Question:
If
\[
\frac{2x^3+x-3}{x^4-5x^2+4}
\]
then partial fraction form is
If
\[
\frac{2x^3+x-3}{x^4-5x^2+4}
\]
then partial fraction form is
Show Hint
After factorization, substitute roots of denominator directly to obtain constants quickly.
Updated On: Jun 15, 2026
- \(\frac{5(x-1)}{4(x^2-3x+2)}+\frac{3x+1}{4(x^2+3x+2)}\)
- \(\frac{5(x+1)}{4(x^2-3x+2)}+\frac{3x-1}{4(x^2+3x+2)}\)
- \(\frac{2}{x-1}+\frac{5}{4(x-2)}-\frac{1}{x+1}+\frac{7}{4(x+2)}\)
- \(\frac{5}{4(x-2)}-\frac{1}{x+1}+\frac{7}{4(x+2)}\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Factor denominator completely and decompose into partial fractions.
Step 1: Factor denominator.
\[ x^4-5x^2+4 \] \[ =(x^2-1)(x^2-4) \] \[ =(x-1)(x+1)(x-2)(x+2) \]
Step 2: Assume decomposition.
\[ \frac{2x^3+x-3}{(x-1)(x+1)(x-2)(x+2)} \] Assume \[ =\frac{A}{x-1} +\frac{B}{x+1} +\frac{C}{x-2} +\frac{D}{x+2} \]
Step 3: Substitute values.
Put \[ x=1 \] \[ A=2 \] Put \[ x=-1 \] \[ B=-1 \] Put \[ x=2 \] \[ C=\frac54 \] Put \[ x=-2 \] \[ D=\frac74 \] Thus \[ = \frac2{x-1} -\frac1{x+1} +\frac5{4(x-2)} +\frac7{4(x+2)} \] Hence \[ \boxed{ \frac2{x-1} +\frac5{4(x-2)} -\frac1{x+1} +\frac7{4(x+2)} } \]
Step 1: Factor denominator.
\[ x^4-5x^2+4 \] \[ =(x^2-1)(x^2-4) \] \[ =(x-1)(x+1)(x-2)(x+2) \]
Step 2: Assume decomposition.
\[ \frac{2x^3+x-3}{(x-1)(x+1)(x-2)(x+2)} \] Assume \[ =\frac{A}{x-1} +\frac{B}{x+1} +\frac{C}{x-2} +\frac{D}{x+2} \]
Step 3: Substitute values.
Put \[ x=1 \] \[ A=2 \] Put \[ x=-1 \] \[ B=-1 \] Put \[ x=2 \] \[ C=\frac54 \] Put \[ x=-2 \] \[ D=\frac74 \] Thus \[ = \frac2{x-1} -\frac1{x+1} +\frac5{4(x-2)} +\frac7{4(x+2)} \] Hence \[ \boxed{ \frac2{x-1} +\frac5{4(x-2)} -\frac1{x+1} +\frac7{4(x+2)} } \]
Was this answer helpful?
0
0
Top TS EAMCET Mathematics Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
View More Questions
Top TS EAMCET Integration by Partial Fractions Questions
- If $\frac{x^{2} + 5}{\left(x^{2} + 1 \right)\left(x-2\right)} = \frac{A}{x-2} + \frac{Bx+C}{x^{2} + 1 } , $ then $A + B + C = $
- If \[ \frac{3x^4 - 2x^2 +1}{(x-2)^4} = A + \frac{B}{x-2} + \frac{C}{(x-2)^2} + \frac{D}{(x-2)^3} + \frac{E}{(x-2)^4}, \] then \(2A + 3B - C - D + E =\;?\)
- 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 \(\frac{x^2}{2x^2 + 7x + 6} = \frac{Ax + B}{x^2 + a} + \frac{Cx + D}{ax^2 + 3}\), then \(A + B + C - 2D =\)
- If $\frac{x+3}{(x+1)(x^2+2)} = \frac{a}{x+1} + \frac{bx+c}{x^2+2}$ then $a-b+c =$
View More Questions
Top TS EAMCET Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
View More Questions