^3=(-2)^3=-8 \]
Step 5: Evaluate the denominator.
Now, \[ 2x-1=2(-2)-1=-4-1=-5 \] and \[ x-3=-2-3=-5 \] Thus, \[ (2x-1)(x-3)=(-5)(-5)=25 \]
Step 6: Find \(C\).
Therefore, \[ C=\frac{-8}{25} \]
Step 7: Final conclusion.
Hence, \[ \boxed{-\frac{8}{25}} \]
We have \[ \frac{x^3}{(2x-1)(x+2)(x-3)} = A+\frac{B}{2x-1}+\frac{C}{x+2}+\frac{D}{x-3} \]
Step 2: Isolate the coefficient \(C\).
Since \(C\) is the coefficient of \[ \frac{1}{x+2}, \] multiply both sides by \[ x+2 \] \[ (x+2)\frac{x^3}{(2x-1)(x+2)(x-3)} = (x+2)A+\frac{B(x+2)}{2x-1}+C+\frac{D(x+2)}{x-3} \]
Step 3: Substitute \(x=-2\).
To eliminate all terms except \(C\), put \[ x=-2 \] Then, \[ C=\left.\frac{x^3}{(2x-1)(x-3)}\right|_{x=-2} \]
Step 4: Evaluate the numerator.
At \[ x=-2, \] we get \[ x^3=(-2)^3=-8 \]
Step 5: Evaluate the denominator.
Now, \[ 2x-1=2(-2)-1=-4-1=-5 \] and \[ x-3=-2-3=-5 \] Thus, \[ (2x-1)(x-3)=(-5)(-5)=25 \]
Step 6: Find \(C\).
Therefore, \[ C=\frac{-8}{25} \]
Step 7: Final conclusion.
Hence, \[ \boxed{-\frac{8}{25}} \]
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